Optimal Control Problems with Vector-Valued Impulse Controls and Time Delays

TL;DR

This paper develops a framework for nonlinear control systems with vector-valued impulse controls and time delays, establishing solution existence, representation, and optimality conditions.

Contribution

It introduces a novel notion of solutions for such systems, providing an equivalent formulation and deriving a maximum principle for optimal control.

Findings

Established well-posedness of solutions

Derived a maximum principle for optimality

Provided a new representation formula for solutions

Abstract

We consider a nonlinear control system with vector-valued measures as controls and with dynamics depending on time delayed states. First, we introduce a notion of discontinuous, bounded variation solution associated with this system and establish an equivalent representation formula for it, inspired by the approach known in delay-free impulsive control as the `graph completions' method. Then, thanks to this equivalent formulation, we prove well-posedness properties of these solutions and also derive necessary optimality conditions in the form of a Maximum Principle for an associated minimization problem.