On stabilization of hybrid stochastic equations

A stochastic hybrid (difference-differential) equation with control is considered in the form of a stochastic differential equation with respect to a semimartingale. This equation contains the stochastic Ito equation and the stochastic difference equation as the partial cases. The conditions on the control matrices, and the coefficients of the equation which ensure stability of its trivial solution are obtained. The main technique employed in this paper is the method of the Lyapunov-Krasovskii functionals for the stochastic differential and difference equations, as well as some approaches from the theory of martingales.