Continuity/constancy of the Hamiltonian function in a Pontryagin maximum principle for optimal sampled-data control problems with free sampling times

Abstract

In a recent paper by Bourdin and Trélat, a version of the Pontryagin maximum principle (in short, PMP) has been stated for general nonlinear finite-dimensional optimal sampled-data control problems. Unfortunately their result is only concerned with fixed sampling times, and thus it does not take into account the possibility of free sampling times. The present paper aims to fill this gap in the literature. Precisely we establish a new version of the PMP that can handle free sampling times. As in the aforementioned work by Bourdin and Trélat, we obtain a first-order necessary optimality condition written as a nonpositive averaged Hamiltonian gradient condition. Furthermore, from the freedom of choosing sampling times, we get a new and additional necessary optimality condition which happens to coincide with the continuity of the Hamiltonian function. In an autonomous context, even the constancy of the Hamiltonian function can be derived. Our proof is based on the Ekeland variational principle. Finally a linear-quadratic example is numerically solved using shooting methods, illustrating the possible discontinuity of the Hamiltonian function in the case of fixed sampling times and highlighting its continuity in the instance of optimal sampling times.

Pontryagin-Type Conditions for Optimal Muscular Force Response to Functional Electrical Stimulations

“…[ and u(t) := u * (t) elsewhere, we exactly recover the nonpositive averaged Hamiltonian gradient condition derived in [8][9][10] given by…”

“…, n, where t 0 := 0 and t n+1 := T . We refer to [8][9][10] for the statement of a Pontryagin maximum principle handling sampled-data controls. From now, we assume that U is convex (see Remark 2.2 for details) and our aim is to briefly recall the derivation of the Pontryagin maximum principle for Problem (2) in the case of sampled-data control.…”

“…6and (8) are clearly different according to the nature of the admissible controls set U. We refer to the works [8][9][10] for detailed discussions about this feature.…”

“…In this paper, we consider the problem of minimizing a general cost function of Mayer form, which depends on the final values of the force response and of the fatigue variables. The resulting optimization problem is clearly related to optimal sampled-data control problems, but it does not fit exactly with the framework considered in [8][9][10], in which a Pontryagin maximum principle with first-order necessary optimality conditions is derived. As a consequence, our main objective in the present paper is to adapt the techniques of [8][9][10] to our specific problem in order to derive the corresponding Pontryagin-type conditions.…”

“…Precisely, the physical FES system is set into the framework of sampled-data control system and a general Mayer optimal control problem is considered. A discussion is initiated in order to explain why it does not fit exactly with the framework investigated in [8][9][10]. Then, we adapt the previous works to our particular problem and we establish the corresponding Pontryagin first-order necessary optimality conditions (see Theorem 3.1).…”