Worst Exponential Decay Rate for Degenerate Gradient flows subject to persistent excitation

TL;DR

This paper analyzes the worst-case exponential decay rate of degenerate gradient flows under persistent excitation, providing bounds and solving an open problem related to $L_2$-gain estimates in control systems.

Contribution

It introduces new bounds for decay rates of degenerate gradient flows under persistent excitation and addresses an open problem on $L_2$-gain estimates for time-varying control systems.

Findings

Established upper bounds for decay rates consistent with known lower bounds.

Related decay rate analysis to optimal control problems.

Provided estimates for the worst $L_2$-gain of certain control systems.

Abstract

In this paper we estimate the worst rate of exponential decay of degenerate gradient flows $\dot x = -S x$, issued from adaptive control theory. Under persistent excitation assumptions on the positive semi-definite matrix $S$, we provide upper bounds for this rate of decay consistent with previously known lower bounds and analogous stability results for more general classes of persistently excited signals. The strategy of proof consists in relating the worst decay rate to optimal control questions and studying in details their solutions. As a byproduct of our analysis, we also obtain estimates for the worst $L_2$-gain of the time-varying linear control systems $\dot x=-cc^\top x+u$, where the signal $c$ is persistently excited, thus solving an open problem posed by A. Rantzer in 1999.