Almost Sure Contraction for Stochastic Switched Impulsive Systems

Abstract

This article studies the contraction issue for stochastic switched impulsive systems (SSISs). The notion of almost sure contraction is proposed for SSISs. By using the first variation, the almost sure contraction is equivalent to the almost sure exponential stability of the virtual dynamics of SSISs. By It <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\hat{\text{o}}$</tex-math></inline-formula> 's formula and martingale theory, criteria on almost sure contraction are established for SSISs. The contraction criteria are also derived by the Lyapunov-like function method. All the criteria of contraction are expressed via the conditions of impulse frequency (I.F.). It is shown that these criteria via I.F. are more general and less restrictive than those reported for subclasses of SSISs. Finally, two examples are given to illustrate the effectiveness of the results.