Stabilization of nonlinear diffusion Ito processes with Markov switchings

Consideration is given to a class of systems that are described by the finite set of affine-controlled diffusion Ito processes with jump-like passages between them, definable by the evolution of the uniform Markov chain (Markov switchings). The stochastic version of the notion of the control Lyapunov function (CLF) is introduced. On the assumption of existence of the CLF, the explicit expression is found for the smooth stabilizing scalar control with the state feedback. In the case when a system is under the action of only noises depending on the state, the vector stabilizing control with the state feedback is found under the same assumptions. In the latter case, formulas for the stabilizing control represent the extension of the “universal” Sontag formula for the stabilizing control of a determinate system.