Chapter 3
Sufficient conditions for optimality of minimum time
control-affine systems
Abstract
In this chapter, we prove a sufficient condition for optimality in the case of min- imum time affine control systems with double-input control on a 4 dimensional manifold. The proof is based on symplectic methods, and the condition given can be check via a simple numerical test. No strong Legendre-type condition is required.
In this chapter we deal with the notion of sufficient conditions for optimality of our extremals. This topic is still a very active field of research, and a variety of different approaches exist and have been applied to a large number of prob- lems. Geometric methods hold a fairly important place in that regard. Since we have existence and uniqueness of the solutions of the extremal system, to obtain a global result about their optimality, one would like to apply Filippov’s theorem, see [24]. This can not be generally achieved, the compactness condition being too delicate to prove, and we switch to local point of view. When the extremal flow (and thus the maximized Hamiltonian) is smooth, the theory of conjugate points can be applied, and local optimality holds before the first conjugate time, we recall this result below. The points where the extremal ceases to be globally optimal are cut points, in general it is an extremely delicate task to compute cut points and cut loci, though it can be done numerically, as for the averaged mini- mum energy orbit transfer problem, in the two body case, see [16]. Unfortunately, we rarely encounter the smooth case in practice, and there is a lack of general method overcoming the different kind of singularities. An extension of the smooth case method which uses the Poincaré-Cartan integral invariant, see [4], is easier to generalize to non-smooth cases, and has been used to prove local optimality for L1 minimization of mechanical systems for instance, in [19]. We use a similar technique to prove theorem 3.2, one of the main difference being the type of sin- gularity: L1-minimization of the control creates singularities of codimension one, and the extremal flow is the concatenation of the flow of two regular Hamiltonians.
In our case, we have codimension two (meaning, unstable) singularities, and one non-Lipschitz Hamiltonian. When the control lies in a box, second order condi- tions can be of use through a finite dimensional subsystem given by allowing the switching times to variate, those techniques have been initiated by Stefani and Poggiolini, see [1], for instance. The majority of this work proved local optimality for normal extremal, and a few of them tackle the abnormal case. One can cite for instance [56] where single input systems are handled. One can refer as well to [39]
where theoretical as well as numerical studies are leaded when the control lies in a polyhedron. We will also tackle only the normal case in the following, since the co-dimension two singularity inducted by minimizing the final time is our main focus in this thesis. The recent paper [3] from Agrachev and Biolo, proved local optimality of these broken extremal around the singularity with extra hypothesis on the adjoint state. Our result present the interest of being more global (in the sens that it is viable along a whole trajectory), and easily checked by a simple nu- merical test. Thanks to that optimality analysis, we can investigate the regularity
3.1. THE SMOOTH THEORY
of a upper bound to the value function of this time optimal problem: the final time of extremals and prove that it is piecewise smooth.
3.1 The smooth theory
Let us begin by recalling the classical options when the extremal system is smooth.
Consider an optimal control system
x9 “fpx, uq, uP U (3.1.1)
and assumeHmaxpx, pq “ max
uPU Hpx, p, uqis smooth. We present a method described in [4], and refined in [19] to deal with codimension a one singularity set. Denote
¯
zptq “ px¯ptq,p¯ptqq, tP r0,¯tfs the reference extremal, starting from ¯z0 P T˚M, ¯u its associated control,and consider the variational equation along ¯zptq:
δz9 “J∇2Hpzptqqδz (3.1.2) Solutions of (3.1.2) are called Jacobi fields.
Definition 3.1 (Conjugate times & points)
A time tc is called a conjugate time if there exists Jacobi field δz such that dπpzp0qqδzp0q “dπpzptcqqδzptcq “0
(ie, δxp0q “ δxptcq “ 0). We say δz is vertical at 0 and tc. The point xptcq “ πpzptcqq is a conjugate point.
The following result imply optimality until the first conjugate time.
Theorem 3.1 Assume:
(1) The reference extremal is normal.
(2) BpBx
0pt,z¯0q ‰ 0 for all t Ps0, tfs.
then the reference trajectory x is a local minimizer among all the C0-admissible trajectories with same end points.
Assumption p2q provide disconjugacy along the reference extremal, and can be verified through a simple numerical test. The proof consists in building a La- grangian manifold, and propagating in by the extremal flow, then one can prove the projection is invertible on this manifold: this allows one to lift all the ad- missible trajectories with same end points to the cotangent bundle, and one can compare, using the Poincaré-Cartan invariant, their cost with the one of the refer- ence extremal. We will extend this proof to the non-smooth case of our minimum time affine control systems.