Lyapunov-like techniques for stochastic stability. (English) Zbl 0845.93085

Florchinger, Patrick

SIAM J. Control Optimization 33, No. 4, 1151-1169 (1995).

The author considers the dynamical system \[ x(t) = x_0 + \int^t_0 f \bigl( x(s), u \bigr) ds + \int^t_0 g \bigl( x(s) \bigr) dw(s) \] where \(x \in\mathbb{R}^n\), \(x_0\) is an initial state, \(u \in\mathbb{R}^p\) is a control law, \(f:\mathbb{R}^n \times\mathbb{R}^n \to\mathbb{R}^n\), \(g :\mathbb{R}^n \times\mathbb{R}^p \to\mathbb{R}^n\) are Lipschitz functionals mapping, \(f(0,0) = 0\), \(g(0) = 0\), \(w(t)\) is a standard \(R^n\)-valued Wiener process. He obtains sufficient conditions for the existence of stabilizing feedback laws that are smooth, except possibly at the equilibrium point of the system using stochastic Lyapunov techniques. The main results are obtained for the class of nonlinear stochastic systems that are affine in the control. Moreover, a stabilization result is also proved for a class of stochastic bilinear systems and for some nonlinear stochastic differential systems by means of a slight extension of the stochastic Jurdjevic-Quinn theorem obtained previously. In the last section, the necessary and sufficient conditions for the dynamic stabilization of control nonlinear stochastic systems are given.

Reviewer: Leslaw Socha (Katowice)

MSC:

93E15	Stochastic stability in control theory
93D30	Lyapunov and storage functions
60H10	Stochastic ordinary differential equations (aspects of stochastic analysis)
93D15	Stabilization of systems by feedback
93D20	Asymptotic stability in control theory
93C10	Nonlinear systems in control theory

Keywords:

stabilizing feedback; stochastic Lyapunov techniques; nonlinear; stabilization; bilinear systems; Jurdjevic-Quinn