The length of switching intervals of a stable linear system

The linear switching system is a system of ODE with the time-dependent matrix taking values from a given control matrix set. The system is (asymptotically) stable if all its trajectories tend to zero for every control function. We consider possible mode-dependent restrictions on the lengths of switching intervals which keeps the stability of the system. When the stability of trajectories with short switching intervals implies the stability of all trajectories? To answer this question we introduce the concept of "cut tail points" of linear operators and study them by the convex analysis tools. We reduce the problem to the construction of Chebyshev-type exponential polynomials, for which we derive an algorithm and present the corresponding numerical results.