Attractivity for Non Autonomous Equations
Proving attractivity or asymptotic stability is more difficult in the non autonomous case than in the autonomous one, because in the former, one cannot rely in general on any invariance property of the limit sets. The situation is more complex and the asymptotic properties which can be proved with substantially equivalent hypotheses are weaker. In Section 2, we introduce the one-parameter families of Liapunov functions of L. Salvadori: they do not appear in the statements of the theorems, but are powerful tools of demonstration. We use them to prove a significant extension of Matrosov’s theorem II.2.5, yielding a new and interesting characterization of the uniform asymptotic stability of the origin. By the way, in order to grade the difficulties, the origin, instead of a set, is studied in this Section 2. Section 3 gives another useful extension of Matrosov’s theorem.
KeywordsAsymptotic Stability Auxiliary Function Invariance Principle Regularity Theorem Autonomous Equation
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