Convex optimization methods for computing the Lyapunov exponent of matrices

We introduce a new approach to evaluate the largest Lyapunov exponent of a family of matrices, which describes the stability with probability one of a randomly switching linear system. For positive systems, of particular importance in systems and control, the rate of convergence of our approximation is estimated and the efficiency of the algorithm is demonstrated on particular switching systems of different dimensions. This is done by introducing new upper and lower bounds for the largest Lyapunov exponent of nonnegative matrices. We generalize this approach to arbitrary systems (not necessarily positive), derive a new universal upper bound for the Lyapunov exponent, and show that a similar lower bound, in general, does not exist.