On the relation between dynamic regret and closed-loop stability

TL;DR

This paper explores the connection between bounded dynamic regret and closed-loop stability, establishing conditions under which they imply each other in systems with unknown, time-varying costs.

Contribution

It provides the first formal analysis linking bounded dynamic regret with asymptotic stability in adaptive control systems with unknown costs.

Findings

Bounded dynamic regret implies asymptotic stability for constant costs.

Necessary conditions for bounded regret in asymptotically stable systems.

Sufficient conditions for bounded regret under additional assumptions.

Abstract

In this work, we study the relations between bounded dynamic regret and the classical notion of asymptotic stability for the case of a priori unknown and time-varying cost functions. In particular, we show that bounded dynamic regret implies asymptotic stability of the optimal steady state for a constant cost function. For the case of an asymptotically stable closed loop, we first derive a necessary condition for achieving bounded dynamic regret. Then, given some additional assumptions on the system and the cost functions, we also provide a sufficient condition ensuring bounded dynamic regret. Our results are illustrated by examples.