Stabilization in distribution of periodic hybrid systems by discrete-time state feedback control

Periodic hybrid stochastic differential equations (SDEs) have been widely used to model systems in many branches of science and industry which are subject to the following natural phenomena: (a) uncertainty and environmental noise, (b) abrupt changes in their structure and parameters, (c) periodicity. In many situations, it is inappropriate to study whether the solutions of periodic hybrid SDEs will converge to an equilibrium state (say, 0 or the trivial solution) but more appropriate to discuss whether the probability distributions of the solutions will converge to a stationary distribution, known as stability in distribution. Given a periodic hybrid SDE, which is not stable in distribution, can we design a periodic feedback control in the shift term based on state observations at discrete times so that the controlled SDE becomes stable in distribution? We will refer to this problem as stabilisation in distribution by periodic feedback control. There is little known on this problem so far. This paper initiates the study in this direction.