Stability Of Stochastic Functional Differential Systems With Markovian Switching And Functional Differential Systems With Impulses

In this dissertation we consider the stability of stochastic functional differential systems with Markovian switching and functional differential systems with impulses, respectively. In Chapter 2, the Lyapunov-Razumikhin method used in the classical ordinary differential equations is generalized to the stochastic functional differential systems with Markovian switching. Some sufficient conditions for the any pth monment exponential stability by only one Lyapunov function are obtained. In Chapter 3, we first obtain the comparision principle for the nonlinear stochastic delay differential systems with Markovian switching. Later, using this comparision principle, we obtain some stability criteria. Thus we connect the stochastic functional differential systems with Markovian switching to a functionial differential equation. The stability of the complicated and high dimension stochastic functional differential systems with Markovian switching can be reduced to the stability of one dimension functional differential equation. In Chapter 4, we investigste the robust stability of the stochastic delay differential systems with Markovian switching. We prove that the oringal systems are still exponentially stable and almost-surely stable when the disturbances of the delay, the drift efficient and the diffusion are small sufficiently. In Chapter 5, we obtain the delay-dependent stability criteria for the pth monment exponential stability of nonlinear stochastic delay differential systems with Markovian switching. In Chapter 6, we study the Lyapunov stability of stochastic (delay) differential systems with Markovian switching, and relax the necessary condition that LV is negative defnition in the known references. In Chapter 7.1, we consider the uniform stability and the uniform aymptotic stability of the impulsive functional equations , and extend the stability results in ordinary differential equations to impulsive functional differential equations. In Chapter 7.2, we investigate the the asymptotic behavior of the forced nonlinear delay differential equations with impulses. Our results hold for linear and nonlinear equations, forced and unforced equations, impulsive and nonimpulsive equations. In Chapter 7.3 and 7.4, we deal with the oscillation of nonlinear delay differential equations with impulses and second order nonlinear ordinary differential equations with impulses, respectively. Some results are generalized and improved.