Impulsive optimal control problems with time delays in the drift term

TL;DR

This paper develops a framework for impulsive optimal control problems with time delays in the drift, establishing well-posedness, continuity, and necessary optimality conditions under mild regularity assumptions.

Contribution

It introduces a bounded variation solution concept for nonlinear control systems with delays and impulsive controls, extending optimality conditions to this complex setting.

Findings

Proved well-posedness of solutions with delays and impulses.

Established continuity of the input-output map.

Derived necessary optimality conditions using advanced variational principles.

Abstract

We introduce a notion of bounded variation solution for a new class of nonlinear control systems with ordinary and impulsive controls, in which the drift function depends not only on the state, but also on its past history, through a finite number of time delays. After proving the well posedness of such solutions and the continuity of the corresponding input output map with respect to suitable topologies, we establish necessary optimality conditions for an associated optimal control problem. The approach, which involves approximating the problem by a non impulsive optimal control problem with time delays and using Ekeland principle combined with a recent, nonsmooth version of the Maximum Principle for conventional delayed systems, allows us to deal with mild regularity assumptions and a general endpoint constraint.