AIMS Series on Applied Mathematics
Volume 5
Optimal Control with Applications
in Space and Quantum Dynamics
Bernard Bonnard and Dominique Sugny
A I M S
American Institute of Mathematical Sciences
EDITORIAL COMMITTEE
Editor in Chief: Benedetto Piccoli (USA)
Members: José Antonio Carrillo de la Plata (Spain), Alessio Figalli (USA),
Kennethk Karlsen (Norway), James Keener (USA),
Thaleia Zariphopoulou (UK).
Bernard Bonnard
Institut de Mathématiques
Université de Bourgogne, Dijon, France
and INRIA Sophia Antipolis, France
E-mail: [email protected]
Dominique Sugny
Laboratoire Interdisciplinaire Carnot de Bourgogne
Université de Bourgogne, Dijon, France
E-mail: [email protected]
AMS 2000 subject classifications: 49K15, 49M05, 70F05, 70F07, 81V55
ISBN-10: 1-60133-013-8;
ISBN-13: 978-1-60133-013-0
c 2012 by the American Institute of Mathematical Sciences. All rights reserved.
°
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Preface
The object of this monograph is to present available techniques to analyze optimal control problems of systems governed by ordinary differential equations.
Coupled with numerical methods, they provide efficient tools to solve practical problems from control engineering. This will be illustrated by analyzing
two such case studies, which are the core of this book.
The first problem is the optimal transfer of a satellite between Keplerian
orbits. This standard problem of mechanical engineering has been recently revisited by research projects where electronic propulsion is used and the thrust
is low compared to the gravitational force. Those projects supported by the
French space agency CNES have some expected practical implementations for
instance through the project SMART-1, which is the ESA project of sending
a spacecraft from the Earth to the Moon, using low propulsion. The model
is the three-body problem, but the orbital transfer can be used to compute
parts of the trajectories.
The second problem still under investigation concerns a quantum mechanical system. It describes the population transfer between two energy levels in a
dissipative environment where the dynamics of the system is governed by the
Kossakowsky-Lindblad equation. It is connected to the experimental project
of controlling the molecular rotation by laser fields, where dissipation effects
are due to molecular collisions. It is also a model for the spin 1/2 dynamics
in Nuclear Magnetic Resonance where the control is a magnetic field.
In both cases, the control system is of the form
m
X
dx(t)
= F0 (x(t)) +
ui (t)Fi (x(t)), x ∈ Rn
dt
i=1
(0.1)
where the control u = (u1 , · · · , um ) satisfies the bound |u| ≤ 1, and | · | is the
euclidian norm. The underlying optimal control problems are the minimization
of the time of transfer or the minimization of the energy. They are extensions
of the so-called Riemannian problems of minimization of the transfer time T
from x0 to x1 for a system of the form
VI
Preface
n
dx(t) X
=
ui (t)Fi (x(t))
dt
i=1
where the drift F0 is zero and |u| ≤ 1.
Motivated by the two research projects in space and in quantum mechanics, we have developed the mathematical theory in several directions to get
substantial new results for this class of systems. Combined with numerical
simulations they give a neat analysis of the systems and they open the road
to an experimental implementation of the computed control laws.
Both studies are gathered in a single volume for two reasons. First of
all, they use similar general techniques from optimal control to be handled.
Secondly they depend upon a technical result about conjugate and cut loci
for Riemannian metrics on a two-sphere of revolution.
Besides the exemplary aspect of the monograph, it is based on a series
of lectures given at the graduate level. More precisely, the two chapters devoted to geometric optimal control were used as lectures notes for a series of
courses on non linear optimal control given by the first author at the European
Courses FAP which took place in Paris in 2004-05 whose participants where
PhD students and researchers in control engineering. The two final chapters
are developments of courses given at the University of Bourgogne for PhD
students in mathematics and physics involved in the research projects.
Since our goal is to provide lecture notes on optimal control introducing
recent developments of geometric optimal control theory and to present in
details two case studies, having in mind that they can be useful in several
research projects of control engineering, this guides the style of the book
providing computational tools to handle similar problems. Also some of the
numerical tools developed in the projects, e.g. the CotCot and the Hampath
codes, are of free access.
The organization of the book is the following. The first chapter is an
advanced introduction to optimal control problems analyzed by the maximum
principle. This principle, due to Pontryagin and his co-workers, is the central
result of the theory of optimal control. Through a set of necessary optimality
conditions, it is the starting point to analyze a wide range of optimal problems
using the Hamiltonian formalism. If we consider a specific control system of
the form (0.1), the maximum principle selects minimizers mainly among a set
of smooth extremal curves, solutions of the Hamiltonian vector field defined
by:
H = H0 + (
m
X
Hi2 )q
i=1
where Hi = hp, Fi (x)i are the Hamiltonian lifts of the vector fields Fi and
q = 1/2 for the time minimum problem and q = 1 in the energy minimization
problem relaxing the control bounds. In this smooth framework, we can use
advanced results on second order necessary and sufficient conditions, under
Preface
VII
generic assumptions, based on the concept of conjugate point. Such points
correspond to a point on the reference extremal solution where the optimality
is lost for the C 1 topology on the set of curves. They can be detected as a
geometric property of the extremal flow (they correspond to the concept of
caustic) and they can be easily numerically computed.
The second chapter is devoted to the time-minimum problem for a system of the form (0.1) . If F0 = 0, and m = n, where n is the dimension
of the state, it corresponds to a Riemannian problem and if m < n we are
in the sub-Riemannian case. An extension of the Riemannian case is a Zermelo navigation problem when m = n and the length of F0 is less than 1
for the Riemannian metric defined by taking {F1 , . . . , Fn } as an orthonormal
frame. We recall some results about curvature computations in the Riemannian case and we present the analysis of two SR-cases which will be useful in
our analysis. They correspond to the so-called Heisenberg and Martinet flat
cases. Advanced results describing the structure of the conjugate and cut-loci
concerning Riemannian metrics on a two-sphere of revolution normalized to
g = dφ2 + G(φ)dθ2 were obtained very recently. Extensions are crucial to
analyze both problems from space and quantum mechanics. Indeed in orbital
transfer such a metric can be obtained using an averaging method and for
the problem of controlling a two-level dissipative quantum system, a similar
metric appears for a specific value of the dissipative parameters. This allows
pursuit of the analysis using a continuation method on the set of parameters.
Another important property discussed in the second chapter is the behavior
of extremal curves near the switching surface Σ, Hi = 0, i = 1 · · · m, which
allows to construct broken extremals. It is a crucial and very technical problem. For m = 1, this corresponds to the classification of extremal curves near
the switching surface for single-input control systems. In this case, it is known
that complicated behaviors can occur e.g. Fuller phenomenon whose analysis
is related to singularity analysis. The multi-input case is a non-trivial extension and we present some preliminary results under generic assumptions which
will be sufficient in our case studies.
The third chapter analyzes the optimal transfer between elliptic Keplerian
orbits. This classical problem has been revisited about ten years ago by a
French research group from ENSEEIHT at Toulouse, in a project sponsored
by the French space agency CNES in the case where electo-ionic propulsion
is used and the thrust is very low. As a product of this research activity, a
lot of numerical techniques were developed in optimal control for this specific
problem, based on the maximum principle, with a lot of numerical results.
More recently they were combined with geometric techniques to obtain a neat
analysis of the problem. Most of these results are presented in this chapter.
The first part is a standard geometric analysis of the problem to get appropriate (Gauss) coordinates whose role is to split the coordinates representation
in two parts if low propulsion is used: a fast angular variable which is the longitude and slow variables corresponding to first integrals of the free motion.
This section is completed by Lie brackets computations to analyze the control-
VIII
Preface
lability properties of the system. In a second part of the chapter the problem
of computing a feedback to realize the transfer is analyzed geometrically using
stabilization techniques. It is based on the periodicity property of the solutions of the free motion (Kepler equation) and uses Jurdjevic-Quinn theorem.
In the final part of the chapter the optimal control problem is analyzed. First
of all we present the main results about the time minimal control problem,
when the final orbit is the geosynchronous orbit. An extremal solution can be
numerically computed using a shooting technique combined with a discrete
continuation method on the magnitude of the thrust and conjugate points are
calculated to check optimality. Secondly, the optimal control is analyzed using
averaging techniques. In this case this amounts only to compute the averaged
with respect to the longitude of the Hamiltonian coming from the maximum
principle. Indeed if low propulsion is used the averaged Gauss coordinates are
numerically indistinguishable from the non averaged ones. If averaging in this
framework can be performed for every cost variable, the most regular corresponds to the energy minimization problem, since the averaged Hamiltonian
is associated to a Riemannian problem, whose trajectories and distance are
approximations of the solutions and of the cost of the original problem. In this
case we present two very neat geometric results in the coplanar case where the
initial and final orbits are in the same plane. First of all, for the transfer to the
geosynchronous orbit, the averaged optimal trajectories are straight lines in
suitable coordinates. Secondly for a general transfer, using homogeneity properties of the metric, we can reduce the analysis to a Riemannian metric on a
two-sphere of revolution for which using the results of chapter 2 we can deduce
the conjugate and cut loci. In particular we obtain global optimality results.
Also with this approach we define a distance between elliptic orbits related
to the optimal problem, which is an important property from theoretical and
practical points of view. In the final part of the chapter we extend the results
in several directions: the averaged non coplanar case is computed leading to
a Riemannian metric in a 5-dimensional space, whose analysis is still an open
problem, and the averaged system is computed if the control is oriented in
a single direction e.g. the tangential direction, such study being related to
cone constraints on the control direction, due to electro-ionic technology. The
results of the chapter are rather completed and are useful to analyze other
problems in space mechanics: maximization of the final mass in orbit transfer,
using a continuation method (from L1 to L2 ) on the cost, SMART-1 transfer
mission of a spacecraft from the Earth to the Moon. In a final section, a trajectory of the energy minimization transfer in the Earth-Moon space mission
is computed using a numerical computation method.
The final chapter is devoted to quantum control. We restrict our analysis
to a specific problem which is the time optimal control of a two-level dissipative system, controlled by a laser field, and described by the KossakowskyLindblad equation. This problem motivated by the research project CoMoc is
a new problem, dealing with optimal control problems in quantum systems,
with a control bound and taking into account dissipation. This leads to a
Preface
IX
complicated system where the dimension of the state is three and the system
depends upon three parameters describing all the interactions of the system
with the environment. The first part of the chapter is devoted to the modeling of dissipative quantum systems using the Kossakowsky-Lindblad equation,
which leads to a finite dimensional system where the dimension of the state is
N 2 − 1, where N is the number of levels kept in the approximation. The twolevel case is significant to model some true experimental systems as the spin
1/2 particle in Nuclear Magnetic Resonance, although in the project CoMoc
about twenty levels are relevant. The two-level case is important because it
allows a geometric analysis and numerical simulations can be tested on this
model, before to be extended to more complicated systems. The second part
of the chapter deals with the geometric analysis of the two-level case, with
final numerical simulations. For this problem the system is an affine system in
R3 , where we denote q = (x, y, z) the Cartesian coordinates and the dynamics
is invariant for the Bloch ball | q |≤ 1. The control u is the complex Rabi
frequency of the laser field and assuming the Rotating Wave Approximation
the system can be written
dq(t)
= F0 (q(t)) + u1 (t)F1 (q(t)) + u2 (t)F2 (q(t))
dt
where u = u1 + iu2 is the control, F0 is an affine vector field depending upon
three parameters and describing the interaction with the environment and
F1 , F2 are two linear vector fields tangent to the unit sphere.
Since the Bloch ball is invariant for the dynamics the system can be represented in spherical coordinates (ρ, φ, θ) where ρ is the distance to the origin
and corresponds to the purity of the system, φ is the angle with respect to
the z−axis and θ is the angle of rotation around the same axis. This representation reveals that the time minimum control problem has a symmetry
of revolution around the z−axis. The extremals contained in meridian planes
have an important physical interpretation: they correspond to extremal solutions for a 2D−system, assuming the control field real. Hence a first analysis
is to make the time minimal synthesis for the corresponding 2D-single input
system. This preliminary analysis is discussed in detail and leads to a complicated classification problem depending upon three parameters. Also this study
is important for the whole system since due to the symmetry of revolution it
describes the time optimal control provided the initial state is a pure state
along the z−axis of the form (0,0,±1), of the Bloch sphere. The second step
is to complete the analysis by taking an arbitrary initial state. The analysis
is split into two parts. First of all, it can be observed that for a family of two
parameters the extremal Hamiltonian flow is integrable. Moreover for a one
parameter sub-family, the purity of the system is not controllable and the time
minimal control problem amounts to analyze the Riemannian problem on the
two-sphere of revolution for the metric g = dφ2 + tan φdθ2 with a singularity
at the equator. Still the results of Chapter 2 can be applied to compute the
conjugate and cut loci and solve the optimal control problem in this case. To
X
Preface
analyze the general integrable case we can make a smooth continuation on
the set of parameters. Roughly speaking if we are closed from the sub-family
the conjugate and cut loci are stable and can be determined by perturbation.
Moreover a bifurcation occurs when the drift term on the sphere cannot be
compensated by a feedback. This fits in the geometric framework of Zermelo
navigation problem and we propose in the integrable case a complete mathematical analysis. The integrable case is not stable and in the generic case the
analysis is different. Still we observe two types of behaviors for the extremals
curves, distinguished by their asymptotic properties. Finally making intensive
numerical simulations to compute extremals with their conjugate points the
analysis is presented in the generic case. The robustness with respect to the
dissipation parameters is analyzed using the numerical continuation method.
We also present a similar study for the energy minimization problem. In a final section, some preliminary results about the contrast problem in Magnetic
Resonance Imaging are described.
Dijon, January 2012
Bernard Bonnard, Institut de Mathématiques de Bourgogne and INRIA
Sophia Antipolis
Dominique Sugny, Laboratoire Interdisciplinaire Carnot de Bourgogne.
Acknowledgments
We thank S. J. Glaser and A. Sarychev for many hepful discussions and John
Marriott for a careful reading of the manuscript.
Contents
1
2
Introduction to Optimal Control . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1 Optimal Control and Maximum Principle . . . . . . . . . . . . . . . . . . .
1.1.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1.2 The Weak Maximum Principle . . . . . . . . . . . . . . . . . . . . . .
1.1.3 Geometric Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1.4 Affine Control Systems and Connection with General
Control Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1.5 Computation of Singular Controls . . . . . . . . . . . . . . . . . . .
1.1.6 Singular Trajectories and Feedback Classification . . . . .
1.1.7 Maximum Principle with Fixed Time . . . . . . . . . . . . . . . .
1.1.8 Maximum Principle, the General Case . . . . . . . . . . . . . . .
1.1.9 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1.10 The Shooting Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Second Order Necessary and Sufficient Conditions in the
Generic Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.1 Second Order Conditions in the Classical Calculus of
Variations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.2 Symplectic Geometry and Second Order Optimality
Conditions under Generic Assumptions . . . . . . . . . . . . . . .
1.2.3 Second Order Optimality Conditions in the Affine Case
1.2.4 Existence Theorems in Optimal Control . . . . . . . . . . . . . .
Riemannian Geometry and Geometric Control Theory . . . . .
2.1 Generalities about SR-Geometry . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.1 Optimal Control Theory Formulation . . . . . . . . . . . . . . . .
2.1.2 Computation of the Extremals and Exponential
Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 A Property of the Distance Function . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Classification of SR Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4 Two Cases Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
2
2
3
6
6
7
8
9
11
12
13
14
14
19
31
47
49
50
51
52
54
54
55
55
XII
Contents
2.5
2.6
2.7
2.8
3
2.4.1 The Heisenberg Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.2 The Martinet Flat Case . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.3 The Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.4 A Conclusion about SR Spheres . . . . . . . . . . . . . . . . . . . . .
The Riemannian Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5.1 A Brief Review of Riemannian Geometry . . . . . . . . . . . . .
2.5.2 Clairaut-Liouville Metrics . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5.3 The Optimality Problem . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5.4 Conjugate and Cut Loci on Two-Spheres of Revolution .
An Example of Almost Riemannian Structure: The Grushin
Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.6.1 The Grushin Model on R2 . . . . . . . . . . . . . . . . . . . . . . . . . .
2.6.2 The Grushin Model on S 2 . . . . . . . . . . . . . . . . . . . . . . . . . .
2.6.3 Generalization of the Grushin Case . . . . . . . . . . . . . . . . . .
2.6.4 Conjugate and Cut Loci for Metrics on the
Two-Sphere with Singularities . . . . . . . . . . . . . . . . . . . . . .
2.6.5 Homotopy on Clairaut-Liouville Metrics and
Continuation Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Extension of SR Geometry to Systems with Drift . . . . . . . . . . . .
2.7.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Generic Extremals Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.8.1 An Application to SR Problems with Drift in
Dimension 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
55
58
61
63
63
63
66
68
68
73
74
75
77
78
79
79
79
82
84
Orbital Transfer Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
3.1 The Model for the Controlled Kepler Equation . . . . . . . . . . . . . . 87
3.1.1 First Integrals of Kepler Equation and Orbit Elements . 88
3.1.2 Connection with a Linear Oscillator . . . . . . . . . . . . . . . . . 88
3.1.3 Orbit Elements for Elliptic Orbits . . . . . . . . . . . . . . . . . . . 89
3.2 A Review of Geometric Controllability Techniques and Results 92
3.2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
3.2.2 Basic Controllability Results . . . . . . . . . . . . . . . . . . . . . . . . 93
3.2.3 Controllability and Enlargement Technique . . . . . . . . . . 94
3.3 Lie Bracket Computations and Controllability in Orbital
Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
3.3.1 Lie Bracket Computations . . . . . . . . . . . . . . . . . . . . . . . . . . 96
3.3.2 Controllability Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
3.4 Constructing a Feedback Control Using Stabilization
Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
3.4.1 Stability Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
3.4.2 Stabilization of Nonlinear Systems . . . . . . . . . . . . . . . . . . . 100
3.4.3 Application to the Orbital Transfer . . . . . . . . . . . . . . . . . . 101
3.5 Optimal Control Problems in Orbital Transfer . . . . . . . . . . . . . . 102
3.5.1 Physical Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
3.5.2 Extremal Trajectories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
Contents
XIII
3.6 Preliminary Results on the Time-Minimal Control Problem . . . 107
3.6.1 Homotopy on the Maximal Thrust . . . . . . . . . . . . . . . . . . . 107
3.6.2 Conjugate Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
3.7 Extremals for Single-Input Time-Minimal Control . . . . . . . . . . . 107
3.7.1 Singular Extremals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
3.7.2 Classification of Regular Extremals . . . . . . . . . . . . . . . . . . 109
3.7.3 The Fuller Phenomenon . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
3.8 Application to Time Minimal Transfer with Cone Constraints . 112
3.9 Averaged System in the Energy Minimization Problem . . . . . . . 113
3.9.1 Averaging Techniques for Ordinary Differential
Equations and Extensions to Control Systems . . . . . . . . . 113
3.9.2 Controllability Property and Averaging Techniques . . . . 114
3.9.3 Riemannian Metric of the Averaged Controlled Kepler
Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
3.9.4 Computation of the Averaged System in Coplanar
Orbital Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
3.10 The Analysis of the Averaged System . . . . . . . . . . . . . . . . . . . . . . 120
3.10.1 Analysis of ḡ1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
3.10.2 Integrability of the Extremal Flow . . . . . . . . . . . . . . . . . . . 122
3.10.3 Geometric Properties of ḡ2 . . . . . . . . . . . . . . . . . . . . . . . . . 124
3.10.4 A Global Optimality Result with Application to
Orbital Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
3.10.5 Riemann Curvature and Injectivity Radius in Orbital
Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
3.10.6 Cut Locus on S 2 and Global Optimality Results in
Orbital Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
3.11 The Averaged System in the Tangential Case . . . . . . . . . . . . . . . 129
3.11.1 Construction of the Normal Form . . . . . . . . . . . . . . . . . . . 129
3.11.2 The Metric g1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
3.11.3 The Metric g2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
3.11.4 The Integration of the Extremal Flow . . . . . . . . . . . . . . . . 131
3.11.5 A Continuation Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
3.12 Conclusion in Both Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
3.13 The Averaged System in the Orthoradial Case . . . . . . . . . . . . . . 133
3.14 Averaged System for Non-Coplanar Transfer . . . . . . . . . . . . . . . . 133
3.15 The Energy Minimization Problem in the Earth-Moon Space
Mission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
3.15.1 Mathematical Model and Presentation of the Problem. . 134
3.15.2 The Circular Restricted 3-Body Problem in Jacobi
Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
3.15.3 Jacobi Integral and Hill Regions . . . . . . . . . . . . . . . . . . . . . 136
3.15.4 Equilibrium Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
3.15.5 The Continuation Method in the Earth-Moon Transfer . 137
XIV
4
Contents
Optimal Control of Quantum Systems . . . . . . . . . . . . . . . . . . . . . 149
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
4.2 Control of Dissipative Quantum Systems . . . . . . . . . . . . . . . . . . . 151
4.2.1 Quantum Mechanics of Open Systems . . . . . . . . . . . . . . . . 151
4.2.2 The Kossakowski-Lindblad Equation . . . . . . . . . . . . . . . . . 158
4.2.3 Construction of the Model . . . . . . . . . . . . . . . . . . . . . . . . . . 160
4.3 Controllability of Right-Invariant Systems on Lie Groups . . . . . 162
4.3.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
4.3.2 The Case of SL(2, R) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
4.3.3 Controllability on Sp(n, R) . . . . . . . . . . . . . . . . . . . . . . . . . 173
4.4 Time Minimal Control of the Lindblad Equation . . . . . . . . . . . . 175
4.4.1 Symmetry of Revolution . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
4.4.2 Spherical Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
4.4.3 Lie Brackets Computations . . . . . . . . . . . . . . . . . . . . . . . . . 178
4.4.4 Singular Trajectories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
4.4.5 The Time-Optimal Control Problem . . . . . . . . . . . . . . . . . 181
4.5 Single-Input Time-Optimal Control Problem . . . . . . . . . . . . . . . . 182
4.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
4.5.2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
4.5.3 Four Different Illustrative Examples . . . . . . . . . . . . . . . . . 187
4.5.4 Physical Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
4.5.5 Complete Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
4.6 The Two-Input Time-Optimal Case . . . . . . . . . . . . . . . . . . . . . . . . 196
4.6.1 The Integrable Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
4.6.2 Numerical Determination of the Conjugate Locus . . . . . . 201
4.6.3 Geometric Interpretation of the Integrable Case . . . . . . . 203
4.6.4 The Generic Case γ− 6= 0. . . . . . . . . . . . . . . . . . . . . . . . . . . 204
4.6.5 Regularity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
4.6.6 Abnormal Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
4.6.7 Singular Value Decomposition . . . . . . . . . . . . . . . . . . . . . . . 211
4.6.8 Continuation Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
4.7 The Energy Minimization Problem . . . . . . . . . . . . . . . . . . . . . . . . 218
4.7.1 Geometric Analysis of the Extremal Curves . . . . . . . . . . . 219
4.7.2 The Optimality Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 238
4.7.3 Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256
4.8 Application to Nuclear Magnetic Resonance . . . . . . . . . . . . . . . . 261
4.9 The Contrast Imaging Problem in NMR . . . . . . . . . . . . . . . . . . . . 265
4.9.1 The Model System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267
4.9.2 The Geometric Necessary Optimality Conditions and
the Dual Problem of Extremizing the Transfer Time
to a Given Manifold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268
4.9.3 Second-Order Necessary and Sufficient Optimality
Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270
4.9.4 An Example of the Contrast Problem . . . . . . . . . . . . . . . . 270
Contents
XV
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281
3
Orbital Transfer Problem
The objective of this chapter is to apply geometric control techniques to analyze the transfer of a satellite between two elliptic orbits, taking into account
physical cost functions such as time or mass consumption. This leads to analyze the controlled Kepler equation, which is a standard equation of space
mechanics. Our study is two-fold. First of all, we make a geometric analysis
of the corresponding systems. Secondly, we complete the analysis when low
propulsion is applied using averaging techniques. The final result consists in
the construction of an explicit distance between elliptic orbits which is connected to the energy minimization problem and which can be used in practice
to compute the time optimal control minimizing time or consumption with
smooth continuation methods. In the coplanar case this metric is analyzed in
detail. For the transfer to the geosynchronous orbit the optimal trajectories
are straight lines. For general transfer, the problem is reduced by homogeneity
to a Riemannian metric on a two-sphere of revolution. The conjugate and cut
loci are determined to get a global optimality result.
3.1 The Model for the Controlled Kepler Equation
In this section, we present the model for the system. It is a classical model
(see [146] for the details) adapted to celestial mechanics or space mechanics
when low propulsion is applied. The coordinates are introduced in relation
with the first integrals of Kepler equation.
Let q be the position of the satellite in a reference frame (I, J, K) whose
origin O is the Earth center, the control is the force due to the thrust |F | ≤
Fmax and µ is the gravitational constant normalized to 1. The system is
represented in Cartesian coordinates by
q̈ = −
q
F
+
|q|3
m
where m is the mass of the satellite whose evolution is described by
(3.1)
88
3 Orbital Transfer Problem
ṁ = −δ|F |
where δ > 0 is a parameter which is the inverse of the ejection velocity ve of
the propellant.
Hence the state of the system is (x, m) with x = (q, q̇) ∈ R6 . If low thrust
is applied, the action of the thrust is small compared to gravitation and the
control system is a small perturbation of Kepler equation.
3.1.1 First Integrals of Kepler Equation and Orbit Elements
Proposition 3.1.1. We consider the Kepler equation q̈ = −µ |q|q 3 . We have
the following vectors first integrals
•
•
C = q ∧ q̇ (momentum)
q
L = −µ |q|
+ q̇ ∧ C (Laplace integral).
µ
Moreover the energy H(q, q̇) = 12 q̇ 2 − |q|
is preserved and the following relations
hold
L·C =0
L2 = µ2 + 2HC 2 .
(3.2)
Hence, we have five independent first integrals which allow to compute the
geometric trajectories as conics, which are ellipses if H < 0.
Proposition 3.1.2. For Kepler equation, if the momentum C is zero then q
and q̇ are on a line called a colliding line. If C 6= 0 then we have
•
•
If L = 0 then the motion is circular.
If L =
6 0 and H < 0 then the trajectory is an ellipse given by
|q| =
C2
,
µ + |L| cos(θ − θ0 )
one of the foci being the Earth center O, (r, θ) being polar coordinates and θ0
is the angle of the pericenter corresponding to the point where the distance of
the satellite to the Earth center is minimal.
Definition 3.1.3. The domain Σe = {(q, q̇); H < 0, C 6= 0} called the elliptic
domain is filled by elliptic orbits and to each (C, L) corresponds an unique
oriented ellipse.
3.1.2 Connection with a Linear Oscillator
A first step to understand the controllability of the problem is to use the
following approach due to Lagrange-Binet[97]. We assume that the thrust is
oriented along the osculating plane (q, q̇) so that the orbital plane is fixed. If
3.1 The Model for the Controlled Kepler Equation
89
ur and u0r represent the respective decomposition of the thrust in the radial
and orthoradial direction. Writing the system in polar coordinates, we get:
r̈ − rθ̇2 = − rµ2 +
rθ̈ + 2ṙθ̇ = um0r
ur
m
(3.3)
so that, up to renormalization
r̈ − rθ̇2 = − r12 + ε umr
rθ̈ + 2ṙθ̇ = ε um0r .
(3.4)
If we set v = 1/r and if we parameterize the equations by θ, our system can
be written as
v 00 + v − (v 2 t0 )2 = −εv 2 t02 (ur +
(v 2 t0 )0 = −εv 3 t03 u0r
v0
v u0r )
(3.5)
where 0 denotes the derivative with respect to θ. This representation shows the
analogy with the control of a linear oscillator and is useful to apply averaging
techniques.
3.1.3 Orbit Elements for Elliptic Orbits
Computing the evolution of (C, L), we obtain easily the system
F
Ċ = q ∧ m
L̇ = (F ∧ C) + q̇ ∧ (q ∧
F
m)
(3.6)
which is a five dimensional system, since C and L are orthogonal. A more
detailed representation uses the following parameters. Recall that (q, q̇) are
coordinates in the Cartesian space (I, J, K) where (I, J) can be identified to
the Earth equatorial plane. We introduce in the elliptic domain the following
quantities:
•
•
•
•
•
The oriented ellipse cuts the equatorial plane in two opposite points which
defines the line of nodes and Ω represents the angle of the ascending node.
The angle ω is the argument of the pericenter, that is the angle between
the axis of the ascending node and the axis of the pericenter.
i: inclination of the osculating plane
a: semi-major axis of the ellipse
e: eccentricity
To represent the position of the satellite on the ellipse, we use the longitude
which is the angle between the position q and the axis I. The previous
coordinates are singular for circular orbits or in the case of orbits in the
equatorial case.
90
3 Orbital Transfer Problem
We can define regular coordinates using polar blowing-up. We proceed as
follows. Let e be the eccentricity vector related to Laplace vector by L = µe,
where for ellipse e is oriented along the semi-major axis. If ω̃ is the angle
between I and e, we set
e1 = e cos ω̃, e2 = e sin ω̃
which is zero for circular orbits. To relax the singularity when i = 0, we
introduce the vector h collinear to the line of node and defined by
i
i
h1 = tan( ) cos Ω, h2 = tan( ) sin Ω
2
2
which is zero for equatorial orbits.
Decomposition of the Thrust in Moving Frames
Introducing the vector fields Fi = ∂∂q˙i , i = 1, 2, 3 identified respectively to I,
J and K, the thrust is decomposed as
F =
3
X
ui Fi
i=1
where the ui ’s are the Cartesian components of the control. More physical
decompositions are obtained by writing the thrust in a moving frame attached
to the satellite. In particular, this allows to take into account constraints on
the thrust due to the technology of electro-ionic propulsion. The two standard
frames, which are defined for q ∧ q̇ 6= 0 are:
•
q ∂
The radial / orthoradial frame {Fr , F0r , Fc } where Fr = |q|
∂ q̇ . The vectors
are chosen in the osculating plane such that {Fr , F0r } forms a frame and
C ∂
Fc is perpendicular to the plane Fc = |C|
∂ q̇ .
•
The tangential / normal frame {Ft , Fn , Fc } where Ft =
perpendicular to Ft in the osculating plane.
q̇ ∂
|q̇| ∂ q̇
and Fn are
Using Cartesian frame, the system (3.1) takes the form
3
q̈ = −
X Fi
q
+
ui
3
|q|
m
i=1
which induces in the moving frame similar representation where the control
components are denoted respectively (ur , u0r , uc ) or (ut , un , uc ).
In terms of control systems, this amounts to make a feedback transformation
u = R(x)v, R ∈ SO(3)
3.1 The Model for the Controlled Kepler Equation
91
where the control magnitude is preserved. In particular, we observe that the
tangential / normal frame has a great interest. The component along the speed
is connected to the standard analysis of the drag, in space mechanics where
dissipation due to the atmosphere can occur. Taking also into account cone
constraints, the single-input system corresponding to the action of the control
along the speed is of special interest, and can be compared to a full control.
Controlled Kepler Equation in Gauss Coordinates
We next give two systems where the coordinates are elliptic elements and the
control is expressed in a moving frame. They reveal controllability properties
of the system and are used in the sequel, when low thrust is applied.
System 1
da
dt
de1
dt
de2
dt
dh1
dt
dh2
dt
dl
dt
q
=
=
2
a3 B
ut
m
qµ A
1
a A 2(e1 +cos l)Dut
m
µ D[
B
− 2e1 e2 cos l −
−eq
2 (h1 sin l − h2 cos l)uc ]
1
A 2(e2 +sin l)Dut
= m µa D
[
+ (2e1 e2 sin l +
B
sin l(e21 −e22 )+2e2 sin l
un
B
cos l(e21 −e22 )+2e1 cos l
)un
B
+eq
1 (h1 sin l − h2 cos l)uc ]
1
a AC
= m
cos luc
qµ D 2
1
a AC
= m
µ D 2 sin luc
q
p
2
1
a A
= aµ3 D
A3 + m
µ D (h1 sin l − h2 cos l)uc
where
p
A = p1 − e21 − e22
B = 1 + 2e1 cos l + 2e2 sin l + e21 + e22
C = 1 + h21 + h22
D = 1 + e1 cos l + e2 sin l.
(3.7)
(3.8)
System 2
dP
dt
q
=
1
m
de1
dt
=
1
m
de2
dt
=
1
m
dh1
dt
=
1
m
q
u0r
P
µ (sin lur
P
qµ
+ (cos l +
e1 +cos l
)u0r
W
(− cos lur + (sin l +
P Z C
µ W 2
cos luc
q
1
P Z C
=m
µ W 2 sin luc
q
p µ W2
P Z
1
= P P +m
µ W uc
dh2
dt
dl
dt
P 2P
W
qµ
−
e2 +sin l
)u0r
W
2e2 uc
W )
+
Ze1 uc
W )
(3.9)
92
3 Orbital Transfer Problem
with
W = 1 + e1 cos l + e2 sin l
Z = h1 sin l − h2 cos l.
(3.10)
The relation between the semi-major axis a and P which is called the semiP
latus rectum is a = √1−e
, the apocenter and pericenter being respectively
2
given by
ra = a(1 + e), rp = a(1 − e).
The coplanar transfer corresponds to the case where the osculating plane is
kept fixed, hence uc = 0. It can be identified to the equatorial plane and the
system is described by the previous equations in which h = (h1 , h2 ) = 0.
We observe that the system is periodic with respect to the true longitude,
i.e., it is a smooth system with l ∈ S 1 . If l ∈ R, we take into account the
rotation number, called the cumulated longitude. Moreover, since l˙ > 0, we
can parameterize the trajectories by l instead of t. This point of view is useful
in orbit transfer since the final position of the spacecraft on the orbit is not
specified.
3.2 A Review of Geometric Controllability Techniques
and Results
In this section, we make a short introduction to the controllability results
which are necessary for our analysis. More details can be found in [93].
3.2.1 Preliminaries
We consider a smooth control system of the form
dx
= F (x, u), x ∈ M, u ∈ U.
dt
By density, we can restrict the set of admissible controls to the set of piecewise constant mappings valued in U . The standard definitions needed in the
controllability problem are the following.
Definition 3.2.1. Let us denote x(t, x0 , u) the solution associated to an admissible control u and starting from
S x0 at t = 0. The accessibility set in
time T is the set A+ (x0 , T ) = u(·) x(t, x0 , T ) and the accessibility set is
S
A+ (x0 ) = T >0 A+ (x0 , T ). Reversing time, we can similarly define the set
A− (x0 , T ) corresponding
to points which can be steered to x0 in time T and
S
we have A− (x0 ) = T >0 A− (x0 , T ). The system is controllable in time T if
for each x0 , A+ (x0 , T ) = M and controllable if A+ (x0 ) = M .
Since the set of admissible controls is taken as the set of piecewise constant
mappings, we introduce the following definition.
3.2 A Review of Geometric Controllability Techniques and Results
93
Definition 3.2.2. We call polysystem the set of vector fields
D = {F (x, u), u ∈ U }.
If F ∈ D then we denote {exp tF } the local one-parameter subgroup and we
introduce
X
ST (D) = {exp t1 F1 ◦ · · · ◦ exp tk Fk ; Fi ∈ D, k ∈ N, ti ≥ 0,
ti = T }
i
S
and S(D) = T ST (D). We observe that A+ (x0 , T ) is the set ST (D) · x0
and A+ (x0 ) the action of S(D). Moreover, by construction S(D) is the local
semi-group of diffeomorphims generated by the set {exp tF, F ∈ D; t ≥ 0}.
We denote by G(D) the associated local group which is generated by the set
{exp tF ; t ∈ R}. The polysystem D is controllable if for each x0 , the orbit of
S(D) is M and weakly controllable if for each x0 , the orbit of x0 is the whole
M.
The second property is related to the following infinitesimal action.
Definition 3.2.3. We denote by DL.A. the Lie algebra generated by vector
fields in D. It can be computed by the following algorithm D1 = D, D2 =
D1 ∪ [D1 , D1 ], · · · , DL.A. = Span(∪p≥1 Dp ) where Dp is formed by iterated Lie
brackets of length smaller or equal to p. If E is a subset of smooth vector fields
then it will define a distribution ∆ : x 7→ SpanE(x). It is called involutive if
[∆, ∆] ⊂ ∆. An integral manifold N is such that for each y ∈ N , Ty N = ∆(y).
3.2.2 Basic Controllability Results
We first present the Nagano-Sussmann theorem [139].
Theorem 3.2.4. Let D be an analytic polysystem on M . If p is the rank
of DL.A. (x0 ) then through x0 , there exists locally an integral manifold (of
dimension p) N (x0 ) of the distribution DL.A. . Moreover, it can be uniquely
extended to a global integral manifold.
We next recall Chow’s theorem.
Theorem 3.2.5. Let D be a C ∞ polysystem on a connected manifold M . We
assume that for each x ∈ M , the rank condition is satisfied: DL.A. (x) = Tx M
then
G(D)(x) = G(DL.A. )(x)
for each x ∈ M .
We deduce a first controllability result.
Proposition 3.2.6. Let D be a smooth polysystem on a connected manifold.
Assume D is symmetric, i.e., if F ∈ D then −F ∈ D. If the rank condition
DL.A. (x) = Tx M is satisfied then D is controllable. Moreover in the analytic
case, the condition is also necessary.
94
3 Orbital Transfer Problem
In the general case, the following local result is true.
Proposition 3.2.7. Let D be a smooth polysystem on M such that
dimDL.A. (x) = dimM
for each x ∈ M . Then for each neighborhood V of x, there exists a non-empty
open set U contained in V ∩ A+ (x).
Proof. The following simple proof highlights the structure of the accessibility
set. Let x ∈ M , if dim M ≥ 1 then there exists F1 ∈ D such that F1 (x) 6=
0, otherwise the rank condition is not satisfied. Let α1 be the curve: {t 7→
(exp tF1 )(x); t ≥ 0} . If dim M ≥ 2 then in every neighborhood V of x, we can
find a point y = (exp t1 F1) (x), t1 ≥ 0 and a vector field F2 ∈ D such that F1
and F2 are not collinear at y, otherwise the rank condition is not satisfied. We
consider the mapping α2 : {(t1 , t2 ) 7→ (exp t2 F2 ◦ exp t1 F1 )(x); t1 , t2 ≥ 0}. If
dimM ≥ 2, we have a vector field F3 in V transverse to the image of α2 near
the point y where the image of α2 is 2-dimensional. Iterating the construction,
this gives a non-empty open set U contained in V ∩ A+ (x).
¤
3.2.3 Controllability and Enlargement Technique
To obtain more general controllability results, we use an algorithm which was
formalized in [93, 94]. We start on a connected manifold M with a smooth
polysystem M satisfying the rank condition. We enlarge D with operations
on vector fields which preserve the controllability.
Lemma 3.2.8. The polysystem D (satisfying the rank condition) is controllable if and only if the adherence of S(D) · x is M for every x ∈ M .
Proof. Let x, y be two points on M . Using proposition 3.2.7 with reversed
time, we deduce that there exists in every neighborhood V of y a non-empty
open set U in V ∩ A− (x). By assumption, there exists y1 in U such that x can
be steered to y1 and the conclusion follows since we can steer y1 to y.
¤
Definition 3.2.9. Let D, D0 be two polysystems satisfying the rank condition.
They are called equivalent if for each x ∈ M , S(D)(x) = S(D0 )(x). The union
of all polysystems equivalent to D is called the saturate of D and is denoted
satD.
We observe that by definition, D is controllable if and only if satD is controllable.
Construction of satD:
We define the operations preserving controllability.
Proposition 3.2.10. The convex cone generated by D is equivalent to D.
3.2 A Review of Geometric Controllability Techniques and Results
95
Proof. If F ∈ D then using reparametrization, for each λ > 0, λF ∈ satD.
Now, from Baker-Campbell-Hausdorff formula, if F, G ∈ D, we have
(exp
F
G
exp )n = exp(F + G) + o(1/n).
n
n
Hence, taking the limit when n 7→ +∞, we have F + G ∈ satD.
¤
Definition 3.2.11. Let F be a smooth vector field on M . The point x0 is
Poisson stable if for every T > 0 and every neighborhood V of x0 , there exists
t1 , t2 ≥ T such that (exp t1 F )(x0 ) and (exp −t2 F )(x0 ) belong to V . The vector
field F is called Poisson stable if the set of Poisson stable points is dense in
M.
Proposition 3.2.12. If F is a Poisson stable vector field in D then −F ∈
satD.
Proof. Let x, y ∈ M such that y = (exp −T F )(x), for a T > 0. We observe
that if F is periodic, there exists T 0 > 0 such that y = (exp T 0 F )(x). More
generally, if x is Poisson stable then for every neighborhood Vy of y, there
exists T 0 > 0 such that (exp T 0 F )(x) ∈ Vy . If x is not Poisson stable, by
density every neighborhood Vx of x contains a point x0 which is Poisson stable
and is used to reach Vy in positive time. The result is proved.
¤
Proposition 3.2.13. Assume ±F, ±G ∈ D then ±[F, G] ∈ satD.
Proof. Using Baker-Campbell-Hausdorff formula, we have
exp tF exp tG exp −tF exp −tG = exp(t2 [F, G] + o(t2 )).
Hence the direction [F, G] in the Lie algebra can be reached.
¤
A very powerful operation is next introduced.
Definition 3.2.14. Let F be a smooth vector field on M and let φ be a smooth
diffeomorphism. A change of coordinates defined by φ transforms F into the
image of F : φ ∗ F = dφ(F ◦ φ−1 ). The associated one-parameter group is
φ ◦ exp tF ◦ φ−1 . If D is a polysystem, the normalizer N (D) of D is the set of
diffeomorphisms φ on M such that for every x ∈ M , φ(x) and φ−1 (x) belongs
to the adherence of S(D)(x).
By definition, we have the following proposition.
Proposition 3.2.15. If F ∈ D, φ ∈ N (D) then φ ∗ F belongs to satD. Moreover if ±G ∈ D then for each λ ∈ R, (exp λG) ∗ F ∈ satD.
Proposition 3.2.16. If D is a polysystem then the closure of D for the topology of uniform convergence on compact sets belongs to satD.
96
3 Orbital Transfer Problem
Proof. From the definition of the topology, Fn → F when n → +∞ and
exp tFn → exp tF , when n → +∞ on each compact set and the assertion
follows.
¤
One consequence of the enlargement technique is a straightforward proof of
the following theorem.
Theorem 3.2.17. Let M be a connected manifold and consider the smooth
system
n
X
dx(t)
= F0 (x(t)) +
ui (t)Fi (x(t))
dt
i=1
where ui takes its value in {−ε, +ε}, ε > 0 for i = 1, · · · , n. We assume
(1) dim{F0 , F1 , · · · , Fn }L.A. (x) = dimM for every x ∈ M .
(2) The vector field F0 is Poisson stable.
Then the system is controllable on M . The rank condition (i) is also necessary
in the analytic case.
Proof. Let D be the associated polysystem. We observe that DL.A. = {F0 , F1 ,
· · · , Fn }L.A. and the rank condition is satisfied. Hence controllability is
equivalent to controllability of satD. By cone convexity, F0 ∈ satD and
since F0 is Poisson stable then ±F0 ∈ satD. Again using cone convexity,
{±F0 , ±F1 , · · · , ±Fn } ∈ satD. Hence DL.A. ∈ satD which proves the result.
¤
3.3 Lie Bracket Computations and Controllability in
Orbital Transfer
3.3.1 Lie Bracket Computations
We consider the orbital transfer where we assume the mass constant. In order
to make the geometric analysis, we have to investigate the Lie structure of the
systems. This requires computations of Lie brackets. They are lengthly but
straightforward in Cartesian coordinates. Recall that if x = (q, q̇), q ∧ q̇ 6= 0,
we have set:
∂
F0 = q̇ ∂q
− µ |q|q 3 ∂∂q̇
q̇ ∂
Ft = |q̇| ∂ q̇
q∧q̇ ∂
Fc = |q∧
q̇| ∂ q̇
Fn =
(q∧q̇)∧q̇ ∂
|(q∧q̇)∧q̇| ∂ q̇ .
(3.11)
3.3 Lie Bracket Computations and Controllability in Orbital Transfer
97
Tangential direction
[F0 , Ft ](x)
=
[F0 , [F0 , Ft ]](x) =
µ(q·q̇)Ft (x)
q̇)∧q̇ ∂
1
− 2µ (q∧
|q̇| F0 (x) +
|q̇|3 |q̇|2
|q|3 |q̇|3 ∂ q̇
q̇)∧q̇ ∂
− 2µ(q∧
|q|3 |q̇|3 ∂q + a1 F0 (x) + a2 Ft (x)
+a3 [F0 , Ft ](x)
(q·q̇)Ft
= − |q̇|1 2 F0 − µ |q|
3 |q̇|3 +
[Ft , [F0 , Ft ]]
(3.12)
1
|q̇| [F0 , Ft ]
with
a1 =
a2 =
a3 =
µ(q·q̇)
3q q̇
|q|3 |q̇|3 − |q|2 |q̇|
2
q̇)2 −|q∧q̇|2
− |q|µ3 + µ (q·|q|
6 |q̇|4
µ(q·q̇)
3q q̇
+
− |q|
3 |q̇|2
|q|2 | .
(3.13)
Normal direction
q̇)∧q̇ ∂
µ|q∧q̇| ∂
[F0 , Fn ](x) = (q∧
|q̇||q∧q̇| ∂q + |q|3 |q̇|3 ∂ q̇
[F0 , [F0 , Fn ](x) = c1 F0 (x) + c2 Fn (x)
q̇|Fn
[Fn , [F0 , Fn ]] = |q̇|1 2 F0 − 2µ |q∧
|q|3 |q̇|3
(3.14)
with
c1 =
c2 =
2µ|q∧q̇|
|q|3 |q̇|3
|q∧q̇|2
−3µ2 |q|
6 |q̇|4
2
− µ 3(q·q̇)|q|−2|q|
5 |q̇|2
2
|q̇|2
.
(3.15)
Momentum direction,
[F0 , Fc ](x) =
(q∧q̇) ∂
|q∧q̇| ∂q
2
¡ µq
q̇)(q∧q̇)∧q̇ ∂
(q∧q̇)q̇ ¢ ∂
[Fc [F0 , Fc ]](x) = |q̇| F0 + (q·|q∧
q̇|2 |q̇|2 ∂q + |q|3 |q̇|2 + |q∧q̇|2 ∂ q̇
[F0, , [Fc , [F0 , Fc ]]](x) = 0
|q|2
q·q̇
[Fc , [Fc , [F0 , Fc ]]](x) = − |q∧
q̇|2 [F0 , Fc ](x) − |q∧q̇|2 Fc (x).
(3.16)
We deduce the following proposition.
Proposition 3.3.1. For x = (q, q̇) ∈ R6 , q ∧ q̇ 6= 0, we have:
•
(i) The dimension of {F0 , Ft }L.A. (x) is 4 and F0 , Ft , [F0 , Ft ], [F0 , [F0 , Ft ]]
form a frame.
• (ii) The dimension of {F0 , Fn }L.A. (x) is 3 and F0 , Fn , [F0 , Fn ] form a
frame.
• (iii) The vectors F0 , Fc and [F0 , Fc ] are independent and (a) if L(0) 6=
0, dim{F0 , Fc }L.A. (x) = 4 and the vectors F0 , Fc , [F0 , Fc ], [Fc , [F0 , Fc ]]
form a frame. (b) If L(0) = 0 then the Lie algebra {F0 , Fc } is a finitedimensional Lie algebra of dimension 3.
98
3 Orbital Transfer Problem
3.3.2 Controllability Results
Using our Lie brackets computations and the representation of the system in
Gauss coordinates, we can compute the orbits corresponding to the control
oriented in a single direction, which are the integral manifolds of the associated
Lie algebra. This gives controllability results in the elliptic domain, since the
trajectories of the free motion are periodic.
Proposition 3.3.2. If we restrict the system to the elliptic domain with a
single thrust direction then the orbits are as follows:
•
Direction Ft : The orbit is the whole 2D-elliptic domain corresponding to
the elliptic domain for coplanar transfer.
• Direction Fn : The orbit is of dimension 3 and is the intersection of the
2D-elliptic domain with a = a(0).
• Direction Fc : The orbit is of dimension 4 if L(0) 6= 0 (resp. 3 if L(0) = 0)
and is given by a = a(0), |e| = |e(0)|.
Similar results can be used from the radial / orthoradial frame. Moreover,
with full control, we have the following proposition.
Proposition 3.3.3. If we restrict the system to the elliptic domain with a full
control, we have:
•
The Lie algebra is of dimension 6 and the vectors F0 (x), Ft (x), Fn (x),
Fc (x), [F0 , Fc ](x), [F0 , Fn ](x) form a frame.
• The orbit is the whole elliptic domain.
Proposition 3.3.4. For the system restricted to the elliptic domain (with full
control or a control oriented in a single direction) every point of the orbit is
accessible.
Proof. On the elliptic domain, the system is analytic. Restricting the system
to the corresponding orbit, we obtain a system whose drift is Poisson stable
since every trajectory of the free motion is periodic which satisfies the rank
condition. Hence, we can apply theorem 3.2.17.
¤
Corollary 3.3.5. We consider the controlled Kepler equation with constant
mass. Then if we restrict the system to the elliptic domain, we can transfer
every state (x̄0 , l0 ) to every state (x̄1 , l1 ) of the domain, where x̄ are orbits
elements and l is the cumulated longitude.
3.4 Constructing a Feedback Control Using Stabilization
Techniques
The aim of this section is to present a method to construct simple feedback
controls using stabilization techniques. The construction is standard for mechanical systems with first integrals. It is based on the theorem of JurdjevicQuinn [96] which is an application to control analysis of La Salle theorem on
stability [124].
3.4 Constructing a Feedback Control Using Stabilization Techniques
99
3.4.1 Stability Results
Definition 3.4.1. Let ẋ = X(x) be a smooth differential equation on an
open set U ⊂ Rn and let x0 ∈ U be an equilibrium point. We say that x0
is stable if ∀ε > 0, ∃η > 0, |x1 − x0 | ≤ η ⇒ |x(t, x1 ) − x0 | ≤ ε, ∀t ≥ 0,
where x(t, x1 ) is the solution issued from x1 . The attraction basin of x0 is
D(x0 ) = {x1 ; x(t, x1 ) 7→ x0 , t → +∞}. The point x0 is exponentially stable if
x0 is stable and D(x0 ) is a neighborhood of x0 . Moreover if D(x0 ) = U then
x0 is globally asymptotically stable.
Definition 3.4.2. Let V : U → R be a smooth function. It is called a Lyapounov function if locally V > 0 for x 6= x0 and V̇ = Lx V ≤ 0; V is called
strict if V̇ < 0 for x 6= x0 .
Theorem 3.4.3. (Lyapunov) If there exists a Lyapunov function, if x0 is
stable and if V is strict then x0 is exponentially stable.
Lyapunov functions are important tools to check stability. This method is
called the direct Lyapunov stability method. In many applications, x0 is exponentially stable but we can only easily construct Lyapunov functions which
are not strict. Still we can conclude by estimating the ω-limit set of x0 .
Definition 3.4.4. Assume that the solution x(t, x1 ) is defined for t ≥ 0. The
point y is a ω-limit point if there exists a sequence tn → +∞ such that
x(tn , x1 ) → y when n → +∞. The set of ω-limit points of x1 is denoted
Ω + (x1 ).
The following results are standard [124].
Lemma 3.4.5. If Ω + (x1 ) is non-empty and bounded then x(t, x1 ) tends to
Ω + when t → +∞.
Lemma 3.4.6. If the positive trajectory {x(t, x1 ); t1 ≥ 0} is bounded then
Ω + (x1 ) is non-empty and compact.
Lemma 3.4.7. The set Ω + (x1 ) is an invariant set, i.e., it is formed by an
union of trajectories.
Proposition 3.4.8. Let V : U → R, V̇ = LX V ≤ 0 on U then for each
x1 ∈ U , V is constant on Ω + (x1 ).
Proposition 3.4.9. (La Salle) Let K be a compact subset on U and V such
that LX V ≤ 0 on K. Let E = {x ∈ K; LX V = 0} and M the largest invariant subset in E. Then for each x1 such that x(t, x1 ) ∈ K for every t ≥ 0,
x(t, x1 ) → M when t → +∞.
Proof. Since V is constant on Ω + (x1 ) and this set is invariant, V̇ = 0 on
Ω + (x1 ). Hence Ω + (x0 ) ⊂ M . Since K is compact, Ω + (x1 ) ⊂ K is compact.
Moreover x(t, x1 ) → Ω + (x0 ) when t → +∞.
¤
100
3 Orbital Transfer Problem
Corollary 3.4.10. (La Salle, global formulation) Let ẋ = X(x) be a differential equation on Rn , X(0) = 0. Assume that there exists a function V such
that V > 0 for x 6= 0, LX V ≤ 0 and V (x) 7→ +∞ when |x| → +∞. Let M be
the largest invariant set contained in E = {x; LX V = 0}. Then all solutions
are bounded and converge to M when t → +∞.
3.4.2 Stabilization of Nonlinear Systems
The La Salle theorem with Lie brackets computations give important stabilization results with simple feedbacks. This is the Jurdjevic-Quinn method
which is stated in the single-input case, the general case being similar.
Theorem 3.4.11. We consider a smooth system on Rn of the form ẋ =
F0 (x) + uF1 (x), F (0) = 0. We assume that:
There exists V : Rn → R, V > 0 on Rn \{0}, V (x) → +∞ when |x| → +∞
such that (a) ∂V
∂x 6= 0 for x 6= 0 and (b) LF0 V = 0, i.e., V is a first integral.
• E(x) = Span{F0 (x), F1 (x), [F0 , F1 ](x), · · · , adn F0 · F1 (x), · · · } = Rn for
x 6= 0.
•
Then the canonical feedback û(x) = −LF1 V (x) stabilizes globally and asymptotically the origin.
Proof. Plugging û(x) in the system, we get an ordinary differential equation
ẋ = F0 (x) + û(x)F1 (x). We have
V̇ (x) = LF0 +ûF1 (V ) = LF0 V + ûLF1 V = −(LF1 V (x))2 ≤ 0.
Using the La Salle theorem, x(t) → M when t → +∞ where M is the largest
invariant set in LF1 V = 0. We can evaluate this set. Indeed since M is invariant if x(0) ∈ M , x(t) ∈ M . Moreover on M , û(x) = 0 and x(t) is solution
of the free motion ẋ = F0 (x). Hence, differentiating with respect to time
V̇ (x) = (LF1 V )(x) = 0, we get
d
LF V (x(t)) = LF0 LF1 V (x(t)) = 0.
dt 1
Since LF0 = 0, we deduce
L[F0 ,F1 ] (V (x(t))) = 0.
Iterating the derivation, one gets
LF0 V = LF1 V = L[F0 ,F1 ] V = · · · = Ladk F0 (F1 ) (V ) = 0.
Hence we obtain
∂V
(x) ⊥ E(x)}.
∂x
Since E(x) = Rn for x 6= 0 and ∂V
∂x 6= 0 except at x = 0, we obtain M = {0}
and the result is proved.
¤
M ⊂ {x;
3.4 Constructing a Feedback Control Using Stabilization Techniques
101
Remark 4 The second condition of the theorem has the following interpretation. If Span{adk F0 · F1 (x); k ≥ 0} = Rn then from results of Chapter 1, the
end-point mapping near u = 0 is an open mapping. Adding F0 (x) corresponds
to adding a time variation. Hence this condition means that the end-point
mapping, when the time varies, is an open mapping for u = 0 and the extremity point x(T ) = (exp T F0 )(x0 ) is interior to the accessibility set A+ (x). Hence
from x0 , we can reach every neighboring point of x(T ) and in particular, we
can make the energy decrease.
3.4.3 Application to the Orbital Transfer
The following stabilization method can be applied to design local feedback
transfer law. Indeed, the system projects in the coordinates C and L into
F
Ċ = q ∧ m
L̇ = F ∧ C + q̇ ∧ (q ∧
F
m ).
(3.17)
Suppose that the final orbit is (CT , LT ) and introduce the function
V (q, q̇) =
1
(|C(q, q̇) − CT |2 + |L(q, q̇) − LT |2 )
2
where | · | is the euclidian norm. Hence V represents the distance to the
d
final orbit. We shall choose a thrust F such that dt
V (q, q̇) ≤ 0 along the
trajectories. If we denote ∆L = L − LT and ∆C = C − CT then a simple
d
F
computation gives dt
V (q, q̇) = m
·W with W = ∆C ∧q+C ∧∆L+(∆L∧ q̇)∧q.
Hence a canonical choice to satisfy V̇ ≤ 0 is
F
= −f (q, q̇)W
m
with an arbitrary f > 0. We deduce that
d
V (q, q̇) = −f (q, q̇)W 2 .
dt
This corresponds to the application of the feedback constructed in the proof of
Jurdjevic-Quinn theorem. To conclude one must prove that the the trajectory
converges exponentially towards the final orbit represented by (CT , LT ). The
proof is geometric: if d represents the distance induced by V = 12 (|C(q, q̇) −
CT |2 + |L(q, q̇) − LT |2 ), we denote
Bl = {(C, L); d((C, L), (CT , LT )) ≤ l}.
We choose l0 small enough such that Bl0 is contained in the elliptic domain.
Hence, if Kl0 = Π −1 (Bl0 ) where Π : (q, q̇) → (C, L), then the set Kl0 is a
compact set corresponding to the fiber product of S 1 with Bl0 . Hence, from
La Salle theorem, each trajectory starting from Kl0 tends when t → +∞ to
102
3 Orbital Transfer Problem
the largest invariant set contained in V̇ = 0, that is W = 0. We shall prove
that it is the orbit (CT , LT ). This can be obtained by Lie brackets computations (second condition of theorem 3.4.11) or using the following geometric
reasoning. The set W = 0 is
∆C ∧ q + C∆L + (∆L ∧ q̇) ∧ q = 0.
(3.18)
Hence, taking the scalar product with q, we get
q · (C ∧ ∆L) = 0 ⇔ ∆L · (q ∧ C) = 0.
We observe that the trajectory q(t) is an ellipse which is contained in a plane
perpendicular to C defined by Π = Span{q(t)∧C}. Thus, using ∆L·(q ∧C) =
0, we have ∆L = λC where λ is constant. Therefore from (3.18), we obtain
(∆C − λ(q̇ ∧ C)) ∧ q = 0.
q
Using L = (q̇ ∧ C) − µ |q|
, we deduce that
(∆C − λL) ∧ q = 0.
Hence the constant vector ∆C − λL is parallel to the non-zero vector q(t)
which sweeps an ellipse. We have therefore
∆C = λL ⇔ CT = C − λL.
Using ∆L = λC, we get LT = L − λC and
0 = CT · LT = −λ(C 2 + L2 ).
Since C 6= 0, we deduce that λ = 0 and CT = C, LT = L.
Remark 5 This gives a local stabilization result on a ball Bl0 in the elliptic
domain. To get a global result to transfer (CI , LI ) to (CT , LT ), we choose
a path γ : [0, 1] → Σe joining the two points and we cover the image by
a finite set of points (Ci , Li ), i = 1, · · · , N such that we can transfer two
consecutive points (Ci , Li ), (Ci+1 , Li+1 ) using the previous feedback. Another
method is to reshape V in such a way that the corresponding ball with radius
dV ((CI , LI ), (CT , LT )) is entirely contained in the domain Σe . Mathematically, this amounts to choose V proper on Σe with V → +∞ when C → 0
and |L| → µ, corresponding to the boundary.
3.5 Optimal Control Problems in Orbital Transfer
3.5.1 Physical Problems
In orbit transfers, we are concerned by two optimal problems.
3.5 Optimal Control Problems in Orbital Transfer
•
•
103
Time optimal control : The problem is to minimize the transfer time.
Maximizing the final mass: Since ṁ = −δ|u|, this problem is equivalent to
RT
minimize the consumption minu(·) 0 |u(t)|2 dt, where T is fixed.
For mathematical reasons, we also consider the following problems.
•
•
Replace min T by min l where l is the cumulative longitude.
Replace the L1 -norm on the control by the L2 -norm, that is
Z T
|u(t)|2 dt,
min
u(·)
0
where T is fixed. This corresponds to a standard energy minimization
problem.
We can relax the constraint |u| ≤ 1 induced by the thrust, choosing a posteriori
the transfer time large enough to satisfy the constraints.
Optimal control problems can be analyzed using a continuation method
at two levels.
•
The maximal amplitude of the thrust Fmax can be taken as a continuation
parameter, especially if low thrust is applied because for Fmax large enough
the optimal control problems are simpler, the limit case being impulse
controls.
• We can make a continuation on the cost, for instance a standard homotopy
path is defined from L2 to L1 by
Z T
min
(λ|u| + (1 − λ)|u|2 )dt, λ ∈ [0, 1].
u(·)
0
The complexity of each problem is described in the next section by using the
maximum principle.
3.5.2 Extremal Trajectories
Time minimal case
We can assume the mass constant since a straightforward computation gives
that the modulus of the thrust is constant and maximal in this case. Hence,
neglecting the mass variation and restricting to the coplanar case for simplicity, the system can be represented in Cartesian coordinates by
ẋ = F0 (x) +
2
X
ui Fi , |ui | ≤ M
i=1
where x = (q, q̇). The state q = (q1 , q2 ) belongs to a plane identified to the
equatorial plane and we assume |q| 6= 0 to avoid collision. The drift F0 is deduced from Kepler equation and Fi = ∂∂q˙i . The state space is a 4-dimensional
104
3 Orbital Transfer Problem
manifold and we denote Σe the 2D-elliptic domain filled by elliptic trajectories
of Kepler equation.
To analyze the extremal curves, we need the following Lie brackets computations.
Lemma 3.5.1. On X, the four dimensional vector fields F1 , F2 , [F0 , F1 ]
and [F0 , F2 ] are linearly independent and D = Span{F1 , F2 } forms a 2dimensional involutive distribution.
Extremal curves
The pseudo-Hamiltonian takes the form H̃ = H + p0 · 1 where H = H0 +
P
2
i=1 ui Hi and Hi = hp, Fi (x)i, i = 0, 1, 2. To complete the analysis, we use
the previous lemma and results from chapter 2. Let Σ be the switching surface
defined by H1 = H2 = 0. Outside Σ, the maximization condition gives
ûi (z) = M p
Hi (z)
H12 + H22
, i = 1, 2
and plugging û into H defines the true Hamiltonian
v
u 2
uX
Ĥ(z) = Ĥ0 (z) + M t
Hi2 .
i=1
The following proposition is straightforward.
Proposition 3.5.2. The solutions of Ĥ are smooth responses to smooth controls with maximal thrust M and Ĥ depends smoothly upon M . The solutions
parameterize the singularities of the end-point mapping, when u is restricted
to the sphere |u| = M .
In order to complete the analysis, we use the classification of Chapter 2, which
exhausts all the connections of the solutions of Ĥ through Σ.
Proposition 3.5.3. The extremals are solutions of Ĥ with a finite number
of crossings of the switching surface Σ at points where the control rotates
instantaneously of an angle of π.
Combined with controllability and existence results, we obtain:
Proposition 3.5.4. If q = (x̄, l), where x̄ is the vector representing the first
integral and l is the cumulative longitude, then for each pair of points (x0 , x1 )
in the elliptic domain, there exists a trajectory transferring x0 to x1 . If r0 is
the distance to a collision of this trajectory, then there exists a time minimal
trajectory such that |q| ≥ r0 . Every optimal trajectory not meeting the boundary r = r0 is bang-bang with maximal thrusts, the switching being points where
the control rotates instantaneously through an angle of π.
The result can be extended to the cases where the mass is not assumed
constant and in the non-coplanar transfer.
3.5 Optimal Control Problems in Orbital Transfer
105
Minimization of the energy
RT
In this case, the cost is 0 |u|2 dt where the transfer time T is fixed (but large
enough to ensure controllability properties) and we relax the uniform bound
|u| ≤ M . Moreover, we assume that we are in the coplanar case and that the
mass is constant. The pseudo-Hamiltonian takes the form
H̃(z, u) = H0 +
2
X
ui Hi + p0
i=1
0
2
X
u2i
i=1
0
where p < 0 in the normal case and p = 0 in the abnormal case.
Lemma 3.5.5. There exist no abnormal extremals.
H̃
Proof. Assume p0 = 0 then ∂∂u
= 0 gives H1 = H2 = 0. Differentiating with
respect to time, we obtain {H0 , H1 } = {H0 , H2 } = 0. From Lemma 3.5.1, we
deduce that p = 0 and thus a contradiction.
¤
We consider now the normal case where p0 is normalized to − 12 . The condition
∂ H̃
∂u = 0 gives us ûi = Hi and plugging ûi into H̃ leads to the true Hamiltonian
2
1X 2
H .
H̃(z) = H0 +
2 i=1 i
Hence, we have:
Proposition 3.5.6. The extremal curves associated to the energy minimization problem are the solutions of the smooth Hamiltonian vector field with
Hamiltonian
2
1X 2
H̃(z) = H0 +
H .
2 i=1 i
Proposition 3.5.7. Let x0 and x1 be in the elliptic domain and assume that
there exists an admissible trajectory transferring x0 to x1 in time T and satisfying |q| ≥ r0 . Then if we impose |q| ≥ r0 , the energy minimization problem
has a solution.
Proof. We apply the existence theorem for optimal control without magnitude
constraints (Proposition 1.2.57). Recalled that the controlled Kepler equation
is q̈ = − |q|q 3 + u. If |q| ≥ r0 then we have |q̈| ≤ r12 + |u| and by integration we
0
obtain
Z T
T
|q̇(T ) − q̇(0)| ≤ 2 +
|u|dt.
r0
0
Hence, there exists an increasing function Φ such tat
Z T
|x(T )| ≤ Φ(
|u|dt).
0
The result is proved.
¤
106
3 Orbital Transfer Problem
The application of the maximum principle for the time minimal problem and the energy minimization one leads to an extremal system which is
smooth, excepted at isolated singularities in the time minimal case. Hence,
they are good candidates to be computed numerically using a shooting method
combined with second order optimality test explained in Chapter 1. On the
opposite, the computations for the minimum fuel consumption reveal more
complexity and a lack of smoothness explained in the next section.
Maximization of the final mass
The system is written as
q̇ = v
v̇ = − |q|q 3 + uε
m
ṁ = −βε|u|, |u| ≤ 1.
The cost function is
RT
0
(3.19)
|u|dt and the associated pseudo-Hamiltonian is
H = (p0 − βεpm )|u| + hv, pq i + hpv , −
uε
q
+
i.
|q|3
m
We consider the normal case with p0 6= 0. Normalizing it to −1, we must
maximize over |u| ≤ 1, the function
−(1 + βpm )|u| + hpv ,
Introducing
ψ = −(1 + βpm ) +
uε
i.
m
ε
|pv |,
m
we have:
Assume |pv | 6= 0 then the maximum of the pseudo-Hamiltonian is:
•
•
If ψ > 0 then u = |ppvv | which corresponds to a maximum thrust
If ψ < 0 then the maximum is given by u = 0.
Hence a generic extremal control is a concatenation of controls with maximal
thrusts and zero controls. The problem is with non smooth extremals which
leads to technical difficulties to compute second order optimality conditions.
In order to take into account cone constraints whose limit case is single
input- system, we must analyze the extremals for the single-input case. The
analysis differs only for the time minimal case. We recall next some results
which will be used in the sequel.
3.7 Extremals for Single-Input Time-Minimal Control
107
3.6 Preliminary Results on the Time-Minimal Control
Problem
In this section we present preliminary results concerning the time-minimal
orbit transfer. They are mainly obtained by numerical simulations and are
two-fold. First of all a continuation method on the magnitude of the maximal
thrust can be applied. In practice a discrete homotopy is sufficient. This leads
to computation of an extremal solution from a low eccentric orbit to the
geosynchronous orbit. Secondly, the Hampath code is used to check optimality.
3.6.1 Homotopy on the Maximal Thrust
We denote Fmax the maximal thrust and a discrete homotopy consists in
picking a finite sequence λ0 = 0 < · · · < λk < · · · λN = 1 to make the
0
2
using the convex homotopy Fmax = (1 −
continuation from Fmax
to Fmax
1
0
λ)Fmax + λFmax .
Variable
µ
1/ve
m0
Fmax
Value
5165.8620912
1.42e − 2
1500
3
Mm3 ·h−2
Mm−1 ·h
kg
N
Table 3.1. Physical constants.
Proposition 3.6.1. The value function Fmax 7→ T (Fmax ) mapping to each
positive maximum thrust the corresponding minimum time is right continuous
for the transfer problem (2D or 3D, constant mass or not).
3.6.2 Conjugate Points
The existence of conjugate points is detected using the Hampath code as can
be seen in Fig. 3.2.
3.7 Extremals for Single-Input Time-Minimal Control
The system takes the form ẋ = F0 (x) + uF1 (x), |u| ≤ 1.
3 Orbital Transfer Problem
2
0
−2
q
3
108
40
40
20
20
0
0
−20
−20
−40
q2
−40
q1
40
2
1
q3
q2
20
0
0
−1
−20
−2
−40
−60
−40
−20
0
20
40
−40
−20
q1
0
q2
20
40
Fig. 3.1. Three dimensional transfer for 3 Newtons. The arrows indicate the action
of the thrust. The main picture is 3D, the other two are projections. The duration
is about twelve days.
3.7.1 Singular Extremals
According to chapter 1, they are contained in the subset Σ1 :
H1 = {H1 , H0 } = 0.
The case where {{H1 , H0 }, H1 } 6= 0 is called of minimal order and the singular control is given by
û = −
{{H1 , H0 }, H0 }
(z).
{{H1 , H0 }, H1 }
The true Hamiltonian Ĥ(z) = H0 (z) + û(z)H1 (z) defines a smooth Hamiltonian vector field on Σ1 restricting the standard symplectic form. The singular
control has to be admissible, i.e., |û(z)| ≤ 1 and the case |û(z)| = 1 is called
saturing.
Singular extremals are split into two categories: normal case if Ĥ0 > 0
and abnormal one if Ĥ0 = 0. Moreover, in order to be time-minimal, the
generalized Legendre-Clebsch condition has to be satisfied:
{{H1 , H0 }, H1 }(z(t)) ≥ 0.
3.7 Extremals for Single-Input Time-Minimal Control
109
q3
10
0
−10
100
40
50
20
0
0
−20
−40
−50
q1
20
1
3
2
0
q
q2
q2
40
−20
0
−1
−40
−2
−50
0
50
100
−40
q
−20
1
20
40
2
−4
4
0
q
−3
x 10
3
x 10
2.5
2
σk
arcsh det(δ x)
2
0
1.5
1
−2
0.5
−4
0
1
2
3
0
0
t/T
1
2
3
t/T
Fig. 3.2. An extremal, which is roughly the same as in fig. 3.1 (the difference being
the fixed final longitude), is extended until 3.5 times the minimum time. Bottom
left, the determinant, bottom right, the smallest singular value of the Jacobi fields
associated to the extremal. There, two conjugate times are detected. The optimality
is lost about three times the minimum time.
3.7.2 Classification of Regular Extremals
Definition 3.7.1. Let (z, u) be an extremal defined on [0, T ]. It is called regular if u(t) = sign H1 (z(t)). A time s is called a swithching time if it belongs
to the closure of the set of t ∈ [0, T ] where z(·) is not C 1 . A regular extremal is bang-bang if the number of switchings is finite. The set of switchings
points forms the switching subset and it is a subset of the switching surface
Σ : H1 (z) = 0. Let z be any smooth solution of H0 + uH1 corresponding
to a smooth control. The switching function Φ is the mapping t 7→ H1 (z(t))
110
3 Orbital Transfer Problem
evaluated along z(·). If u = +1 (resp. −1) then we set z = z + and Φ = Φ+
(resp. z = z − and Φ = Φ− ).
Lemma 3.7.2. The first two derivatives of the switching mappings are
Φ̇(t) = {H1 , H0 }(t)
Φ̈(t) = {{H1 , H0 }, H0 }(z(t)) + u(t){{H1 , H0 }, H1 }(z(t)).
(3.20)
Normal Switching Points
Let Σ be the surface H1 = 0 and Σ1 be the subset of Σ with {H1 , H0 } = 0.
Let z0 = (x0 , p0 ) and assume F1 (x0 ) 6= 0, z0 ∈ Σ\Σ1 . The point z0 is called
a normal switching point. From the previous lemma, we have:
Lemma 3.7.3. Let t0 be the switching time defined by z + (t0 ) = z − (t0 ). Then
the following equation holds
Φ̇+ (t0 ) = Φ̇− (t0 ) = {H1 , H0 }(z0 )
and the extremal passing through z0 is of the form
z = γ+ γ−
if {H1 , H0 }(z0 ) < 0 and
z = γ− γ+
if {H1 , H0 }(z0 ) > 0 (γ1 γ2 represents the arc γ1 follows by the arc γ2 ).
The fold case
Let z0 ∈ Σ1 and assume Y (x0 ) 6= 0 and Σ1 be a smooth surface of codimension
2. If H+ and H− are the Hamiltonian vector fields associated to H0 ± H1 then
Σ = {z, H+ = H− } and at z0 ∈ Σ1 , both vector fields are tangent to Σ. We
set
λ± = {{H1 , H0 }, H0 }(z0 ) ± {{H1 , H0 }, H1 }(z0 )
(3.21)
and we assume that both λ± 6= 0. We deduce that the contact of H+ and H−
with Σ is of order 2 and we distinguish three cases
•
•
•
λ+ λ− > 0: parabolic case
λ+ > 0, λ− > 0: hyperbolic case
λ+ < 0, λ− < 0: elliptic case
The respective behavior of regular extremals are represented in Fig. 3.3 and
we have the following result.
In the parabolic case, there exists a neighborhood V of z0 such that each
extremal in V has at most two switchings. It is the case if {{H1 , H0 }, H0 } = 0
or {{H1 , H0 }, H0 } 6= 0 and the singular extremal of minimal order through
z0 is not admissible. In the hyperbolic and elliptic cases, a singular extremal
3.7 Extremals for Single-Input Time-Minimal Control
111
passes through z0 with a control satisfying |u| < 1. The generalized LegendreClebsch condition is satisfied only in the hyperbolic case. In this case, in
a neighborhood V of z0 , every extremal has at most one switching. In the
elliptic case, the situation is more complex because every regular extremal in
a neighborhood V of z0 has a finite number of switchings, but with a nonuniform bound on this number.
γ+
γ+
γs
γ+
Σ
γ−
γ−
γ−
(a)
(b)
(c)
Fig. 3.3. different behaviors of regular extremals in the fold case.
Hence from this analysis, we deduce the classification of all extremals near
a fold point.
Proposition 3.7.4. Let z0 be a fold point. Then there exists a neighborhood
V of z0 such that:
•
•
•
In the hyperbolic case, each extremal trajectory has at most two switchings
and is of the form γ± γs γ± where γs is a singular arc.
In the parabolic case, each extremal arc is bang-bang with at most two
switchings and has the form γ+ γ− γ+ or γ− γ+ γ− .
In the elliptic case, each extremal arc is bang-bang but with no uniform
bound on the number of switchings.
3.7.3 The Fuller Phenomenon
In the elliptic case, the main problem when analyzing the extremals is to
prove that every extremal on [0, T ] has a finite number of switchings. One of
the main contribution in geometric control was to prove that is not a generic
situation. This result based on the Fuller example is due to Kupka [104].
112
3 Orbital Transfer Problem
Definition 3.7.5. An extremal (z, u) defined on [0, T ] is called a Fuller extremal if the switching times form a sequence 0 ≤ t1 ≤ · · · ≤ T such
that tn → T when n → +∞ and if there exists k > 1 with the property
tn+1 − tn ∼ k1n as n → +∞.
Fuller example
We consider the following problem:
ẋ = y, ẏ = u, |u| ≤ 1
R +∞
with the cost function minu(·) 0 x2 dt. This problem is a linear quadratic
problem where u is not penalized in the cost (the problem is called cheap).
The normal Hamiltonian is
H(x, p, u) = −x2 + p1 y + p2 u
and the extremal control is defined by u(t) = sign p2 (t). An associated trajectory converges to zero as t → +∞ and the adjoint vector satisfies the
transversality conditions
p1 (+∞) = p2 (+∞) = 0.
It turns out that the optimal synthesis is characterized by a switching locus
given by the equation
x + hy|y| = 0
where h ' 0.4446 andqevery non-trivial optimal solution exhibits a Fuller
2
1+2φ
1
phenomenon with k = 1−2φ
> 1 where φ is the positive root of x4 + x12 − 18
=
0. Such optimal trajectories provide Fuller extremals for the time-minimal
problem, where the system is the cost extended previous system.
Hence, we have Fuller extremals for time-optimal control problem in R3 ,
but the example is not stable. The contribution of Kupka was to find a stable
model. The difficulty lies in the determination of semi-algebraic conditions
for which the result is true. These conditions involve the Poisson brackets of
H± = H0 ± H1 at z0 up to order 5 and all the Poisson brackets up to order 4
has to be zero. The Fuller example satisfies these conditions at x0 = (0, 0, 1)
and p0 = (0, 0, −1). To summarize, we have the following theorem.
Theorem 3.7.6. If the dimension of the state space is large enough then there
exists a stable model (F0 , F1 ) exhibiting Fuller extremals.
3.8 Application to Time Minimal Transfer with Cone
Constraints
A non-trivial application of the previous section, together with the Lie brackets computation is to analyze the structure of the time-minimal control for
the coplanar transfer
3.9 Averaged System in the Energy Minimization Problem
113
ẋ = F0 + ut Ft , |ut | ≤ ε
where the control is oriented in the tangential direction.
Proposition 3.8.1. Every time-optimal trajectory of the system ẋ = F0 +
ut Ft , |ut | ≤ ε, ε > 0 is bang-bang.
Proof. We first compute the singular extremals solutions of
Ht = {Ht , H0 } = 0
{{Ht , H0 }, H0 } + u{{Ht , H0 }, Ht } = 0
(3.22)
where Ht = hp, Ft i. From the Lie brackets computations of Section 3.3.1, we
observe that {{Ht , H0 }, Ht } can be written −λH0 mod{Ht , {Ht , H0 }} where
λ > 0. Hence, if {{Ht , H0 }, Ht } = 0 then H0 = 0 and every singular extremal
not of minimal order is abnormal. Moreover, we must have {{Ht , H0 }, H0 } =
0. The relations are not compatible since {F0 , Ft , [F0 , Ft ], [F0 , [F0 , Ft ]]} form
a frame. Hence every singular extremal is of minimal order. Again, using
{Ht , {Ht , H0 }} = −λH0 , λ > 0 and H0 > 0, we deduce that every singular extremal does not satisfy the generalized Legendre-Clebsch condition. We
must now analyze the regular extremals using our classification. According to
the classification of fold points, we can have elliptic or parabolic points but
not hyperbolic points. Moreover, we can have contacts of order 3 where
Ht = {Ht , H0 } = 0
{{Ht , H0 }, H0 } ± ε{{Ht , H0 }, Ht } = 0
(3.23)
for one extremal arc γ+ or γ− but not for both, otherwise
Ht = {Ht , H0 } = {{Ht , H0 }, H0 } = {{Ht , H0 }, Ht } = 0
which is excluded since {F0 , Ft , [F0 , Ft ], [F0 , [F0 , Ft ]]} form a frame. Hence,
the Fuller phenomenon cannot occur.
¤
If we assume that the thrust is oriented in the orthoradial direction only, the
system restricted to the 2D domain remains controllable but the analysis is
more intricate because there exists singular trajectories which can be elliptic, hyperbolic or abnormal. Hence the structure of an optimal trajectory is
complex.
3.9 Averaged System in the Energy Minimization
Problem
3.9.1 Averaging Techniques for Ordinary Differential Equations
and Extensions to Control Systems
We recall the averaging technique for ordinary differential equation [97] and
the straightforward extension to control systems. We consider an equation of
the form
114
3 Orbital Transfer Problem
dx
= εF (x, t, ε), x ∈ Rn
(3.24)
dt
where F is smooth and 2π-periodic with respect to t. Expanding F as
F (x, t, ε) = F0 (x, t) + o(ε), we introduce the following definition.
Definition 3.9.1. The averaged differential equation is
Z 2π
ε
ẋ = εM (x) =
F0 (x, t)dt
2π 0
and we have the standard result.
Proposition 3.9.2. Provided we stay in a compact subset K, let x and x̄ be
the respective solutions of ẋ = εF (x, t, ε) and x̄˙ = εM (x̄) with the same initial
condition x0 . Then |x(t) − x̄(t)| → 0 when ε > 0 uniformly on any subinterval
of length O(1/ε).
This technique can be applied to a control system of the form
ẋ = ε(F (x, t)u + εg(x, t, u)), |u| ≤ 1
where we restrict the set of controls to the set of smooth and 2π-periodic
controls u(·), |u| ≤ 1. For such a control, we can define the averaged differential
equation
x̄˙ = εMu (x̄),
and introducing I = {M (x̄)}, we can consider the differential inclusion
x̄˙ ∈ εI(x̄(t)) whose solutions are trajectories x̄(t) such that there exists an
integrable mapping u(t, θ), 2π-periodic with respect to θ, bounded by 1 such
that
Z T
Z 2π
1
x̄(t) = x̄(0) + ε
F (x(t), θ)u(t, θ)dθ.
0 2π 0
The approximation result concerning differential equations can be easily extended to such differential inclusions.
3.9.2 Controllability Property and Averaging Techniques
We consider an analytic control system of the form:
m
X
dx(t)
= F0 (x) +
ui (t)Fi (x(t)), |ui | ≤ 1, x ∈ Rn .
dt
i=1
From the analysis of Chapter 1, we recall the following lemma.
Lemma 3.9.3. The singular control u = 0 is regular on [0, T ] if and only if
E1 (t) = Span{adk F0 · Fi ; k ≥ 0, i = 1, · · · , m} = Rn .
3.9 Averaged System in the Energy Minimization Problem
115
Proposition 3.9.4. Assume that the control system is of the form
m
dx X
=
ui (t)Fi (x, l))
dt
i=1
dl
=1
dt
where Fi is 2π-periodic with respect to l. Then
Span{adk F0 · Fi } = Span{
∂ k Fi
; k ≥ 0}.
∂lk
Corollary 3.9.5. The averaged differential inclusion associated to the system
k
is of full rank if and only if Span{ ∂∂tFk ; k ≥ 0} is of full rank.
Proof. The system is written as
dx
= εF (x, t)u + o(ε2 )
dτ
dt
= 1, t ∈ S 1 .
dτ
Neglecting o(ε2 ), it can be written in the extended space y = (x, t) as
m
X
dy
= F0 + ε
ui Fi
dτ
i=1
where F0 =
∂
∂t
and from the previous lemma
Span{adk F0 · Fi } = Span{
∂ k Fi
; k ≥ 0}.
∂tk
More precisely, from [87], using constant control perturbations of u = 0 up to
o(ε), we deduce that only the directions Span{adk F0 · Fi ; k ≥ 0, i = 1, · · · , m}
are tangent vectors directions in the accessibility set along the reference trajectory corresponding to u = 0 in fixed time. In particular, since any 2π-periodic
control u(t) can be approximated by such controls, we deduce the results. ¤
3.9.3 Riemannian Metric of the Averaged Controlled Kepler
Equation
Preliminaries
Let X be a n-dimensional smooth manifold and let Fi (x, l), i = 1, · · · , m be
smooth vector fields parameterized by l ∈ S 1 . We consider the control system
116
3 Orbital Transfer Problem
m
dx X
=
ui (t)Fi (x, l))
dt
i=1
dl
= g0 (x, l)
dt
where g0 is a smooth 2π-periodic function with respect to l and g0 > 0. We
consider the minimum energy problem
Z T X
m
(
u2i (t)dt).
min
u(·)
0
i=1
The control is rescaled with u = εv to introduce the small parameter ε and
the trajectories parameterized by l are solutions of
Pm
vi Fi (x, l)
dx
= ε i=1
dl
g0 (x, l)
whenever the cost takes the form
Z l(T ) X
m
vi2 (l)
2
ε
dl.
l(0) i=1 g0 (x, l)
We assume that l(0) and l(T ) are fixed. The cost extended system takes the
form
Pm
Pm 2
v (l)
dx
d c
i=1 vi Fi (x, l)
=ε
,
= ε i=1 i
dl
g0 (x, l)
dt ε
g0 (x, l)
and we rescale c into c̄ = εc .
The associated pseudo-Hamiltonian is
H̃(x, p, l, v) =
m
X
ε
(p0 |v|2 +
vi Hi (x, p, l))
g0 (x, l)
i=1
where Hi (x, p, l) = hp, Fi (x, l)i, i = 1, · · · , m are the 2π-periodic Hamiltonian
lifts and p0 ≤ 0 is a constant. We consider the normal case p0 < 0, which can
be normalized to p0 = − 12 and the Hamiltonian takes the form
H̃(x, p, l, v) =
m
X
ε
1
(− |v|2 +
vi Hi (x, p, l)).
g0 (x, l) 2
i=1
H̃
Since v is valued in the whole Rm , the maximum principle gives ∂∂v
=0
and we get vi = Hi . Plugging such v into H̃, we obtain the true Hamiltonian
n
H(x, p, l) =
X
1
H 2 (x, p, l)
2g0 (x, l) i=1 i
where ε is omitted to simplify the notations. We observe that since g0 is
positive, H can be written as a sum of squares.
3.9 Averaged System in the Energy Minimization Problem
117
Lemma 3.9.6. The function H is a non-negative quadratic form in p which
is denoted w(x, l).
Definition 3.9.7. The averaged Hamiltonian is
Z
1 2π
H̄(x, p) =
H(x, p, l)dl.
2 0
The following result is clear.
Lemma 3.9.8. The averaged Hamiltonian defines a non-negative quadratic
form in p denoted w̄(x). Moreover
ker w̄(x) = ∩l∈S 1 ker w(x, l).
Remark 6 According to this lemma, the rank of w̄(x) is not smaller than m
if the Fi0 s are m independent vector field and we can only expect it to increase.
The geometric interpretation is straightforward. From the maximum principle,
an extremal control is computed as a mapping of the form u(x, p, l) which is
2π-periodic with respect to l. The oscillations induced by l which act as a fast
variable generate new control directions, namely Lie brackets in the linear
space E1 (t) = Span{adk F0 · Fi ; k ≥ 0, i = 1, · · · , m}. Moreover, generically
we can expect to generate all the Lie brackets in E1 (t) to provide an averaged
system of full rank.
Definition 3.9.9. The averaged system is said to be regular if the rank of
w̄(x) is a constant k.
In this case, there exists an orthogonal matrix R(x) such that if P = R(x)p
then w̄(x) is written as a sum of squares
k
1X
λi (x)Pi2
2 i=1
where λ1 , · · · , λk are the non-negative eigenvalues of the symmetric matrix
S(x) defined by
1
w̄(x) = t pS(x)p.
2
Hence, we can write
w̄(x) =
k
k
1X p
1X
( λi Pi )2 =
hp, Fi i2
2 i=1
2 i=1
where the Fi ’s are smooth vector fields on X. This gives of formal proof of
the following proposition.
118
3 Orbital Transfer Problem
Proposition 3.9.10. If the averaged system is regular of rank k, the averaged
Hamiltonian H̄ can be written as a sum of squares regular
k
H̄ =
1X 2
P , Pi = hp, Fi i
2 i=1 i
and is the Hamiltonian of the SR-problem
ẋ =
k
X
i=1
Z
ui Fi (x), min
u(·)
0
T
k
X
u2i (t)dt
i=1
where k is not smaller than n. If k = n = dim X then H̄ is the Hamiltonian
of a Riemannian problem.
Remark 7 For this new optimal control problem, the extremal controls are
not related to the previous ones, but still the true extremal control u(x, p, l)
can be approximated by u(x̄, p̄, l), where x̄ and p̄ are the averaged values.
Moreover, if we apply Proposition 3.9.4 to the cost extended system, we deduce:
Proposition 3.9.11. The extremals of the averaged Hamiltonian systems are
approximations of the true extremal trajectories of order o(ε) for a length of
order o(1/ε) and the cost of the SR-problem is an approximation of the true
cost up to order o(ε2 ).
Remark 8 If we consider the SR-problem, it is equivalent
Pk to a time-minimal
control problem where the controls satisfy the bounds i=1 u2i (t) = 1, which
amounts to fixing the level set of the Hamiltonian to 1/2. By homogeneity,
rescaling u into εu rescales the transfer time from t to t/ε. Therefore if tf is
the transfer time lf − l0 , we have tf ² = M constant where M is the length of
the curve. It gives an estimate of the transfer time with respect to ² where ²
is the maximum thrust amplitude, see [50].
3.9.4 Computation of the Averaged System in Coplanar Orbital
Transfer
Preliminaries
We consider the coplanar constant mass case. In the elliptic domain Σe , the
state of the system is represented by a polar angle l which corresponds to the
longitude and three first integrals of Kepler equation which are the geometric
parameters of an osculating ellipse. For instance, we have x = (P, ex , ey )
where P is the semi-latus rectum, e = (ex , ey ) is the eccentricity vector whose
direction is the semi-major axis and whose length e is the eccentricity. The
elliptic domain is {P > 0, e < 1} where e = 0 corresponds to circular orbits
3.9 Averaged System in the Energy Minimization Problem
119
and e = 1 corresponds to parabolic orbits. To simplify the computation, the
control is decomposed in the radial-orthoradial frame. Applying the previous
process, the true Hamiltonian in the normal case is H = 21 (P12 + P22 ) where
P1 =
P2 =
P 5/4
W (pex sin l − pey cos l)
ex +cos l
P 5/4
2P
)
W [pp W + pex (cos l +
W
+ pey (sin l +
ey +sin l
)]
W
(3.25)
with W = 1 + ex cos l + ey sin l.
The computation of the averaged system requires evaluations of integrals
of the form
Z 2π
Q(cos l sin l)
dl
Wk
0
where Q is a polynomial and k is an integer between 2 and 4. Such integrals
are computed by means of the residue theorem. Using the complex notation
e = ex + iey , the poles are
√
−1 ± 1 − e2
z± =
ē
and only z+ belongs to the unit disk.
An inspection of the Hamiltonian shows that the following averages are
required.
√
Lemma 3.9.12. With δ = 1/ 1 − e2 :
•
•
•
•
•
•
•
•
1/W 2 = δ 3
cos l/W 3 = −(3/2)ex δ 5 , sin l/W 3 = −(3/2)ey δ 5
cos2 l/W 3 = 1/2(δ 3 + 3e2x δ 5 ), sin2 l/W 3 = 1/2(δ 3 + 3e2y δ 5 )
cos l sin l/W 3 = 3/2ex ey δ 5
1/W 4 = 1/2(2 + 3e2 )δ 7
cos l/W 4 = (−1/2)ex (4 + |e|2 )δ 7 , sin l/W 4 = (−1/2)ey (4 + |e|2 )δ 7
cos2 l/W 4 = 1/2(δ 5 + 5e2x δ 7 ), sin2 l/W 4 = 1/2(δ 5 + 5e2y δ 7 )
cos l sin l/W 4 = 5/2ex ey δ 7
Substituting these expressions, we obtain the averaged Hamiltonian.
Proposition 3.9.13. We have
H̄(x, p) =
P 5/2
[4p2 P 2 (−3 + 5(1 − e2 )−1 )
4(1 − e2 )5/2 p
+p2ex (5(1 − e2 ) + e2y ) + p2ey (5(1 − e2 ) + e2x )
−20Pp p2ex − 20Pp p2ey − 2pex pey ex ey ]
120
3 Orbital Transfer Problem
3.10 The Analysis of the Averaged System
At this point, to identify the metric, H̄ has to be written as a sum of squares.
More precisely, we make the following change of variables
P =
1 − e2
, ex = e cos θ, ey = e sin θ
n2/3
where n is the so-called mean motion related to the semi-major axis by
n = a−3/2 . Such a transformation is singular for circular orbits. On the Hamiltonian, this amounts to the Mathieu transformation: x = φ(y), p = q ∂φ
∂y where
q is the new adjoint variable. In the new coordinates, we have:
Proposition 3.10.1. In the coordinates (n, e, θ), the averaged Hamiltonian is
H̄ =
5 − 4e2 2
1
[18n2 p2n + 5(1 − e2 )p2e +
pθ ]
5/3
e2
8n
where the singularity e = 0 corresponds to circular orbits. In particular,
(n, e, θ) are orthogonal coordinates for the Riemannian metric associated to
H̄
dn2
2n5/3 dθ2
2n5/3 de2
ḡ = 1/3 +
+
.
2
5(1 − e )
5 − 4e2
9n
The main step in the analysis is to use further normalization to obtain a
geometric interpretation.
Proposition 3.10.2. In the elliptic domain, we set
r=
2 5/6
n , φ = arcsin e
5
and the metric is isometric to
ḡ = dr2 +
with c =
p
2/5 and G(φ) =
r2
(dφ2 + G(φ)dθ2 )
c2
5 sin2 φ
1+4 cos2 φ .
Geometric interpretation
This normal form captures the main properties of the averaged orbital transfer. Indeed, we extract from ḡ two 2D-Riemannian metric
ḡ1 = dr2 + r2 dψ 2
with ψ = φ/c which is associated to orbital transfer where θ is kept fixed and
the metric
ḡ2 = dφ2 + G(φ)dθ2
which represents the restriction to r2 = c2 .
We next make a complete analysis of ḡ1 and ḡ2 .
3.10 The Analysis of the Averaged System
121
3.10.1 Analysis of ḡ1
θ is a cyclic coordinate and pθ a first integral. If pθ = 0 then θ is constant. The
corresponding extremals are geodesics of the 2D-Riemannian problem defined
by dθ = 0. We extend the elliptic domain restriction to
Σ0 = {n > 0, e ∈] − 1, +1[, e = ex , ey = 0}
and in polar coordinates (r, ψ), Σ0 is defined by {r > 0, ψ ∈] − π/2c, π/2c[}.
This extension allows to go through the singularity corresponding to circular
orbits.
Geometrically, this describes transfer where the angle of the semi-major
axis is kept fixed and pθ = 0 corresponds to the transversality condition. Such
a policy is clearly associated of steering the system towards circular orbits
where the angle θ of the pericenter is not prescribed. An important physical
subcase is when the final orbit is geostationary.
In particular in the domain Σ0 , the metric ḡ1 = dr2 + r2 dψ 2 is a polar
metric isometric to the flat metric dx2 + dz 2 if we set x = r sin ψ and z =
r cos ψ.
We deduce the following proposition.
Theorem 3.10.3. The extremals of the averaged coplanar transfer are straight
lines in the domain Σ0 in suitable coordinates, namely
x=
23/2 5/6
1
23/2 5/6
1
n sin( arcsin e), z =
n cos( arcsin e)
5
c
5
c
p
with c = 2/5. Since c < 1, the domain is not convex and the metric ḡ1 is
not complete.
Proof. The extremals are represented in Fig. 3.4 in the physical coordinates
(n, ex ) (ey is fixed to 0) and in the flat coordinates.
L1
1
L3
n
2
3
S
L2
ex
Fig. 3.4. Geodesics of the metric ḡ1 in (n, ex ) and flat coordinates.
The axis ex = 0 corresponds to circular orbits. Among the extremals, we
have two types: complete curves of type 1 and non-complete curves of type
122
3 Orbital Transfer Problem
2 when meeting the boundary of the domain. The domain is not geodesically
convex and the existence theorem fails. For each initial condition, there exists
a separatrix S which corresponds to a segment line in the orbital coordinates
which is meeting n = 0 in finite time. Its length gives the bound for a sphere
to be compact.
¤
In order to complete the analysis of ḡ and to understand the role of ḡ2 , we
present now the integration algorithm.
3.10.2 Integrability of the Extremal Flow
The integrability property is a consequence of the normal form only
g = dr2 + r2 (dφ2 + G(φ)dθ2 )
and the associated Hamiltonian is decomposed into
H=
1
p2
1 2
1
pr + 2 H 0 , H 0 = (p2φ + θ ).
2
r
2
G(φ)
Lemma 3.10.4. The Hamiltonian vector field H admits three first integrals
in involution: H, H 0 and pθ and is Liouville integrable.
To get a complete parameterization, we proceed as follows. We use the (e, n, θ)
coordinates and we write
H=
with H 00 = 5(1 − e2 )p2e +
1
[18n2 p2n + H 00 ]
4n5/3
5−4e2 2
e2 pθ .
Lemma 3.10.5. Let s = n5/3 then s(t) is a polynomial of degree 2: s(t) =
c1 t2 + ṡ(0)t + s(0) with s(0) = n5/3 (0), ṡ(0) = 15n(0)pn (0) and c1 = 25
2 H.
Lemma 3.10.6. Let dT = dt/4n5/3 then if H 00 (0) 6= 0,
1
[arctan L(s)]t0
T (t) = p
2 |∆|
00
√ , a = c1 , b = ṡ(0) and ∆ = − 25
where L(t) = 2at+b
2 H (0) is the discriminant
|∆|
of s(t).
This allows to make the integration. Indeed if H 00 = 0, pe = pθ = 0 and the
trajectories are straight lines (the line S in Fig. 3.4). Otherwise, we observe
that n5/3 (t) is known and depends only upon n(0), pn (0) and H which can be
fixed to 1/2 by parameterizing by arc-length. Hence, it is sufficient to integrate
the flow associated to H 00 using the parameter dT = 4ndt5/3 where T is given
by the previous Lemma.
3.10 The Analysis of the Averaged System
123
We proceed as follows. Let H 00 = c23 and pθ = c2 . Using pe = e/10(1 − e2 ),
we obtain
20(1 − e2 )
ė2 =
[c3 e2 − (5 − 4e2 )c22 ].
e2
To integrate, we set for e ∈]0, 1[, w = 1 − e2 and the equation takes the form
dw
= Q(w)
dT
where
Q(w) = 80w[(c23 − c22 ) − (c23 + 4c22 )w]
with positive discriminant. Hence the solution is
q
w=
√
1 c23 − c22
[1 + sin(4/ 5
2
2
2 c3 + 4c2
c23 + 4c22 )T + K],
K being a constant. We deduce that
Z
θ(T ) = θ(0) + 2c2
0
T
1 + 4w(s)
ds
1 − w(s)
where θ(0) can be set to 0 by symmetry. To conclude, we must compute
p
R T 1+4w(s)
ds with w = K1 (1 + sin x) and x = √45 c23 + 4c22 s + K. Therefore,
0 1−w(s)
we must evaluate an integral of the form
Z
A + B sin x
dx
C + D sin x
which is a standard exercise. More precisely, the formula is
Z
Z
A + B sin x
B
dx
dx = x + AD − BC
C + D sin x
D
C + D sin x
with
Z
dx
2
C tan(x/2) + D
=√
arctan( √
)
2
2
C + D sin x
C −D
C 2 − D2
for C 2 − D2 > 0 in our case. The previous lemmas and computations give:
Proposition 3.10.7. For H 00 6= 0, the solution of H can be computed using
elementary functions and we get
2
5/3
n(t) = ( 25
(0))3/5
2 Ht + 15n(0pn (0)t + n
1/2
e(t) = (1 − K1 (1 + sin K2 (t)))
(3.26)
(1−K1 ) tan(x/2)−K K2 (t)
pθ
10
θ(t) = θ(0) + 2|p
K
[−4x
+
arctan(
)]
3
K
K3
K3
θ|
2
with K = arcsin( 1−e(0)
− 1), K1 =
K1
00
2
1 H (0)−pθ
2 H 00 (0)+4p2θ ,
124
3 Orbital Transfer Problem
q
4
K2 (t) = √ ( H 00 (0) + 4p2θ T (t) + K)
5
r
and K3 =
5p2θ
.
H 00 (0)+4p2θ
For H 00 = 0, they are straight lines.
Remark 9 The above formulas give the complete solution of the associated
Hamilton-Jacobi equation.
3.10.3 Geometric Properties of ḡ2
The previous integration algorithm shows that the extremals of this metric
describe the evolution of the angular variables θ and φ, parameterized by T
dt
2
with dT = r(t)
is a second order polynomial whose coefficients
2 where r(t)
depend only upon the energy level H fixed to 1/2, r(0) and pr (0). We next
describe some basic properties of ḡ2 .
Lemma 3.10.8. The metric ḡ2 can be extended to an analytic metric on the
whole S 2 , where θ and φ are spherical coordinates with two polar singularities
at φ = 0, π corresponding to e = 0. The equator corresponds to e = 1 and θ is
an angle of revolution. The meridians are projections on S 2 of the extremals
of ḡ1 .
Lemma 3.10.9. The metric is isometric for the two transformations
(φ, θ) 7→ (φ, −θ)
and
(φ, θ) 7→ (π − φ, θ).
This induces the following symmetries for the extremal flow.
•
•
If pθ 7→ −pθ then we have two extremals with the same length symmetric
with respect to the meridian.
If pφ 7→ −pφ then we have two extremals of same length intersecting on
the antipodal parallel φ = π − φ(0).
Such properties are shown by the following one-parameter family of metrics.
Metrics induced by the flat metric on oblate ellipsoid of revolution
We consider the flat metric of R3 : g = dx2 +dy 2 +dz 2 restricted to the ellipsoid
defined by
x = sin φ cos θ, y = sin φ sin θ, z = µ cos φ
where µ ∈]0, 1[. A simple computation leads to
3.10 The Analysis of the Averaged System
125
π−φ
φ
φ
0
φ0
θ
0
θ
Fig. 3.5. Action of the symmetry group on the extremals
g2 = Eµ (φ)dφ2 + sin2 φdθ2
where Eµ (φ) = µ2 + (1 − µ2 ) cos2 φ. Computing for ḡ2 = dφ2 + G(φ)dθ2 ,
5 sin2 φ
G(φ) = 1+4
cos2 φ we can write
ḡ2 =
1
(Eµ (φ)dφ2 + sin2 φdθ2 )
Eµ (φ)
√
where µ = 1/ 5. We deduce the following lemma:
Lemma 3.10.10. The metric ḡ2 is conformal to the flat
√ metric restricted to
an oblate ellipsoid of revolution with parameter µ = 1/ 5.
3.10.4 A Global Optimality Result with Application to Orbital
Transfer
In this section, we consider an analytic metric on R+ × S 2
g = dr2 + (dφ2 + G(φ)dθ2 )
(3.27)
and let H be the associated Hamiltonian. We fix the parameterization to
arc-length by restricting to the level set H = 1/2. Let x1 , x2 be two extremal
curves starting from the same initial point x0 and intersecting at some positive
t̄. We get the relations
r1 (t̄) = r2 (t̄), φ1 (t̄) = φ2 (t̄), θ1 (t̄) = θ2 (t̄)
and from lemma 3.10.5, we deduce the following lemma.
Lemma 3.10.11. Both extremals x1 and x2 share the same pr (0) and for
each t, r1 (t) = r2 (t).
0
If we consider now the integral curves of H 0 where H = 12 p2r + H
r 2 on the fixed
induced level and parameterizing these curves using dT = rdt2 , we deduce the
following characterization.
126
3 Orbital Transfer Problem
Proposition 3.10.12. The following conditions are necessary and sufficient
to characterize extremals of H 0 6= 0 intersecting with the same length
φ1 (T̄ ) = φ2 (T̄ ), θ1 (T̄ ) = θ2 (T̄ )
with the compatibility condition
Z
T̄ =
0
t̄
dt
2
= [ √ arctan L(t)]t̄t=0 .
r2 (t)
∆
Theorem 3.10.13. A necessary global optimality condition for an analytic
metric on R+ × S 1 normalized to
g = dr2 + r2 (dφ2 + G(φ)dθ2 )
is that the injectivity radius be greater than or equal to π on the sphere r = 1,
the bound being reached by the flat metric in spherical coordinates.
Proof. We observe that in the flat case, the compatibility condition cannot be
satisfied. Moreover, the injectivity radius on S 2 is π corresponding to the halflength of a great circle. Let us now complete the proof. For the analytic metric
on S 2 , the injectivity radius is the length of the conjugate point at minimum
distance of the half-length of a closed geodesic (see [56]). The conjugate point
is, in addition, a limit point of the separating line. Hence, if the injectivity
radius is smaller than π, we have two minimizers for the restriction of the
metric on S 2 which intersects with a length smaller than π. We shall show
that it corresponds to a projection of two extremals x1 and x2 which intersect
with the same length.
For such extremals r(0) = 1, we set pr (0) = ε, H = 1/2 and we get
2H 0 = p2φ (0) +
p
p2θ (0)
= λ2 (ε), λ(ε) = 1 − ε2 .
G(φ(0))
If t1 is the injectivity radius on the level set H 0 = 1/2 which corresponds to
2
pr (0) = ε = 0. For H 0 = λ 2(ε) and pr (0) = ε, it is rescaled as T1 = t1 /λ(ε).
The compatibility relation for T̄ = T1 gives
T1 = arctan[
t̄ + ε
ε
] − arctan[
].
λ(ε)
λ(ε)
Clearly, the maximum of the right member is π, taking ε < 0, |ε| → 1. Hence,
it can be satisfied since t1 < π. The flat case shows that it is the sharpest
bound.
¤
By homogeneity, we deduce the following corollary.
Corollary 3.10.14. If the metric is normalized to dr2 +
then the bound for the injectivity radius on r2 = c2 is cπ.
r2
2
c2 (dφ
+ G(φ)dθ2 )
3.10 The Analysis of the Averaged System
127
3.10.5 Riemann Curvature and Injectivity Radius in Orbital
Transfer
Using the formulae of Chapter 2, we have the following proposition.
Proposition 3.10.15. Let g be a smooth metric of the form dr2 + r2 (dφ2 +
G(φ)dθ2 ) with x = (x1 , x2 , x3 ) = (r, θ, φ) the coordinates. Then the only nonzero component of the Riemann tensor is
R2323 = r2 [−
G0 (φ)2
G00 (φ)
− G(φ) +
]
2
4G(φ)
which takes the form R2323 = −r2 F (F 00 + F ) if we set G(φ) = F 2 (φ). We
have therefore R2323 = 0 if and only if F (φ) = A sin(φ + φ0 ) which is induced
by the flat case in spherical coordinates.
Hence, the main non-zero sectional curvature of the metric is
K=
R2323
∂
∂ 2
| ∂θ
∧ ∂φ
|
and computing this term in the case of orbital transfer, we get:
Lemma 3.10.16. The sectional curvature in the plane (φ, θ) is given by
KV =
(1 − 24 cos2 φ − 16 cos4 φ)
r2 (1 + 4 cos2 φ)2
and KV → 0 as r → +∞.
Proposition 3.10.17. The Gauss curvature of the metric on S 2 , ḡ2 = dφ2 +
5 sin2 φ
G(φ)dθ2 with G(φ) = 1+4
cos2 φ is
KV =
5(1 − 8 cos2 φ)
.
(1 + 4 cos2 φ)2
Theorem 3.10.18. The Gauss curvature of ḡ2 is negative near the poles and
√
maximum (constant equal to 5) at the equator. The injectivity radius is π/ 5
and is reached by the shortest conjugate point along the equator.
Proof. Clearly K is maximum and constant equal to 5 along the equator which
is an extremal solution. Hence a direct computation
√ gives that the shortest
conjugate point is along the equator with length π/ 5. It corresponds to the
injectivity
√ radius if the half-length of a shortest periodic extremal is greater
than π/ 5. Simple closed extremals are computed in [21] using the integrability property but a simple reasoning gives that the shortest corresponds to
meridians with length 2π. Hence the result is proved.
¤
p
√
Corollary 3.10.19. Since π/ 5 < π 2/5, the necessary optimality condition of theorem 3.10.18 is not satisfied in orbital transfer for the extension of
the metric to R+ × S 2 .
128
3 Orbital Transfer Problem
3.10.6 Cut Locus on S 2 and Global Optimality Results in Orbital
Transfer
From the previous section, the computation of the injectivity radius for the
metric on S 2 is not sufficient to conclude about global optimality. A more
complete analysis is necessary to evaluate the cut locus. This analysis requires
numerical simulations. The explicit analytic representation of the extremal
flows is given in [21]. The main results of this analysis are:
Proposition 3.10.20. For the metric ḡ2 on S 2 , they are exactly five simple closed extremals modulo rotations around the poles, the shortest being
√ a
meridian with length 2π and the longest being the equator with length 2π 5.
Theorem 3.10.21. (1) Except for poles, the conjugate locus is a deformation
of a standard astroid with axial symmetry and two cusps located on the
antipodal parallel.
(2) Except for poles, the cut locus is a simple segment, located on the antipodal
parallel with axial symmetry and whose extremities are cusps points of the
conjugate locus.
(3) For a pole, the cut locus is reduced to the antipodal pole.
Proof. The proof is made by direct analysis of the extremal curves, see also
Section 2.5.4 for a more general framework. The main problem is to prove that
the separating line is given by points on the antipodal parallel, where due to
the isometry φ → π − φ, two extremals curves with same length intersect.
This property cannot occur before. The results are represented in Fig. 3.6. ¤
φ
π−φ0
φ0
θ
Fig. 3.6. Conjugate and cut loci in averaged orbital transfer
Geometric interpretation and comments
The metric is conformal to the restriction of the flat metric to an oblate
ellipsoid of revolution. For such a metric, the cut locus is given by Proposition
2.5.24 and is similar to the one represented on Fig. 3.6. It is a remarkable
3.11 The Averaged System in the Tangential Case
129
property that there is no bifurcation of the cut locus when the metric is
deformed by the factor Eµ (φ) although the properties of the metric are quite
different. For instance, in orbital transfer, the Gauss curvature is not positive.
The mathematical proof requires a thorough analysis of the extremal flow.
A similar result can be obtained with numerical simulations. Indeed on S 2 ,
relations between the conjugate and cut loci allow to deduce the cut locus from
the conjugate locus. Also a domain bounded by two intersecting minimizing
curves must contain a conjugate point. The same result can be obtained using
Theorem 2.6.7, the first return mapping being evaluated using the explicit
parameterization of the extremal curves.
In this case, the conjugate locus can be easily computed using the Cotcot
code presented in [26]. In such a situation, it can also be deduced by inspecting
the extremal flow only, the conjugate locus being an envelope. The structure
of the conjugate locus is also a consequence of Theorem 2.6.7.
Finally, we observe that in order to have intersecting minimizers, we must
cross the equator φ = π for which e = 1. The same is true for conjugate
points. Hence we deduce:
Theorem 3.10.22. Conjugate loci and separating lines of the averaged Kepler
metric in the spaces of ellipses for which e ∈ [0, 1[ are always empty.
3.11 The Averaged System in the Tangential Case
An interesting question is to analyze if the averaged system in the tangential
case where the control is oriented along Ft conserves similar properties. The
first step is to compute the corresponding averaged system.
Proposition 3.11.1. If the control is oriented along Ft only, the averaged
Hamiltonian associated to the energy minimization problem is
H̄t =
1
4(1 − e2 ) p2θ
4(1 − e2 )3/2 2
2 2
√
√
p
+
[9n
p
+
]
e
n
2n5/3
1 + 1 − e2
1 + 1 − e2 e2
which corresponds to the Riemannian metric
√
√
n5/3 1 + 1 − e2 2 1 + 1 − e2 2 2
dn2
ḡt = 1/3 +
(
de +
e dθ
4
(1 − e2 )
9n
(1 − e2 )3/2
where (n, e, θ) are orthogonal coordinates.
3.11.1 Construction of the Normal Form
We proceed as in Section 3.10. We set
r=
p
2 5/6
n , e = sin φ 1 + cos2 φ.
5
130
3 Orbital Transfer Problem
The metric takes the form
r2
2
(dφ2 + G(φ)dθ2 ), c2 = < 1
c2
5
g = dr2 +
and
G(φ) = sin2 φ(
1 − (1/2) sin2 φ 2
) .
1 − sin2 φ
Hence the normal form is similar to the full control case. We introduce the
metrics
g1 = dr2 + r2 dψ 2 , ψ = φ/c
and
g2 = dφ2 + G(φ)dθ2 .
Next we make the analysis by comparing with the full control case. The main
difference will concern the singularities of G.
3.11.2 The Metric g1
The metric corresponds again to transfer to circular orbits and is the polar
form of the flat metric dx2 + dz 2 , if x = r sin ψ and z = r cos ψ.
3.11.3 The Metric g2
The normal form reveals the same homogeneity property between the full
control and the tangential case, the metric g2 can be used to make a similar
optimality analysis, evaluating the conjugate and cut locus. But the metric g2
cannot be interpreted as a smooth metric on S 2 . This can be seen by computing
the Gauss curvature.
Proposition 3.11.2. The Gauss curvature of g2 is given by
K=
(3 + cos2 φ)(cos2 φ − 2)
.
(1 + cos2 φ) cos2 φ
In particular K → −∞ when φ → π/2 since K < 0 and the conjugate locus
of a point is empty.
Nevertheless, the extremals can be smoothly extended through the singular
boundary of the domain where φ = π/2 and we get a similar picture than for
the full transfer case represented in Fig. 3.6. This corresponds to a Grushin
type singularity discussed in Chapter 2.
3.12 Conclusion in Both Cases
131
3.11.4 The Integration of the Extremal Flow
The algorithm based on the normal form is similar to the bi-input case, but
we compare the respective transcendence. The Hamiltonian is written as
H=
1
[18n2 p2n + H 00 ]
4n5/3
where H 00 takes now the form
H 00 =
8(1 − e2 ) p2θ
8(1 − e2 )3/2 2
√
√
pe +
.
1 + 1 − e2
1 + 1 − e2 e2
√
2
1−e )
We set H 00 = c23 , pθ = c2 and from pe = 4n5/3 e(1+
, we obtain with
16(1−e2 )3/2
√
2
w = 1−e
Q(w)
dw 2
(
) =
dT
(1 + w)2
where T is as in the bi-input case and Q is the fourth-order polynomial
Q(w) = 32w[c23 (1 − w2 )(1 + w) − 8c22 w2 ].
Hence, the integration requires the computation of the elliptic integral
Z
dw(1 + w)
p
Q(w)
which is an additional complexity.
3.11.5 A Continuation Result
On Fig. 3.7, we show the convergence of the continuation method from the
non-averaged trajectory to the averaged one, in the tangential case (boundary
conditions are GTO towards GEO orbits), represented in flat and orbital
coordinates.
3.12 Conclusion in Both Cases
The previous analysis shows that the full control case and the tangential one
admit an uniform representation in the coordinates (φ, θ). In particular, it
allows in such coordinates to make a continuation between the respective
Hamiltonians, i.e., between the respective G(φ). A correction has to be made
between orbit elements e which are respectively defined by
e = sin φ
and
132
3 Orbital Transfer Problem
Trajectory in flat coordinates
−0.1
Averaged
ε = 1.000000e−02
ε = 5.000000e−03
ε = 1.000000e−03
−0.2
u sin(v)
−0.3
−0.4
−0.5
−0.6
0.22
0.24
0.26
0.28
0.3
u cos(v)
0.32
0.34
0.36
0.38
0.4
Trajectory in cartesian coordinates
Averaged
ε = 1.000000e−02
2
1.5
1
0.5
0
−0.5
−1
−1.5
−2
−2.5
−3
−2
−1
0
1
2
Fig. 3.7. Convergence of the continuation method between non averaged and averaged trajectories
p
e = sin φ 1 + cos2 φ.
The flows in the two cases are presented on Fig. 3.8 and reveal the similar
structure. In both cases optimality is lost after having crossed the equator, as
deduced by the computations of cut points which are located on the antipodal
parallel.
3.14 Averaged System for Non-Coplanar Transfer
133
3.13 The Averaged System in the Orthoradial Case
We assume that the control is oriented in the orthoradial direction. Still in
this case the computation of the averaged system is explicit and we have:
Proposition 3.13.1. In the coordinates (n, e, θ), the averaged Hamiltonian is
1
[a(e)(npn )2 + 2b(e)(npn )pe + c(e)p2e + d(e)p2θ ],
4n5/3
√
p
6(1 − e2 )(1 − 1 − e2 )
2
,
a(e) = 18 1 − e , b(e) =
e
"
#
√
2(1 − e2 )(1 − 1 − e2 )
2
c(e) = (1 − e ) 5 −
,
e2
Hor =
d(e) = (5 − 4e2 ) − (1 − e2 )(1 + A/e2 ),
where A = (1 + 2/δ)(−1 + 1/δ)2 , with δ = (1− | e |2 )−1/2 .
Still θ is a cyclic variable and extremals such that pθ = 0 are associated to
transfer to circular orbits but the situation is much more complex as shown
by the curvature computation.
Proposition 3.13.2. The Gauss curvature underlying transfers towards circular orbits is given by
K=−
5
n−5/3
√
[18(1 − e2 )5/2 + 75(1 − e2 )2 + 96(1 − e2 )3/2
8 (3 1 − e2 + 5)2
−78(1 − e2 ) + 70(1 − e2 )1/2 + 75].
In particular K is strictly negative in the domain.
3.14 Averaged System for Non-Coplanar Transfer
Neglecting in the averaging the action of the control on the longitude, in
non-coplanar transfer the averaged Hamiltonian is approximated by H =
1
2
2
2
2 (P1 + P2 + P3 ), where P1 , P2 are given by the coplanar case while
P 5/4
C
C
(−Zpex ey + Zpey ex + phx cos l + phy sin l),
W
2
2
2
where Z = hx sin l − hy cos l, C = 1+ | h | .
As in the bi-input case, we use (n, r, θ) as coordinates and we make a polar
representation of h, hx = σ cos Ω, hy = σ sin Ω, where the angle Ω is the longitude of the ascending node. In such coordinates the averaged Hamiltonian is
P3 =
134
3 Orbital Transfer Problem
the sum of the term associated to coplanar transfer and the term corresponding to the action of the control component uc orthogonal to the osculating
plane, which is
·
¸
1 (σ 2 + 1)2 1 + 4r2
pθΩ 2
pθΩ 2
(cos
ωp
+
sin
ω
)
+
(−
sin
ωp
+
cos
ω
)
σ
σ
2
1 − r2
σ
σ
8n5/3
where ω = θ − Ω is the angle of the pericenter and where we have set
pθΩ =
2σ 2
pθ + pΩ .
+1
σ2
From which we deduce:
Theorem 3.14.1. • The averaged Hamiltonian of the non-coplanar transfer
is associated with a five-dimensional Riemannian metric.
• The averaged Hamiltonian corresponding to the action of the control perpendicular to the osculating plane corresponds to a SR-problem in dimension three defined by the contact distribution,
3
3
2.5
2.5
2
2
φ
φ
(σ 2 + 1)dω − (σ 2 − 1)dΩ = 0.
1.5
1.5
1
1
0.5
0.5
0
0
0.5
1
1.5
θ
2
2.5
3
0
0
0.5
1
1.5
θ
2
2.5
3
Fig. 3.8. Extremal flow of g2 in the full control and tangential cases, in the (φ, θ)
coordinates, starting from φ = π/6
3.15 The Energy Minimization Problem in the
Earth-Moon Space Mission
3.15.1 Mathematical Model and Presentation of the Problem.
In this section, we follow mainly [112], see also [120] and [140].
3.15 The Energy Minimization Problem in the Earth-Moon Space Mission
135
The N-Body Problem
Consider N point masses m1 , . . . , mN moving in a Galilean reference system R3 where the only forces acting being their mutual attraction. If q =
(q1 , . . . , qN ) ∈ R3N is the state and p = (p1 , . . . , pN ) being the momentum
vector, the equations of the motion are
q̇ =
∂H
∂H
, ṗ = −
∂p
∂q
where the Hamiltonian is:
H=
N
X
k pi k2
− U , U (q) =
2mi
i=1
N
X
1≤i<j≤N
Gmi mj
.
k qi − qj k
A restricting case is the coplanar situation where the N masses are in a plane
R2 . In this case the Galilean reference frame can be replaced by a rotating
frame defined by
µ
¶
µ
¶
0 1
cos ωt sin ωt
K=
, exp(ωtK) =
−1 0
− sin ωt cos ωt
and introducing a set of coordinates wich uniformly rotates with frequency ω
defines the symplectic transformation:
ui = exp(ωtK)qi , vi = exp(ωtK)
and a standard computation gives the Hamiltonian of the N -body problem in
rotating coordinates:
H=
N
N
X
k v k2 X t
−
w ui Kvi −
2mi
i=1
i=1
N
X
1≤i<j≤N
Gmi mj
.
k qi − qj k
In particular, the Kepler problem in rotating coordinates up to a normalization
has the following Hamiltonian
H=
k p k2 t
1
− qKp −
.
2
kqk
3.15.2 The Circular Restricted 3-Body Problem in Jacobi
Coordinates
Recall the following representation of the Earth-Moon problem. In the rotating
frame, the Earth which is the biggest primary planet with mass 1−µ is located
at (−µ, 0) while the Moon with mass µ, is located at (1 − µ, 0) with the small
parameter µ ' 0.012153. We note z = x + iy the position of the spacecraft,
%1 , %2 are the distances to the primaries
136
3 Orbital Transfer Problem
q¡
¢
%1 =
(x + µ)2 + y 2 ,
q¡
¢
%2 =
(x − 1 + µ)2 + y 2 .
The equation of the motion takes the form
z̈ + 2iż − z = −(1 − µ)
z+µ
z−1+µ
−µ
%31
%32
which can be written:
∂V
∂x
∂V
ÿ + 2ẋ − y =
∂y
ẍ − 2ẏ − x =
where −V is the potential of the system defined by V =
can be written using Hamiltonian formalism by setting
1−µ
%31
+ %µ3 . The system
2
q1 = x, q2 = y, p1 = ẋ − y, q2 = ẏ + x
and the Hamiltonian describing the motion takes the form
H0 (q1 , q2 , p1 , p2 ) =
1 2
1−µ
µ
(p1 + p22 ) + p1 q2 − p2 q1 −
− .
2
%1
%2
3.15.3 Jacobi Integral and Hill Regions
The Jacobi integral using Hamiltonian formalism is simply the Hamiltonian
H0 which gives
H(x, y, ẋ − y, ẏ + x) =
ẋ2 + ẏ 2
− Ω(x, y)
2
where
Ω(x, y) =
1−µ
µ
1 2
(x + y 2 ) +
+ .
2
%1
%2
Hence solutions are confined on the level set
ẋ2 + ẏ 2
− Ω(x, y) = h
2
where h is a constant.The Hill domain for the value h is the region where the
motion can occur, that is {(x, y) ∈ R2 , Ω(x, y) + h ≥ 0}.
3.15 The Energy Minimization Problem in the Earth-Moon Space Mission
137
3.15.4 Equilibrium Points
The equilibrium points of the problem are well known. They split in two
different types:
•
Euler points. They are the collinear points denoted L1 ,L2 and L3 located
on the line y = 0 defined by the primaries. For the Earth-Moon problem
they are given by
x1 ' −1.0051, x2 ' 0.8369, x3 ' 1.1557.
•
Lagrange points. The two points L4 and L5 form with the two primaries
an equilateral triangle.
Some important informations about the stability of the equilibrium points are
provided by the eigenvalues of the linearized system. The linearized matrix
evaluated at points L1 ,L2 or L3 admits two real eigenvalues, one being strictly
positive and two imaginary eigenvalues. The collinear points are consequently
non stable. In particular, the eigenvalues of the linearized matrix evaluated at
L2 with µ = 0.012153 are ±2.931837 and ±2.334248i. When it is evaluated at
L4 or L√5 , the linearized matrix has two imaginary eigenvalues when µ < µ1 =
69
1
2 (1 − 9 ). So in the Earth-Moon system, the points L4 and L5 are stable
since µ < µ1 (according to Arnold stability theorem, see [112]).
Considering the Earth-Moon system, eigenvalues of the linearized matrix
evaluated at L2 are ±2.931837 and ±2.334248i.
3.15.5 The Continuation Method in the Earth-Moon Transfer
The (mathematical) continuation method in the restricted circular problem
is omnipresent in Poincaré work, in particular for the continuation of circular orbits. Geometrically, it is simply a continuation of trajectories of Kepler
problem into trajectories of the three-body problem. It amounts to consideration of µ as a small parameter, the limit case µ = 0 being Kepler problem in
the rotating frame, writing
H0 =
k p k2 t
1
− qKp −
+ o(µ)
2
kqk
and the approximation for µ small is valid, a neighborhood of the primaries
being excluded. In the Earth-Moon problem, since µ is very small, the Kepler
problem is clearly a good approximation of the motion in a large neighborhood
of the Earth. This point of view is important in our analysis, as indicated by
the status report of the SMART-1 mission since most of the time mission is
under the influence only of the Earth attraction, see [122] and [123].
138
3 Orbital Transfer Problem
The Control Problem
The control system in the rotating frame is deduced from the previous model
and can be written in the Hamiltonian form
−
−
→
−
→
dz →
= H 0 (z) + u1 H 1 (z) + u2 H 2 (z)
dt
→
−
−
→ −
→
−
→
where z = (q, p), H 0 is the free motion and H 1 , H 2 are given by H i = −qi ,
i =1,2. As for the Kepler problem, the mass variation of the satellite can be introduced in the model dividing ui by m(t) and adding the equation ṁ = −δ|u|.
Again, it will be not taken into account in the problem. Moreover
R t we still restrict our analysis to the energy minimization problem: minu(.) t0f u21 + u22 dt,
where the transfer time tf is fixed and the control valued in R2 . The physical
problem which corresponds to the maximization of the final mass can be analyzed using a (numeric) continuation method.
It is worthwhile to point out that a lunar mission using low-propulsion
called SMART-1 was realized by ESA and the practical details of the mission,
in particular the description of the trajectory, are reported in [122] and [123].
Next we present a trajectory analysis based on our geometric and numeric
techniques. For simplicity, we have fixed the boundary conditions to circular
orbits, the one around the Earth corresponding to the geostationary one. But
everything can be applied to other boundary conditions, like the GTO ellipse
as the initial orbit, as described in the report status of the mission SMART-1.
Our analysis is based on a numerical continuation, taking into account
second-order optimality condition, where µ is the parameter of the continuation.
Boundary Conditions and Shooting Equation of the Earth-L2
Transfer
As a first approach we choose to simulate the Earth-L2 transfer in the restricted three-Body problem. Indeed, at the limit case µ = 0, the Moon and
the point L2 are identical. Moreover, in the Earth-Moon system, the point L2
and the Moon are located very close to each other. As a result, the first phase
of an Earth-Moon transfer is comparable to an Earth-L2 transfer. Solving the
shooting function associated to the Earth-L2 transfer is consequently useful
to provide a good approximation of the solution of the Earth-Moon transfer
shooting function.
The Numerical Continuation Method for the Earth-L2 Transfer
According to the report status of ESA, we fixed the transfer time to 121 time
units of the restricted 3-body problem which approximatively corresponds to
3.15 The Energy Minimization Problem in the Earth-Moon Space Mission
139
the transfer time from the Earth to the point L2 during the SMART-1 mission
(about 17 months). In addition we considered a constant spacecraft mass of
350 kg, see [122] and [123]. Setting µ = 0, we computed an extremal using
the simple shooting method. Then we made the parameter µ vary from 0 to
0.012153 with an initial step of 10−3 that could be automatically reduced by
the continuation algorithm if necessary.
At each step, the first conjugate time t1,c along the extremal has been
computed to ensure the necessary condition of convergence of the continuation
method t1,c > tf . Additionally, the Euclidian norm of the extremal control
has been plotted to stand a comparison between the control bound and the
maximal thrust allowed by electro-ionic engines used while the SMART-1 mission. Figures 3.9 to 3.16 present the computed spacecraft trajectories in both
rotating and fixed frames, as well as the first conjugate time and the norm of
control along trajectories in both Kepler case and Earth-Moon system.
Fig. 3.9. Earth-L2 trajectory in rotating frame, µ = 0.
We checked that the first conjugate time along each extremal is higher
than the transfer time, that was needed for the convergence of the method.
Moreover the second order optimality conditions ensure that the computed
extremals are locally energy minimizing in L∞ ([0, tf ]). We also note that in
both cases µ = 0 and µ = 0.012153, the maximum value reached by the
norm of the extremal control is highly inferior to the bound k u k≤ 0.08
which corresponds to the maximal thrust of the SMART-1 electro-ionic engine
140
3 Orbital Transfer Problem
0.8
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
−0.5
0
0.5
1
Fig. 3.10. Earth-L2 trajectory in fixed frame, µ = 0.
100
arcsh det(δ x)
80
60
40
20
0
0
100
200
300
400
500
600
700
400
500
600
700
t
10
8
σn
6
4
2
0
0
100
200
300
t
Fig. 3.11. First conjugate time, Earth-L2 transfer, µ = 0.
(0.073 N). As one can see, we actually found a maximal bound of the norm
of the extremal control twice lower than the one associated to SMART-1, the
transfer time being the same.
3.15 The Energy Minimization Problem in the Earth-Moon Space Mission
141
Fig. 3.12. Norm of the extremal control, Earth-L2 transfer, µ = 0.
Fig. 3.13. Earth-L2 trajectory in rotating frame, µ = 0.012153.
Boundary conditions and shooting function for the Earth-Moon
transfer.
The second part of our trajectories analysis has been devoted to the EarthMoon transfer. We used the same dynamics and initial condition as previously.
142
3 Orbital Transfer Problem
0.8
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
−0.5
0
0.5
1
Fig. 3.14. Earth-L2 trajectory in fixed frame, µ = 0.012153.
Fig. 3.15. First conjugate time, Earth-L2 transfer, µ = 0.012153.
In this case, the target is the circular orbit around the Moon, denoted OL and
defined by
3.15 The Energy Minimization Problem in the Earth-Moon Space Mission
143
Fig. 3.16. Norm of the extremal control, Earth-L2 transfer, µ = 0.012153.


(x1 − 1 + µ)2 + x22 = 0.0017 

x23 + x24 = 0.2946
.
OL = (x1 , x2 , x3 , x4 ) ∈ R4 ,


h(x1 − 1 + µ, x2 ), (x3 , x4 )i = 0
From Pontryagin maximum principle, an optimal trajectory has to satisfy the
transversality condition and the shooting equation is modified accordingly.
Numerical continuation method for the Earth-Moon transfer.
The transfer time was fixed to 124 time units of the restricted three-body
problem and the spacecraft mass remains 350 kg, see [122] and [123]. The
extremal trajectory corresponding to µ = 0 was computed using the simple
shooting method and initializing p0 with the initial costate vector associated
to the Earth-L2 transfer. The step of the variation parameter µ is 10−3 .
Since the target is a manifold of codimension one, the concept of conjugate
point is replaced by the concept of focal point. At each step, the first focal
time tf oc,1 along extremal has been computed to ensure the necessary condition of convergence of the continuation method tf oc > tf . The Earth-Moon
trajectories in both rotating and fixed frames, the first focal time and the
norm of extremal control are presented from Fig. 3.17 to Fig. 3.24 for µ = 0
and µ = 0012153.
Once again, we compute an extremal trajectory of the energy minimization
Earth-Moon transfer thanks to the continuation method. In both cases µ = 0
and µ = 0.012153, the first focal time along extremals tf oc,1 is higher than
144
3 Orbital Transfer Problem
3
2 tf ,
ensuring local optimality. The maximal bound of the norm of extremal
control is 0.045, that approximatively corresponds to the half of the maximal
thrust allowed during the mission SMART-1.
It is interesting to notice that the Earth-L2 Keplerian trajectory greatly
differs from the Earth-Moon Keplerian trajectory. This difference illustrates
the restricting role of the transversality condition provided by the maximum
principle when the target is a submanifold. On the contrary, for µ = 0.012153
the first phase of the Earth-Moon transfer matches the Earth-L2 transfer. It
underlines the crucial role of the neighborhood of the point L2 where Earth
and Lunar attractions are compensating. It is worth to point out that the best
available numeric codes are necessary in this case to get the numerical results.
0.8
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
Fig. 3.17. Earth-Moon trajectory in rotating frame, µ = 0.
Notes and Sources
The geometric analysis in orbital transfer is due to [25]. For the stabilization
analysis, see [17]. The averaging technique with preliminary computations
has been introduced in orbital transfer by [68]. For the complete analysis,
see [20],[21] and [30] for the analysis in the tangential case. We have in both
cases make the analysis using the explicit parameterization of the extremal
flow combined with numerical simulations. For the presentation we keep the
original analysis but it can be shorten if we use the results from [24] which
where motivated by the orbital transfer problem. The computations of the
averaged non-coplanar case are from [20]. The analysis of the corresponding
3.15 The Energy Minimization Problem in the Earth-Moon Space Mission
145
0.8
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
−0.5
0
0.5
1
Fig. 3.18. Earth-Moon trajectory in fixed frame, µ = 0.
arcsh det(δ x)
50
0
−50
−200
−180
−160
−140
−120
−100
t
−80
−60
−40
−20
0
−180
−160
−140
−120
−100
t
−80
−60
−40
−20
0
1
0.8
σ
n
0.6
0.4
0.2
0
−200
Fig. 3.19. First focal time and norm of extremal control, Earth-Moon transfer,
µ = 0.
metric is still open. For a general reference about the three-body problem,
146
3 Orbital Transfer Problem
Fig. 3.20. Norm of extremal control, Earth-Moon transfer, µ = 0.
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
Fig. 3.21. Earth-Moon trajectory in rotating frame, µ = 0.012153.
see [140]. The SMART-1 mission is described in [123]. The numerical results
about the Earth-Moon transfer come from [23].
3.15 The Energy Minimization Problem in the Earth-Moon Space Mission
0.8
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
−0.5
0
0.5
1
Fig. 3.22. Earth-Moon trajectory in fixed frame, µ = 0.012153.
arcsh det(δ x)
50
0
−50
−250
−200
−150
−100
−50
0
−100
−50
0
t
0.1
0.08
σn
0.06
0.04
0.02
0
−250
−200
−150
t
Fig. 3.23. First focal time, Earth-Moon transfer, µ = 0.012153.
147
148
3 Orbital Transfer Problem
Fig. 3.24. Norm of extremal control, Earth-Moon transfer, µ = 0.012153.