Establishing stability and instability of matrix hypercubes

The problem of establishing stability and instability of a matrix hypercube is considered and some conditions are proposed based on the stability boundary crossing principle and sum of squares relaxations. Specifically, a sufficient and asymptotically necessary condition for the stability is derived which can be checked through convex LMI optimizations. With respect to existing approaches that provide nonconservative conditions, the contribution consists of a significantly smaller computational burden in some cases. Indeed, even among small systems there are cases in which such approaches cannot be used due to the huge computational burden while the proposed technique can be easily applied. Moreover, a sufficient and asymptotically necessary condition for the instability is proposed which amounts to solving a linear algebra problem once that the condition for the stability has been investigated. Such a condition has never been proposed in the literature.