Weak stability of switching dynamical systems and fast computation of the p-radius of matrices

The stability of a switching linear dynamical system is ruled by the so-called joint spectral radius of the set of matrices characterizing the dynamical system. In some situations, the system is not stable in the classical sense, but might still be stable in a weaker meaning. We introduce the new notion of weak stability or L_p-stability of a switched dynamical system based on the so-called p-radius of the set of matrices. The p-radius characterizes the average rate of growth of norms of matrices in a multiplicative semigroup. This quantity has found several applications in the recent years. We analyze the computability of this quantity, and we describe a series of approximations that converge to the p-radius with a priori computable accuracy. For nonnegative matrices, this gives efficient approximation schemes for the p-radius computation. We finally show the efficiency of our methods on several practical examples.