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Adding an integrator for the stabilization problem. (English) Zbl 0747.93072
Summary: We study the relationship between the following two properties: P1: The system \(\dot x=f(x,y)\), \(\dot y=v\) is locally asymptotically stabilizable; and P2: The system \(\dot x=f(x,u)\) is locally asymptotically stabilizable; where \(x\in\mathbb{R}^ n\), \(y\in\mathbb{R}\). Dayawansa, Martin and Knowles have proved that these properties are equivalent if the dimension \(n=1\). Here, using the so-called Control Lyapunov function approach, (a) we propose another more constructive and somewhat simpler proof of Dayawansa, Martin and Knowles’s result; (b) we show that, in general, P1 does not imply P2 for dimensions \(n\) larger than 1; (c) we prove that P2 implies P1 if some extra assumptions are added like homogeneity of the system. By using the latter result recursively, we obtain a sufficient condition for the local asymptotic stabilizability of systems in a triangular form.

MSC:
93D20 Asymptotic stability in control theory
93C15 Control/observation systems governed by ordinary differential equations
93D30 Lyapunov and storage functions
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