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Feedback stabilization and adaptive stabilization of stochastic nonlinear systems by the control Lyapunov function. (English) Zbl 1222.60039
Summary: Our aims of this paper are twofold: On one hand, we study the asymptotic stability in probability of stochastic differential system, when both the drift and diffusion terms are affine in the control. We derive sufficient conditions for the existence of control Lyapunov functions (CLFs) leading to the existence of stabilizing feedback laws which are smooth, except possibly at the equilibrium state. On the other hand, we consider the previous systems with an unknown constant parameters in the drift and introduce the concept of an adaptive CLF for stochastic system and use the stochastic version of Florchinger’s control law to design an adaptive controller. In this framework, the problem of adaptive stabilization of nonlinear stochastic system is reduced to the problem of non-adaptive stabilization of a modified system.

60H10Stochastic ordinary differential equations
93C10Nonlinear control systems
93D05Lyapunov and other classical stabilities of control systems
93D15Stabilization of systems by feedback
93D21Adaptive or robust stabilization
93E15Stochastic stability
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