Second-order necessary conditions in Pontryagin form for optimal control problems

International audienceIn this report, we state and prove first- and second-order necessary conditions in Pontryagin form for optimal control problems with pure state and mixed control-state constraints. We say that a Lagrange multiplier of an optimal control problem is a Pontryagin multiplier if it is such that Pontryagin's minimum principle holds, and we call optimality conditions in Pontryagin form those which only involve Pontryagin multipliers. Our conditions rely on a technique of partial relaxation, and apply to Pontryagin local minima.Dans ce rapport, nous énonçons et prouvons des conditions nécessaires du premier et second ordre sous forme Pontryaguine pour des problèmes de commande optimale avec contraintes pures sur l'état et mixtes sur l'état et la commande. Nous appelons multiplicateur de Pontryaguine tout multiplicateur de Lagrange pour lequel le principe de Pontryaguine est satisfait et parlons de conditions d'optimalité sous forme Pontryaguine si elles ne font intervenir que des multiplicateurs de Pontryaguine. Nos conditions s'appuient sur une technique de relaxation partielle et sont valables pour des minima de Pontryaguine

Higher Order Conditions in Nonlinear Optimal Control

The most widely used tool for the solution of optimal control problems is the Pontryagin Maximum Principle. But the Maximum Principle is, in general, only a necessary condition for optimality. It is therefore desirable to have supplementary conditions, for example second order sufficient conditions, which confirm optimality (at least locally) of an extremal arc, meaning one that satisfies the Maximum Principle. Standard second order sufficient conditions for optimality, when they apply, yield the information not only that the extremal is locally minimizing, but that it is also locally unique. There are problems of interest, however, where minimizers are not locally unique, owing to the fact that the cost is invariant under small perturbations of the extremal of a particular structure (translations, rotations or time-shifting). For such problems the standard second order conditions can never apply. The first contribution of this thesis is to develop new second order conditions for optimality of extremals which are applicable in some cases of interest when minimizers are not locally unique. The new conditions can, for example, be applied to problems with periodic boundary conditions when the cost is invariant under time translations. The second order conditions investigated here apply to normal extremals. These extremals satisfy the conditions of the Maximum Principle in normal form (with the cost multiplier taken to be 1). It is, therefore, of interest to know when the Maximum Principle applies in normal form. This issue is also addressed in this thesis, for optimal control problems that can be expressed as calculus of variations problems. Normality of the Maximum Principle follows from the fact that, under the regularity conditions developed, the highest time derivative of an extremal arc is essentially bounded. The thesis concludes with a brief account of possible future research directions

Caratheodory-Equivalence, Noether Theorems, and Tonelli Full-Regularity in the Calculus of Variations and Optimal Control

We study, in a unified way, the following questions related to the properties of Pontryagin extremals for optimal control problems with unrestricted controls: i) How the transformations, which define the equivalence of two problems, transform the extremals? ii) How to obtain quantities which are conserved along any extremal? iii) How to assure that the set of extremals include the minimizers predicted by the existence theory? These questions are connected to: i) the Caratheodory method which establishes a correspondence between the minimizing curves of equivalent problems; ii) the interplay between the concept of invariance and the theory of optimality conditions in optimal control, which are the concern of the theorems of Noether; iii) regularity conditions for the minimizers and the work pioneered by Tonelli.Comment: 24 pages, Submitted for publication in a Special Issue of the J. of Mathematical Science

Pontryagin Maximum Principle and Stokes Theorem

We present a new geometric unfolding of a prototype problem of optimal control theory, the Mayer problem. This approach is crucially based on the Stokes Theorem and yields to a necessary and sufficient condition that characterizes the optimal solutions, from which the classical Pontryagin Maximum Principle is derived in a new insightful way. It also suggests generalizations in diverse directions of such famous principle.Comment: 21 pages, 7 figures; we corrected a few minor misprints, added a couple of references and inserted a new section (Sect. 7); to appear in Journal of Geometry and Physic

Generalized splines in ℝn and optimal control

We give a new time-dependent definition of spline curves in ℝn, which extends a recent definition of vector-valued splines introduced by Rodrigues and Silva Leite for the time-independent case. Previous results are based on a variational approach, with lengthy arguments, which do not cover the non-autonomous situation. We show that the previous results are a consequence of the Pontryagin maximum principle, and are easily generalized using the methods of optimal control. Main result asserts that vector-valued splines are related to the Pontryagin extremals of a non-autonomous linear-quadratic optimal control problem

Weak Conservation Laws for Minimizers which are not Pontryagin Extremals

We prove a Noether-type symmetry theorem for invariant optimal control problems with unrestricted controls. The result establishes weak conservation laws along all the minimizers of the problems, including those minimizers which do not satisfy the Pontryagin Maximum Principle.Comment: Accepted for presentation (Paper No: 113) at the 2nd International Conference "Physics and Control" (PhysCon 2005), August 24-26, 2005, Saint Petersburg, Russia. To appear in the respective Conference Proceeding