Finite-Time Stability of Impulsive Switched Systems

Abstract

This paper studies finite-time stability ( <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">FTS</i> ) of impulsive switched systems. Some sufficient criteria based on multiple Lyapunov functions coupled with dwell time condition are derived for ensuring the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">FTS</i> property. It shows that when the mode governing continuous dynamic is <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">FTS</i> but the discrete dynamic involves destabilizing impulses, the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">FTS</i> can be guaranteed if the impulses can be effectively restrained by dwell time condition. Conversely, when the mode governing continuous dynamic is not <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">FTS</i> but the discrete dynamic involves stabilizing impulses, the system can be successfully stabilized in <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">FTS</i> sense if the impulses are applied frequently. Moreover, when impulsive switched systems consist of stable and unstable modes, the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">FTS</i> can also be ensured if there is a tradeoff among the activating time of unstable modes, impulsive dynamics, and initial condition. Finally, two examples are proposed to illustrate the efficiency of theoretical results.