Theory of Hybrid Systems: Deterministic and Stochastic by Mohamad S. Alwan & Xinzhi Liu

Author:Mohamad S. Alwan & Xinzhi Liu
Language: eng
Format: epub
ISBN: 9789811080463
Publisher: Springer Singapore

Namely, which implies that

where the first inequality is from (4.8). With the aid of Theorem 4.3, we have

This is a contradiction. It turns out that is uniformly stable in m.s. This completes the proof of (i).

(ii) The assertion of this part can be proved easily; thus, it is left here as an exercise.

Remark 4.2

Assumptions (i) (or (ii)) in Corollary 4.2 is made to ensure that the Lyapunov function V is nonincreasing (or strictly decreasing) in main, which in turn implies that the continuous system is m.s. uniformly stable (or asymptotically stable). To guarantee that the overall behavior of V decreases for all time (including the impulse moments), we assume that V is nonincreasing at these moments, because, otherwise, the reduction of V may not compensate the jump increases. This condition is summarized in (4.7).

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