Stability testing of matrix polytopes

Motivated by questions in robust control and switched linear dynamical systems, we consider the problem checking whether every element of a polytope of n×n matrices A is stable. We show that this can be done in polynomial-time in n when the number of extreme points of A is constant, but becomes NP-Hard when the number of extreme points grows as Θ(n). This result has two useful corollaries: (i) for the case when A is a line, we give a stability-testing algorithm considerably faster than the best currently known algorithms (ii) we show that verifying the absolute asymptotic stability of a continuous-time switched linear system with n - 1 n × n matrices A_i satisfying 0 Υ A_i + A_i^T is NP-hard.