Remarks on Dwell Time Solutions and Stability of Families of Nonlinear Vector Fields

In this note, we discuss the problem of stability of (finite or infinite) families of continuous vector fields, all of them asymptotically stable but, in general, not exponentially stable. Under a multiple Liapunov function condition and an average dwell time constraint, we prove that the system possesses a form of stability, weaker than the standard one. The advantage of our results is that the conditions imposed on the Liapunov functions can be verified a priori, with no previous knowledge of the integral curves of the family.

I. Introduction

Families of vector fields have been fruitfully employed as a model for representing input systems both in geometric control theory and in the recent theory of switched systems. Here we consider a family of finite dimensional, continuous vector fields , which possess a common equilibrium position at the origin. In general terms, by a solution of we mean any continuous, piecewise differentiable curve which is obtained by gluing together integral curves of the vector fields ; however, the solutions of which are interesting from the point of view of applications, are often required to satisfy additional constraints, as it will be explained later.