Necessary and sufficient conditions for finite-time stability of linear systems

In this paper we consider the finite-time stability and finite-time boundedness problems for linear systems subject to exogenous disturbances. The main results of the paper are some necessary and sufficient conditions, obtained by means of an approach based on operator theory; such conditions improve some recent results on this topic. An example is provided to illustrate the proposed technique.

1 Introduction

When dealing with the stability of a system, a distinction should be made between classical Lyapunov Stability and Finite-Time Stability (FTS) (or short-time stability). The concept of Lyapunov Asymptotic Stability is largely known to the control community; conversely a system is said to be finite-time stable if, once we fix a time-interval, its state does not exceed some bounds during this time-interval. Often asymptotic stability is enough for practical applications, but there are some cases where large values of the state are not acceptable, for instance in the presence of saturations. In these cases, we need to check that these unacceptable values are not attained by the state; for these purposes FTS can be used.