The LASALLE Theorem Of Stochastic Jump Systems And The Stability Of Coupled Systems

Stochastic jump differential equations with Markov switching have played moreprominent role in people's daily lives, particularly in the natural sciences, engineeringand many other fields. In recent years, people's attention focused on the stability of thesolution, LaSalle invariance principle is a powerful tool to solve the stability problem,therefore, study the LaSalle invariance principle of stochastic jump differential equation-s with Markov switching have practical significance. In this paper, the author establishthe LaSalle invariance principle of stochastic jump differential equations with Markovswitching. Using the established LaSalle invariance principle, receiving the criterion ofthe stochastic asymptotic stability of stochastic jump differential equations with Markovswitching.Firstly, the paper take advantage of a Lyapunov function and multiple Lyapunovfunctions to study the LaSalle invariance principle of the stochastic asymptotic stability.With the help of the Non-negative semimartingale convergence theorem, basics of thestochastic jump differential equations with Markov switching, and so on, the paper estab-lish the LaSalle invariance principle of stochastic jump differential equations with Markovswitching. Define the LaSalle invariant set of stochastic jump differential equations withMarkov switching, study the properties of it; under the certain conditions, the paper obtainthe conclusion that the compound of the solution and the specific V function will tend to acertain non-empty set, study the LaSalle invariance principle of the Autonomous system.Secondly, the paper study the application of the LaSalle invariance principle of s-tochastic jump differential equations with Markov switching. That is, the application ofthe stability analysis for stochastic coupled systems on networks. Combining graph the-ory and the Lyapunov's second method, the paper establish the stability theorem of thestochastic coupled systems on networks, take advantage of the stability theorem to studythe stability in the large of the stochastic jump oscillation equations.