If a linear switching system with frequent switches is stable, will it be stable under arbitrary switches? In general, the answer is negative. Nevertheless, this question can be answered in an explicit form for any concrete system. This is done by finding the mode-dependent critical lengths of switching intervals after which any enlargement does not influence the stability. The solution is given in terms of the exponential polynomials of least deviation from zero on a segment (“Chebyshev-like” polynomials). By proving several theoretical results on exponential polynomial approximation we derive an algorithm for finding such polynomials and for computing the critical switching time. The convergence of the algorithm is estimated and numerical results are provided.

中文翻译：

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开关间隔的长度如何影响动力系统的稳定性？


如果频繁开关的线性开关系统是稳定的，那么在任意开关下它会稳定吗？一般来说，答案是否定的。然而，对于任何具体系统，这个问题都可以以明确的形式回答。这是通过找到开关间隔的模式相关临界长度来完成的，之后任何放大都不会影响稳定性。解是根据线段上与零的最小偏差的指数多项式（“类似切比雪夫”多项式）给出的。通过证明指数多项式近似的几个理论结果，我们推导出了一种用于查找此类多项式和计算临界开关时间的算法。估计算法的收敛性并提供数值结果。