AN INTRODUCTION TO THE
MOMENT APPROACH FOR
OPTIMAL CONTROL
MATHIEU CLAEYS
DIDIER HENRION
JEAN-BERNARD LASSERRE
LAAS-CNRS and University of Toulouse
This presentation
2

Compute lower bounds on the cost of optimal
control problems
 For
a large class of problems
 Systematically
 On a computer

Solve the problem
 Sometimes
only!
The idea
3
f(x)
f(x)
x
x
f(x)
f(x)
x
x
Presentation outline
4
1.
Introduction
2.
Static optimization
3.
Optimal control: impulsive controls
4.
Perspectives
Static optimization
5
such that
f(x)
x
X
The moment approach
6
Minimization
problem
Measure
problem
Moment problem
LMI problem
Measures: geometric viewpoint
7

Measure space

Notation

Properties
B1
B2
A few measures
8

Lebesgue measure (
a
)
b
B

X
Dirac measure at y (
)
y
A

Probality measure
B
X
Measures: relation with integration
9

The Lebesgue integral
completion of its Riemann counterpart

Integration w.r.t. Dirac measure:
is the
Measures: functional analysis viewpoint
10



Let :

, bounded continuous functions,

, sufficiently regular measures
Riesz: then positive linear functionals on
"are"
We note
Towards a measure problem
11


: take
: by definition
then
What does it measure?
12
Properties of the measure problem
13

The problem is linear!
 The
set of admissible solutions is convex
 The
set of optimal solutions is convex
 The
infimum is a minimum!
The moment approach
14
Minimization
problem
Measure
problem
Moment problem
LMI problem
Moments
15

Moments

Moment sequence

Moment matrix

Localizing matrix at e.g.
Properties of moments matrices
16

Proof:

Is the converse true?
The converse
17

Let
is supported on

Proof: Riesz-Haviland + Putinar’s Positivstellensatz
From a measure to a moment problem
18
such that
such that
The moment approach
19
Minimization
problem
Measure
problem
Moment problem
LMI problem
From a moment to a LMI problem
20

(Lasserre) relaxations: optimize only on moments of
order up to 2r


If at some relaxation
Then μ is k-atomic and we can extract its k support
points
A simple example
21
Minimization
problem
Measure
problem
Moment
problem
LMI
problem
A GloptiPoly snippet
22
>> mpol x; % define optimization variable
>> P = msdp( min((x-1)^2) , x^2 <= 4, 2); % define problem: criterion, constraint, order
>> [flag, obj] = msol(P) % solve problem
flag =
1
obj =
2.8868e-009
>> double(x) % extract solution(s)
ans =
1.0000
>> double(mmat(meas(1))) % show moment matrix
ans =
1.0000
1.0000
0.9999
1.0000
0.9999
0.9999
0.9999
0.9999
0.9998
The moment approach
23
Minimization
problem
Measure
problem
Moment problem
LMI problem
Presentation outline
24
1.
Introduction
2.
Static optimization
3.
Optimal control: impulsive controls
4.
Perspectives
Our problem
25
The moment approach
26
Minimization
problem
Measure
problem
Moment problem
LMI problem
Measure for impulses
27

We embed
in
where
can be decomposed as
Occupation measures
28

Fix

Define

Occupation measure:
B
A

Control measure:
Weak formulation of dynamics
29

For each continuous
:
This is a relaxation!
30
A basic example (1/2)
31
such that
A basic example (2/2)
32
such that
such that
Orbital rendez-vous, l1-induced norm
33
v-
u+
u-
v+
On Carter’s third example
34
d
1
2
3
4
Vd
0.0463
0.0680
0.2188
0.2972
A non-convex example (1/2)
35
A non-convex example (2/2)
36
Cost = 0.682
Presentation outline
37
1.
Introduction
2.
Static optimization
3.
Optimal control: impulsive controls
4.
Perspectives
The method, yet again
38
Minimization
problem
Measure
problem
Moment problem
LMI problem
Critique
39
+
-
Global optimality
Lower bounds
Certificate of infeasibility
Extraction of solutions
Finite convergence ?
Relies on LMI solvers
The main problem…
40


# of monomials of n variables up to degree 2r:
Example with n=6
r
#
0
1
1
28
2
210
3
924
4
3003
5
8008
6
18564
Thank you!
41

Presentation available at:
http://homepages.laas.fr/mclaeys