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Lie-algebraic stability criteria for switched systems. (English) Zbl 0995.93064

The authors study a switched system and its exponential stability under the condition that the Lie algebra generated by given matrices can be decomposed into a sum of a solvable ideal and a subalgebra with a compact Lie group; the local stability result for a nonlinear switched system is also established. Infinitely fast switching, which calls for a concept of generalized solution, is not considered in this paper, and jumps of the solution at the switching instant are not allowed.

The main algebraic problem treated here can be summerized as follows:

Given a matrix Lie algebra that contains the identity matrix, is it true that any set of stable generators gives rise to a switched system that is exponentially stable?

The authors state and prove four theorems.

The paper is recommended for scientists working in control theory and using Lie algebraic methods for solving problems.

MSC:
93D20Asymptotic stability of control systems
93B12Variable structure systems
93B29Differential-geometric methods in systems theory (MSC2000)
17B30Solvable, nilpotent Lie (super)algebras