Stabilization in Distribution of Hybrid Systems by Intermittent Noise

For many stochastic hybrid systems in the real world, it is inappropriate to study if their solutions will converge to an equilibrium state (say, 0 by default) but more appropriate to discuss if the probability distributions of the solutions will converge to a stationary distribution. The former is known as the asymptotic stability of the equilibrium state while the latter the stability in distribution. This article aims to determine whether or not a stochastic state feedback control can make a given nonlinear hybrid differential equation, which is not stable in distribution, to become stable in distribution. We will refer to this problem as stabilisation in distribution by noise or stochastic stabilisation in distribution. Although the stabilisation by noise in the sense of almost surely exponential stability of the equilibrium state has been well studied, there is little known on the stabilisation in distribution by noise. This article initiates the study in this direction.