Stabilization of stochastic regime-switching Poisson jump equations by delay feedback control

Abstract

Abstract This paper is concerned with the stabilization of stochastic regime-switching Poisson jump equations (also known as stochastic differential equations with Markovian switching and Poisson jumps, abbreviated as SDEwMJs). The aim of this paper is to design a feedback controller with delay δ ( δ > 0 $\delta >0$ ) to make an unstable SDEwMJ become stable. It is proved that the delay δ is bounded by a constant δ̄. Moreover, an implicit lower bound for δ , ¯ $\bar{\delta ,}$ which can be computed numerically, is provided. As a product, the almost sure exponential stability of the controlled SDEwMJ is obtained. Besides, an example is given to demonstrate the theoretical results.

Keywords

• Almost sure exponential stability
• Markov’s chain
• Poisson jumps
• Delay feedback control