Stability of Invariant Sets of Itô Stochastic Differential Equationswith Markovian Switching

Invariant sets of dynamic systems play an important role in many situations when the dynamic behavior is constrained in some way. Knowing that a set in the state space of a system is invariant means that we have bounds on the behavior. We can verify that pre-specified bounds which originate from, for example, safety restrictions, physical constraints, or state-feedback magnitude bounds are not invalidated. There is significant literature devoted to the invariant sets of ordinary differential equations, functional differential equations, and stochastic differential equations, and we here mention [2, 4, 15–18]. Recently, much work has been done on stochastic differential equations with Markovian switching [1, 3, 5–14, 19, 20]. In particular, we here highlight Mao’s significant contribution [6, 11, 12]. However, to the best of the author’s knowledge to date, the problem of the invariant sets of equations of this kind, has not been investigated yet. The aim of the present paper is to study the invariant sets of nonlinear Itô stochastic differential equations with Markovian switching. Similar to the result of [18], which investigates the usual stochastic differential equations, some sufficient conditions for the invariance and stochastic stability of invariant sets of equations of this kind are derived. At the same time, we establish some conditions for stochastic asymptotic stability and instability of the invariant sets, which are not discussed in [18] even in the case of equations without Markovian switching.