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Stabilization of affine in control nonlinear systems. (English) Zbl 0662.93055
Asymptotic and practical stabilization is dealt with of single-input affine control systems by a feedback law. A main result states that a system \(\dot x=f(x)+u g(x)\) is asymptotically and practically stabilizable, if there exists a Lyapunov function V such that \(L_ gV=0\) implies \(L_ fV<0\) \((L_ f\) stands for the Lie derivative with respect to f).
It is not out of the place to mention here that the relevance of a condition like the above for deriving stabilizing feedbacks is well known in the literature, not only for single-input, but also for multi-input control systems [cf. E. B. Lee and L. Markus, Foundations of optimal control theory (1967; Zbl 0159.132) or V. M. Kuntsevich and M. M. Lychak, Synthesis of automatic control systems via Lyapunov functions (Russian) (1977; Zbl 0451.93001)].
Reviewer: K.Tchon

MSC:
93D15 Stabilization of systems by feedback
93C10 Nonlinear systems in control theory
93C15 Control/observation systems governed by ordinary differential equations
93D20 Asymptotic stability in control theory
93D05 Lyapunov and other classical stabilities (Lagrange, Poisson, \(L^p, l^p\), etc.) in control theory
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