The problem of the adaptive tracking for a class of stochastic nonlinear systems with stationary Markovian switching is considered in this note. An Ito formula is proposed for stochastic integral equations with an integral about martingale measure. An adaptive backstepping controller is designed such that the closed-loop system has a unique solution that is globally bounded in probability and L₄-norm of the tracking error converges to an arbitrarily small neighborhood of zero. A simulation example demonstrates the efficiency of the proposed scheme.

In this paper, the problem of stability on stochastic systems with state-dependent switching is investigated. To analyze properties of the switched system by means of Itô's formula and Dynkin's formula, it is critical to show switching instants being stopping times. When the given active-region set can be replaced by its interior, the local solution of the switched system is constructed by defining a series of stopping times as switching instants, and the criteria on global existence and stability of solution are presented by Lyapunov approach. For the case where the active-region set can not be replaced by its interior, the switched systems do not necessarily have solutions, thereby quasi-solution to the underlying problem is constructed and the boundedness criterion is proposed. The significance of this paper is that all the results presented depend on some easily-verified assumptions that are as elegant as those in the deterministic case, and the proofs themselves provide design procedures for switching controls.

Stochastic differential equations (SDEs) are widely adopted to describe systems with stochastic disturbances, while they are not necessarily the best models in some specific situations. This paper considers the nonlinear systems described by random differential equations (RDEs). The notions and the corresponding criteria of noise-to-state stability, asymptotic gain and asymptotic stability are proposed, in the m-th moment or in probability. Several estimation methods of stochastic processes are presented to explain the reasonability of the assumptions used in theorems. As applications of stability criteria, some examples about stabilization, regulation and tracking are considered, respectively. A theoretical framework on stability of RDEs is finally constructed, which is distinguished from the existing framework of SDEs.

In the deterministic case, a significant improvement on stability analysis of nonlinear systems is caused by introducing Barbalat's lemma into control area after Lyapunov's second method and LaSalle's theorem were established. This note considers the extension of Barbalat's lemma to the stochastic case. To this end, the uniform continuity and the absolute integrability are firstly described in stochastic forms. It is nevertheless a small generalization upon the existing references since our result can be used to adapted processes which are not necessarily Itô diffusions. When it is applied to Itô diffusion processes, many classical results on stochastic stability are covered as special cases.