How to Cut the Tail Right?

Several problems in the theory of linear dynamic systems lead to the same question: given a linear differential equation with a constant matrix $x'(t) = A x(t)$, $x(0) = x_0$, find the time $T$ when its trajectory $x(t)$ enters inside its (symmetrized) convex hull. In this case, the entire "tail" $\{x(t): t\ge T\}$ will not leave its bounds. The answer depends only on the spectrum of the matrix $A$ and is given in terms of exponential best approximation polynomials. These are polynomials not in powers of the variable $t$, but in a system of complex exponentials. This system is not Chebyshev, and practically nothing is known about approximations by such systems. Nevertheless, it is possible to define the concept of "generalized alternance" for them and construct an efficient method for computing the nearest polynomial. We then consider an application to the stability of linear systems with switching: $x'(t) = A(t)x(t)$, where the matrix $A(t)$ is a controllable parameter taking values in a compact set $U$. It will be shown that if the system is stable under the condition that the lengths of all switching intervals do not exceed $T$, then it will remain stable without this restriction.