On the Nonuniqueness and Instability of Solutions of Tracking-Type Optimal Control Problems

TL;DR

This paper investigates the nonuniqueness and instability of solutions in certain tracking-type optimal control problems, showing that solutions are not always unique and the solution set can be highly sensitive to data variations.

Contribution

It provides a theoretical analysis demonstrating the inherent nonuniqueness and instability in a class of nonlinear optimal control problems with weak control-to-state mappings.

Findings

Solutions are nonuniquely solvable for certain data choices.

The solution set does not admit a continuous selection, indicating instability.

The analysis applies to problems with non-affine, weak-to-weak continuous control-to-state mappings.

Abstract

We study tracking-type optimal control problems that involve a non-affine, weak-to-weak continuous control-to-state mapping, a desired state $y_d$, and a desired control $u_d$. It is proved that such problems are always nonuniquely solvable for certain choices of the tuple $(y_d, u_d)$ and instable in the sense that the set of solutions (interpreted as a multivalued function of $(y_d, u_d)$) does not admit a continuous selection.