How do the lengths of switching intervals influence the stability of a dynamical system?

Abstract: 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.