The Minkowski–Lyapunov equation for linear dynamics: Theoretical foundations

Abstract

We consider the Lyapunov equation for the linear dynamics, which arises naturally when one seeks for a Lyapunov function with a uniform, exact decrease. In this setting, a solution to the Lyapunov equation has been characterized only for quadratic Lyapunov functions. We demonstrate that the Lyapunov equation is a well-posed equation for strictly stable dynamics and a much more general class of Lyapunov functions specified via Minkowski functions of proper C-sets, which include Euclidean and weighted Euclidean vector norms, polytopic and weighted polytopic (1,∞)-vector norms as well as vector semi-norms induced by the Minkowski functions of proper C-sets. Furthermore, we establish that the Lyapunov equation admits a basic solution, i.e., the unique solution within the class of Minkowski functions associated with proper C-sets. Finally, we provide a characterization of the lower and upper approximations of the basic solution that converge pointwise and compactly to it, while, in addition, the upper approximations satisfy the classical Lyapunov inequality.