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Control Lyapunov function for feedback stabilization of affine in the control stochastic time-varying systems. (English) Zbl 1238.93079
Necessary and sufficient conditions of Lyapunov’s function type are established for a feedback stabilization problem of nonuniform in time global asymptotic stability in probability [see, e.g., I. Karafyllis and J. Tsinias, “A converse Lyapunov theorem for nonuniform in time global asymptotic stability and its application to feedback stabilization”, SIAM J. Control Optimization 42, No. 3, 936–965 (2003; Zbl 1049.93073)] of affine stochastic time-varying systems with control parameters $$u^z$$ and with $$v$$ as input: $$dx= f(t,x,v) dt+ \sum_{z=1}^{z=p} g_z (t,x)u^z dt + \sum_{k=1}^{k=m}h_k (t,x) dw, x \in R^n, v \in R^l,u \in R^p,t \geq 0$$. Based on the Lyapunov function associated to the system an explicit formula for a time-varying feedback stabilizer is proposed together with the remark that, even for a class of autonomous systems, it is possible to achieve nonuniform in time globally asymptotic stabilization in probability by smooth time-varying feedback, although a smooth time-independent feedback exhibiting uniform in time stabilization does not exist. A numerical example is provided.

##### MSC:
 93D15 Stabilization of systems by feedback 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 93C10 Nonlinear systems in control theory 93D05 Lyapunov and other classical stabilities (Lagrange, Poisson, $$L^p, l^p$$, etc.) in control theory 93D21 Adaptive or robust stabilization 93E15 Stochastic stability in control theory
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