Pontryagin maximum principle for finite dimensional nonlinear optimal control problems on time scales (Q2873854)

In this paper, the authors derive a strong version of the Pontryagin Maximum Principle (PMP) for general nonlinear optimal control problems on time scales in finite dimension. Optimal control theory is concerned with the analysis of controlled dynamical systems, where one aims at steering such a system for a given configuration to some desired target by minimizing or maximizing some criterion. The objective of the present paper is to state and prove a strong version of the Pontryagin maximum principle on time scales, valuable for general nonlinear dynamics, and without assuming any unnecessary Lipschitz or convexity conditions. Here the authors statement is as general as possible and encompasses the classical continuous-time PMP. General constraints on the initial and final values of the state variable are considered, and the resulting transversality conditions are derived. The aim is to fill a gap in the literature and to derive a general strong version of the PMP on time scales. There exist several proofs of the continuous-time PMP in the literature. Mainly they can be classified as variants of two different approaches: the first of which consists of using a fixed-point argument, and the second consists of using Ekeland's variational principle. In all cases needle-like variations are used to generate the co-called Pontryagin cone, serving as a first-order convex approximation of the reachable set. The adjoint vector is then constructed by propagating backward in time a Lagrange multiplier which is normal to this cone. Roughly, needle-like variations are kinds perturbations of the reference control in the \(L^1\) topology (perturbations with arbitrary values, over small intervals of time) which generate perturbations of the trajectories in the \(C^\circ\) topology.NEWLINENEWLINEMain result: The proof of the PMP given in this paper is based on Ekeland's variational principle, which permits avoiding the proposed obstructions and happens to be well adapted for the proof of a general PMP on time scales. After having shown that the set of admissible controls is open, the authors define needle-like variations at right-dense and right-scattered points and derive some properties. Ekeland's variational principle is applied to a well-chosen functional in an appropriate complete metric space and then the PMP is proved.