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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.

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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