A characterization of solutions for general copositive quadratic Lyapunov inequalities

This article provides answers to an open question raised in [1] with regard to checking existence of a solution for general copositive Lyapunov inequalities. We consider homogeneous LTI systems that preserve a proper cone C.* A necessary and sufficient condition for stability of such a system is the existence of a quadratic Lyapunov solution for the associated copositive Lyapunov inequality. This article provides a computationally efficient alternative necessary and sufficient condition for stability of the cone-invariant LTI system, in which geometric algebraic conditions for the stability of an equilibrium state are established from the concepts of dual and polar cones. The conditions are polynomial-time verifiable, provided C is a proper cone in a Hilbert space and has a polynomial-time evaluable self-concordant barrier function. We show that the feasible solutions of those conditions can be used to characterize the extreme rays of the set of solutions for copositive Lyapunov inequalities.