Is switching systems stability harder for continuous time systems?

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 τ for which the corresponding discrete time system with stepsize τ is stable implies the stability of the LSS. Our main goal is to obtain a converse statement, that is to estimate the discretization stepsize τ > 0 up to a given accuracy ε > 0. This would lead to a method for deciding the stability of a continuous time LSS with a guaranteed accuracy. As a first step towards the solution of this problem, we show that for systems of matrices with real spectrum the parameter τ can be effectively estimated. We prove that in this special case, the discretized system is stable if and only if the Lyapunov exponent of the LSS is smaller than - C τ, where C is an effective constant depending on the system. The proofs are based on applying Markov-Bernstein type inequalities for systems of exponents