ИСТИНА

Linear switching dynamical systems are systems of ODE x’(t) = A(t)x(t) with the matrix-valued function A(t) that takes values from a given control set U. This is a linear system with a matrix control. The system is stable if all its trajectories tends to zero. The stability problem has been studied in great details starting with pioneering works of Molchanov, Pyatnicky, Opoitsev, etc., due to many engineering applications. Even for systems with two retinal matrices in U, the stability problem is in general algorithmically undecidable (Blondel, Tsitsiclis, 2000). It can be solved approximately by the Lyapunov function, which diverges along every trajectory. It is known that every stable system does possess a Lyapunov norm, which is, moreover, convex, i.e., a Lyapunov norm. Methods of its approximate computations has been addressed in many works, including polyhedral (piecewise linear) approximations. While the general problem of approximating a unit ball of a Banach space by polyhedral is notoriously hard, this problem can be efficiently solved for Lyapunov norms. We suggest an iterative method and estimate its precision and rate of approximation. Some of theoretical results are obtained by applying the exponential Chebyshev polynomials and an exponential version of Markov-Bernstein inequality. We illustrate our results by numerical examples and formulate some open problems.