Practical stability of nonlinear stochastic hybrid parabolic systems of Itô-type: Vector Lyapunov functions approach

Lyapunov’s second method is widely recognized as a fundamental tool not only in the theory of stability but also in studying other qualitative properties of solutions of differential equations. The main characteristic of this method is the utilization of a function, namely the Lyapunov function, together with the theory of differential and integral inequalities to develop a very general comparison principle under much less restrictive conditions to investigate qualitative and quantitative properties of nonlinear differential equations. The most important advantage of this technique is that it does not require the explicit knowledge of solutions. This paper is devoted to extend this technique to investigate the practical stability of the solution process of stochastic hybrid parabolic partial differential equations of Itô type under Markovian structural perturbations. The jump nonlinear system is a hybrid system with state vector that has two components x(t)x(t) and η(t)η(t). Here x(t)x(t) is referred to as the state and η(t)η(t) is referred as the mode. During the operation, the system can jump from one mode to another in a random way, which makes this class of systems a stochastic one. The switching between the modes is governed by a Markov process with discrete and finite state space. When the system mode is fixed it evolves like a deterministic nonlinear system. This kind of system can be used to describe abrupt phenomena, such as component and interconnection failures.