Continuity of Pontryagin extremals with respect to delays in nonlinear optimal control

TL;DR

This paper proves that Pontryagin extremals in nonlinear optimal control problems depend continuously on delays, which enhances the understanding and numerical handling of delayed control systems.

Contribution

It establishes the continuous dependence of Pontryagin extremals on delays, including the adjoint vector, under specific assumptions, aiding numerical methods.

Findings

Continuity of extremals with respect to delays is proven.

The adjoint vector's continuity requires a detailed analysis.

Results support improved numerical methods for delayed control problems.

Abstract

Consider a general nonlinear optimal control problem in finite dimension, with constant state and/or control delays. By the Pontryagin Maximum Principle, any optimal trajectory is the projection of a Pontryagin extremal. We establish that, under appropriate assumptions, Pontryagin extremals depend continuously on the parameter delays, for adequate topologies. The proof of the continuity of the trajectory and of the control is quite easy, however, for the adjoint vector, the proof requires a much finer analysis. The continuity property of the adjoint with respect to the parameter delay opens a new perspective for the numerical implementation of indirect methods, such as the shooting method. We also discuss the sharpness of our assumptions.