Explicit criteria for moment exponential stability and instability of switching diffusions with Lévy noise

This paper is devoted to moment exponential stability of a class of Markovian switching diffusions with Lévy noise. Our objective is to find verifiable conditions for moment exponential stability and instability. By employing the ireducibility of the Markovian switching process and the Fredholm alternative, we derive explicit criteria for moment exponential stability and instability for linear and general nonlinear systems. Explicit contribution of the Markovian switching to the stability and instability is revealed. Using the criteria developed, we can design feedback controls or perturbations to stabilise a given diffusion with Markovian switching and Lévy noise. Several examples are provided to demonstrate our findings.