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Finite time stability and stabilization of a class of continuous systems. (English) Zbl 1131.93043
The paper deals with systems of ODE’s in finite-dimensional space having unique solutions in forward time. It discusses finite-time stability, i.e., the strong version of asymptotic stability when the systems reaches the equilibrium point. The contains two main results. The first result shows that, under appropriate assumptions, existence of a Lyapunov function plus a certain integral property are necessary and sufficient for finite time stability of a system of ODE’s. The second result shows that a control-affine system admits a feedback making it finite time stable if and only if there exists a control Lyapunov function satisfying a certain differential inequality. In that case, the finite time stabilizing feedback is given in explicit form.

MSC:
93D15 Stabilization of systems by feedback
93D30 Lyapunov and storage functions
93C10 Nonlinear systems in control theory
93C15 Control/observation systems governed by ordinary differential equations
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