Lyapunov exponent and joint spectral radius: some known and new results

The logarithm of joint spectral radius of a set of matrices coincides with Lyapunov exponent of corresponding linear inclusions. Main results about Lyapunov exponents of discrete time and continuous time linear inclusions are presented. They include the existence of extremal norm; relations between Lyapunov indices of dual inclusions; maximum principle for linear inclusions; algebraic criteria for stability of linear inclusions; algorithm to find out the sign of Lyapunov exponents. The main result is extended to linear inclusions with delays. The Aizerman problem for three-ordered time-varying continuous time systems with one nonlinearity is solved. The Perron-Frobenius theorem is extended for three-ordered continuous time linear inclusions.