关于周期切换律的稳定性并不意味着任意切换下的全局稳定性
Stability with respect to periodic switching laws does not imply global stability under arbitrary switching
AI总结：
本文构造四阶四状态线性切换系统，证明周期切换律下的指数稳定性不能保证任意切换下的全局一致稳定，否定高维情形下的相关猜想。
摘要：
R. Shorten、F. Wirth、O. Mason、K. Wulff 和 C. King 曾提出如下问题：如果一个线性切换系统已知在任意周期切换律下，每条轨迹都指数收敛到原点，那么该系统是否必然在任意切换下全局一致稳定？此前，对于二阶和三阶线性切换系统，已有对该问题的肯定回答被宣布。我们通过构造一个具有四个切换状态的四阶线性切换系统，对该问题在所有更高阶情形下给出否定回答：该系统并非一致指数稳定，但在任意周期切换律下，每条轨迹都指数收敛到原点。我们非正式地论证，在足够高的维度下，具有这种性质组合的正线性系统很可能存在。
英文摘要：
R. Shorten, F. Wirth, O. Mason, K. Wulff and C. King have asked whether a linear switched system is guaranteed to be globally uniformly stable under arbitrary switching if it is known that every trajectory induced by a periodic switching law converges exponentially to the origin. Positive answers to this question have previously been announced for linear switched systems of order two and three. We answer this question negatively in all higher orders by constructing a fourth-order linear switched system with four switching states which is not uniformly exponentially stable but which has the property that every trajectory defined by a periodic switching law converges exponentially to the origin. We argue informally that positive linear systems with this combination of properties are likely to exist in sufficiently high dimensions.
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