Stabilization and destabilization of impulsive Markovian switching systems via periodic stochastic controls and impulsive controls

Abstract Given an unstable (or stable) impulsive Markovian switching system (IMSS), can it be stabilized (or destabilized) by noise? This problem has not been studied. In view of this question, this article will establish the noise stabilization and destabilization strategies of IMSS, and also study the impulsive control realized by adjusting the impulse intensity. It is worth noting that the impulse jump is the random impulse whose jump value depends on the mode, and the noise stabilization and destabilization are realized by periodic stochastic control based on Lévy noise. In order to establish these new results, we use the time‐inhomogeneous property of coefficients of the controlled system and the ergodicity of discrete‐time Markov chains based on impulse trigger time. In addition, the periodic stochastic control on diffusion and jump terms can be applied to design intermittent stochastic control.