Abstract
We study the asymptotic stability properties of nonlinear switched systems under the assumption of the existence of a common weak Lyapunov function.   We consider the class of nonchaotic inputs, which generalize the different notions of inputs with dwell-time, and the class of general ones. For each of them we provide some sufficient conditions for asymptotic stability in terms of the geometry of certain sets.
Highlights
This paper is concerned with the asymptotic stability properties of nonlinear switched systems defined by a finite collection {f1, . . . , fp} of smooth vector fields in Rd
In particular in [4, 5, 11] the vector fields are linear, the Lyapunov function is quadratic, and the asymptotic stability properties are closely related to the geometry of the union of some linear subspaces of Rd
We introduce in the same way two geometric subsets of Rd which turn out to be fundamental in the sense that they contain all the limit sets for two classes of inputs
Summary
This paper is concerned with the asymptotic stability properties of nonlinear switched systems defined by a finite collection {f1, . . . , fp} of smooth vector fields in Rd. In particular in [4, 5, 11] the vector fields are linear, the Lyapunov function is quadratic, and the asymptotic stability properties are closely related to the geometry of the union of some linear subspaces of Rd. In the present paper, we introduce in the same way two geometric subsets of Rd which turn out to be fundamental in the sense that they contain all the limit sets for two classes of inputs. They consider only dwell-time inputs and their results are improved by the distinction we introduce between the two categories of sets and inputs.
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