  We analyse the problem of stability of a continuous time linear switching
system (LSS) versus the stability of its Euler discretization. It is well-known
that the existence of a positive {\tau} for which the corresponding discrete
time system with step size {\tau} is stable implies the stability of LSS. Our
main goal is to obtain a converse statement, that is, to estimate the
discretization step size {\tau} > 0 up to a given accuracy {\epsilon} > 0. This
leads to a method of deciding the stability of continuous time LSS with a
guaranteed accuracy. As the first step, we solve this problem for matrices with
real spectrum and conjecture that our method stays valid for the general case.
Our approach is based on Markov-Bernstein type inequalities for systems of
exponents. We obtain universal estimates for sharp constants in those
inequalities. Our work provides the first estimate of the computational cost of
the stability problem for continuous-time LSS (though restricted to the
real-spectrum case).
