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Zbl 0995.93064
Agrachev, Andrei A.; Liberzon, Daniel
Lie-algebraic stability criteria for switched systems. (English)
[J] SIAM J. Control Optimization 40, No.1, 253-269 (2001). ISSN 0363-0129; ISSN 1095-7138/e

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.\par The main algebraic problem treated here can be summerized as follows:\par 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?\par The authors state and prove four theorems.\par The paper is recommended for scientists working in control theory and using Lie algebraic methods for solving problems.
[Constantin Vârsan (Bucureşti)]

MSC 2000:: *93D20 Asymptotic stability of control systems
93B12 Variable structure systems
93B29 Differential-geometric methods in systems theory
17B30 Solvable, nilpotent Lie algebras

Keywords: switched system; asymptotic stability; Lie algebra

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