CIMPA School on Optimization and Control
Centro Internacional de Encuentros Matemáticos
Castro Urdiales (Cantabria, Spain)
August 28 – September 8, 2006
SPEAKERS
First week, August 28 – September 1
Joseph Frédéric Bonnans (INRIA, Rocquencourt, France)
Optimal Control of Ordinary Differential Equations
Jean-Baptiste Hiriart-Urruty (Université Paul Sabatier, France)
First Steps in Optimization
Pierre Rouchon (Ecole de Mines de Paris, France)
Differentially Flat Control Systems
Second week, September 4 – 8
Eduardo Casas (Universidad de Cantabria, Spain)
Optimal Control of PDE. Theory and Numerical Analysis
Karl Kunisch (University of Graz, Austria)
Numerical Analysis and Algorithms for Optimal Control
of PDE with Control and State Constraints
Enrique Zuazua (Universidad Autónoma de Madrid, Spain)
Controllability of PDE and its Numerical Approximation Scheme
ORGANIZERS
Eduardo Casas
and
[email protected]
Information and Registration:
CIMPA
Le Dubellay, 4 avenue Joachim
Bat. B, 06100 Nice, FRANCE
http://www.cimpa-icpam.org
[email protected]
Tel. +33 4 92 07 79 30
Fax +33 4 92 07 05 02
Michel Théra
[email protected]
Sponsors:
Castro Urdiales Council
Centre International de Mathématiques Pures et Apliquées (CIMPA)
Centre National de la Recherche Scientifique (CNRS)
Centro Internacional de Encuentros Matemáticos (CIEM)
Conseil Régional du Limousin
International Council for Industrial and Applied Mathematics (ICIAM)
Universidad de Cantabria
Université de Limoges
PROGRAM
First week (August 28 - September 1, 2006)
Joseph Frédéric BONNANS (INRIA-Rocquencourt, France)
Title of the Course: Optimal Control of Ordinary Differential Equations
The course will present the derivation of optimality conditions of constrained optimal control problems, and discuss
the error estimates obtained when discretizing by one step schemes. The course will be divided into three parts:
1. Optimality conditions
a.
Modelling: some examples
b.
Pontryagin’s principle: proofs and consequences
c.
Junction conditions for state constrained problems
2. One step (Runge Kutta) discretization schemes
a.
Order conditions: partitioned and symplectic scheme
b.
Analysis of junction points
c.
Shooting methods
3. Advanced topics
a.
Interior-point methodology
b.
Multiphase and stochastic problems
c.
The predictive control approach to feedback problems
The aim of the course is to present some approaches developed in the recent years, for solving optimal control
problems of ODE’s. We start by a discussion of modelling problems: which variables are control and which are states?
In general there is a cascade on more or less refined models, whose solutions are related by the (difficult and incomplete)
theory of singular perturbations for optimal control problems. Our theoretical and numerical approach is based on
trajectory optimization, i.e., the main object is a candidate trajectory whose optimality can be tested by checking
if Pontryagin’s principle, or maybe only the first-order optimality conditions (their counterpart after discretization)
are satisfied. We discuss some relatively simple proofs of Pontryagin’s principle, and then specialize the study to
the analysis of junction points. We introduce one step (Runge Kutta) discretization schemes, and show that (for
unconstrained problems with strongly convex Hamiltonians) the resulting optimality conditions may be interpreted as
those of a partitioned Runge Kutta scheme of symplectic type. As a consequence the error order can be computed,
using the computation on trees introduced in the theory of (partitioned) Runge Kutta schemes. This allows to optimize
refinement procedures. We describe the limited results that can be obtained when analyzing constrained problems,
especially in the neighbourhood of a junction. This allows a discussion of shooting methods in which the junction times
are introduced as (unknown) parameters. The last part of the course introduces to some more advanced topics. The
interior-point methodology is introduced. It has many computational advantages, but leads to difficult questions as far
as error estimates are concerned. We show how it can be adapted to multiphase problems, and discuss how to take
into account uncertain events. Finally we show how the optimal control methodology can be used in order to obtain
feedback controls. It appears that the sequential quadratic programming methodology is specially well suited to this
case.
Pierre ROUCHON (Ecole des Mines de Paris, France)
Title of the Course: Differentially Flat Control Systems
More than 10 years ago, Michel Fliess and co-workers introduced a special class of non-linear control systems
described by ordinary equations: differential flat systems form a special class of underdetermined differential systems
(also called implicit control systems or differential/algebraic control systems) for which systematic control methods are
available once a flat-output is explicitly known. Flatness is related to state feedback linearization and in fact has a long
history going back to Hilbert with his work on the general solution of Monge equations and also Elie Cartan. The first
aim of this course is to provide the basic mathematical definitions and engineering motivating examples. The second aim
is to propose some open-problems and new directions of possible researches around the flatness characterization problem
and extensions to control systems described by PDE’s. The course will be illustrated by several Matlab simulations and
divided into the following three parts:
1. Finite dimensional flat systems (3 hours). Definition, relation with dynamic feedback linearisation, trajectory
generation and tracking algorithm, examples (juggling robot, car with trailers), a bit of history (Hilbert, Cartan,
Pfaffian contact systems and nonholonomic control systems).
2. Some results around the flatness Characterization problem: endogenous versus exogenous dynamic feedback, CNS
for linearization via static feedback, Driftless systems with two controls, Pfaffian systems of codimension 2, the
ruled-manifold criterion.
3. Examples of infinite dimensional flat systems: symbolic computation in the Laplace domain; waves and systems
with delay (water tank and heavy chain), reaction/diffusion where the flat output is a Gevrey functions of time
(heat, beams, tubular chemical reactor, quantum particle in a moving box....).
Jean-Baptiste HIRIART URRUTY (Université Paul Sabatier)
Title of the Course: First Steps in Optimization.
Tentative program (level to be adapted for the background of the audience). This introductory course is devoted to
explaining the main underlying key-ideas governing modern Optimization. Here is the sketch of the different foreseen
chapters:
1. The role of constraints, especially inequality constraints, in Optimization: necessary conditions for optimality (first,
second (of the Karush-Kuhn-Tucker type), possibly higher order but in the unconstrained case only). between an
optimization problem and its ”closest relative ” : the dual problem. ” closest relative ”: the dual problem.
2. Important classes of optimization problems: the ones with affine data (Linear programming), quadratic data, the
so-called SDP optimization (a recent subject of research, where some matricial constraints are expressed as the
positive semidefiniteness of some matrices).
3. Multicriteria optimization: Pareto optimality, peculiar aspects of this type of optimization.
4. The challenges of global optimization: optimality conditions, problems with a specific structure (like quadraticquadratics).
5. From a difficult problem to a simpler one: the way of relaxing optimization problems. We present the course in the
spirit of a popularization book we wrote some years ago (”L’Optimisation”, Collection ”Que sais-je ? ”, Presses
Universitaires de France).
Second week (September 4 - 8, 2006)
Eduardo CASAS (Universidad de Cantabria, Spain)
Title of the Course: Optimal Control of PDE. Theory and Numerical Analysis.
Programme:
1. Some examples of optimal control problems. A model problem.
2. Results about the existence and uniqueness of a solution.
3. First and second order optimality conditions.
4. Numerical approximation of a control problem. Convergence analysis.
5. Error estimates.
This is an introductory course to Optimal Control of Partial Differential Equations beginning with some examples
of optimal control corresponding to different equations describing different physical situations and setting different
mathematical difficulties. Then we formulate a model control problem that will be completely studied along the course.
This model is described by a semilinear elliptic equation, the cost functional is a general one involving the control and
the state and we will assume the presence of control constraints. First of all we analyze the existence and uniqueness of a
solution, showing the importance of convexity assumptions and the compactness methods. Then we study the first-order
optimality conditions, which allow us to deduce qualitative properties of the optimal controls, such as the bang-bang
behaviour or continuity or even differentiability of the optimal control. Some comments will be given about the case
of state constraints. The second-order optimality conditions are currently investigated because of their importance to
analyze the convergence properties of the numerical algorithms to solve the control problem as well as to deduce the
error estimates of the numerical approximation of the problem. In fact the sufficient second-order conditions are the
useful ones to study the mentioned problems, but the knowledge of the necessary second-order conditions is crucial
to formulate the sufficient ones in a realistic way, such that the unavoidable gap between the necessary and sufficient
conditions is minimal. The last part of the course is devoted to the numerical analysis of the control problem. First we
analyze different forms of discretization by using piecewise constant or linear discrete controls. We study the conditions
on the discretization to be convergent and then we derive some error estimates of the discretizations. Finally some open
problems will be enounced.
Karl KUNISH (University of Graz, Austria)
Title of the Course: Numerical Analysis and Algorithms for Optimal Control of Partial Differential
Equations with Control and State Constraints
Tentative program (level to be adapted for the background of the audience) In these lectures I shall address some
of the fundamental concepts for the numerical solution of open loop optimal control problems for partial differential
equations. Special attention will be paid to optimal control of fluids governed by the Navier Stokes equations. In
particular the following issues will be addressed:
1. Newton versus sequential quadratic programming methods
2. Primal-dual active set method for control and state constraints
3. Path-following methods for regularized problems
4. Semi-smooth Newton methods
The impact of optimal control strongly relies on the ability to compute the controls for practically relevant problems.
Significant developments and achievements on this topic were made during the last decade, for example, in the context
of controlling nonlinear heating/cooling regimes for the steel production process. Another focus area was the optimal
control of fluids aiming, for example, at the minimization of recirculation zones. In these lecture we shall present
some of the most relevant techniques for solving such problems and we shall justify them analytically. Optimal control
methods rely on derivatives of the cost-functional with respect to the controls. We shall therefore describe methods
for characterizing first and second derivatives utilizing a Lagrangian framework. Once derivatives are available, this
will allow to describe the conjugate gradient method (a first-order method) and to move on to second order methods,
describing and analyzing the relative merits of Newton, SQP, and quasi- Newton methods. At the heart of all of these
methods, linear systems must be solved. This is typically done inexactly and hence preconditioning techniques, specific
to the saddle point structure of optimal control problems should be utilized. If control or state constraints enter into
the problem formulation the degree of difficulty rises significantly. The particular challenge consists in finding iterative
methods which still have almost second order convergence. We discuss the primal-dual active set strategy to solve
such problems. For linear quadratic problems it can be shown that it is globally convergent. Introducing the notion
of Newton derivative, it can be argued that the primal-dual active set strategy behaves, in certain cases, which include
the situation of control constraints, as a semi-smooth Newton method. This, in turn, allows to are local super- linear
converge. For state-constrained problems, this analysis must be combined with a regularization method, motivated
by a generalized Moreau-Yosida type approximation of monotone maps. To update the regularization parameter a
path-procedure similar to that known from interior-point methods is developed. We shall also point several research
directions of numerical analysis as well as of algorithm nature.
Enrique ZUAZUA (Universidad Autnoma de Madrid, Spain)
Title of the Course: Controllability of PDE and its Numerical Approximation Scheme
In this series of lectures we shall discuss several topics related with the modelling, analysis, numerical simulation
and control of Partial Differential Equations (PDE) models arising in various contexts of Science and Technology. We
shall focus mainly on the case of flexible structures with special emphasis on strings and beams. We shall be concerned
with the following issues:
1. Networks of flexible strings and beams.
2. Singular limits for classical models in elasticity.
3. Numerical approximation issues.
These problems will be addressed from the point of view of control. Roughly speaking, the problem of controllability
we shall discuss may be formulated as follows: To analyze whether by means of a suitable (and feasible!) controller the
solution can be driven to a desired final configuration (or close to it). Of course this problem can be made precise in
several ways: a) One may for instance analyze the control properties of the projections of solutions in certain frequency
bands; b) address the problem of exact controllability in which one is interested on controlling the whole solution; or
c) the problem of approximate controllability in which one relaxes the control requirement to simply request to reach a
distance from the final target. The appropriate choice of the control problem to be analyzed is part of the modelling work
that needs to be done to make the mathematical results of practical use. It is classical in control theory to attack the
control problem through its dual version. In this case the dual notion of controllability one is the so called observability
problem. It concerns the possibility of measuring or observing by suitable sensors the whole dynamics of the system
through partial measurements made on the region which is accessible to the controllers. This problem, as we said, is
relevant for control purposes but also in other contexts like inverse problems and identification issues. Once again, there
are different degrees of observability. Indeed, the models under consideration are genuinely infinite-dimensional and
whether observability holds or not may strongly depend on the norm we consider. But, if one ignores this fact, there is
a fundamental issue that needs to be address to be able to give a satisfactory answer to the problem observability: Does
the system posses localized vibrations that scape the region where the sensors are placed? Of course, if the answer to
this question is positive, one can immediately conclude that observability fails. there are several reasons for the possible
existence of localized solutions: a) Geometric ones: for instance, if the domains is perforated microvibrations may occur
when waves get trapped between several holes, b) Systems: When the system under consideration takes account of
several components (thermo-elasticity, longitudinal+transversal deformations, multi-structures) this may be naturally
be produced when most of the energy concentrates in one component of the state for some high frequency vibrations;
c) Numerics: in the context of numerical approximation resonances may occur as a consequence of the interaction of
solutions with the discrete medium. In these lectures we shall try to summarize the most relevant work that has been
done in the subject in recent years. We shall first describe and document some of the most relevant applications to
Sciences, Engineering and Technology in which these problems arise. We shall the carefully discuss the three situations
mentioned above. In the case of networks of flexible strings and beams we shall follow our recent work with R. Dager
and show how the geometry of the network and the mutual lengths of the various strings/beams entering on it may
influence the properties of the system in what concerns control and observation. We shall then consider some classical
models for vibrations like the full von Karman system and Tymoshenko’s one. It is well known that the latter is a
singular limit of the first one as a suitable parameter tends to zero. However, we shall show that form a spectral point
of view part of the spectrum passes to the limit while the other one vanishes at infinity. We shall describe how control
properties are affected by this fact and the control properties that are robust under this kind of singular limit process.
Finally, we shall address the problem of the numerical approximation of control problems. To begin with we shall show
that the control and the numerical approximation process do not conmute so that, in general, when controlling a finitedimensional approximation of the continuous model, one does not actually compute an approximation of the control
one is looking for. We shall see what are the possible remedies to these pathologies: space discretizations, numerical
damping, filtering of high frequencies, multi-grid algorithms, etc. The latter fact is of great impact form a modelling
point view. Indeed, numerical approximation schemes may also be used (and they are often used that way) as discrete
models. Our analysis shows that these tow modelling approaches yield different results form a control theoretical point
of view and this should be taken into account. To conclude, we shall present a list of open problems and directions of
possible future research.
CONTACTS
Eduardo Casas
Departamento de Matemtica Aplicada y C.C.
E.T.S.I. Industriales y de Telecomunicación
Universidad de Cantabria
Av. Los Castros s/n 39005 Santander, Spain
Phone: +34 942 201 005, Fax: +34 942 201 829
E-mail: [email protected]
Michel Théra
LACO, UMR-CNRS
6090 Université de Limoges 123
avenue Albert Thomas 87060 Limoges Cedex
Phone: +33 5 55457325, Fax: +33 5 55457322
E-mail: [email protected]