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Stability and stabilization of continuous-time switched linear systems. (English) Zbl 1130.34030

This paper is devoted to the stability and stabilizability of switched systems defined by finite sets of matrices \(A_1,\dots,A_N\). The first result concerns time-switching control laws. Assume that all the matrices \(A_1,\dots,A_N\) are stable, and that each \(A_i\) is associated to a positive definite matrix \(P_i\). Combining continuous time and discrete time Liapunov matrix inequalities, the authors obtain a condition for global asymptotic stability, provided that the switching signal has a sufficiently large dwell time. Therefore, this result is generalized in order to deal with the evaluation of a quadratic cost associated to the system. A third result concerns state-switching control. The authors prove that if a suitable Lyapunov-Metzler matrix inequality is satisfied, then the min-projection strategy \(u=\arg\min x'P_ix\) stabilizes the system in a global sense. The results are illustrated by some examples.

MSC:

34D20 Stability of solutions to ordinary differential equations
93D05 Lyapunov and other classical stabilities (Lagrange, Poisson, \(L^p, l^p\), etc.) in control theory
93D20 Asymptotic stability in control theory
93D15 Stabilization of systems by feedback
34D23 Global stability of solutions to ordinary differential equations
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