Optimal sampled-data control, and generalizations on time scales

TL;DR

This paper extends the Pontryagin maximum principle to optimal control problems where the state and control evolve on arbitrary time scales, unifying sampled-data control with more general dynamic systems.

Contribution

It introduces a generalized framework for optimal control on arbitrary time scales, encompassing sampled-data systems and providing new theoretical insights.

Findings

Derived a Pontryagin maximum principle for systems on time scales.

Unified sampled-data control with general time scale systems.

Established a variational approach using needle-like variations.

Abstract

In this paper, we derive a version of the Pontryagin maximum principle for general finite-dimensional nonlinear optimal sampled-data control problems. Our framework is actually much more general, and we treat optimal control problems for which the state variable evolves on a given time scale (arbitrary non-empty closed subset of R), and the control variable evolves on a smaller time scale. Sampled-data systems are then a particular case. Our proof is based on the construction of appropriate needle-like variations and on the Ekeland variational principle.