Optimal value functions for weakly coupled systems: a posteriori estimates

TL;DR

This paper analyzes how the optimal value function in weakly coupled LQ control problems is sensitive to coupling strength, introducing a coupling-adapted norm and revealing conditions under which the value function changes significantly.

Contribution

It provides new sensitivity estimates for the value function in weakly coupled systems and introduces a coupling-adapted norm for better analysis.

Findings

Weak coupling can destabilize the system and cause drastic changes in the value function.

A coupling-adapted norm improves sensitivity estimates.

A new, simpler proof relates the separation operator to the stability radius.

Abstract

We consider weakly coupled LQ optimal control problems and derive estimates on the sensitivity of the optimal value function in dependence of the coupling strength. In order to improve these sensitivity estimates a "coupling adapted" norm is proposed. Our main result is that if a weak coupling suffices to destabilize the closed loop system with the optimal feedback of the uncoupled system then the value function might change drastically with the coupling. As a consequence, it is not reasonable to expect that a weakly coupled system possesses a weakly coupled optimal value function. Also, for a known result on the connection of the separation operator and the stability radius a new and simpler proof is given.