LMI parametrization of Lyapunov Functions for Infinite-Dimensional Systems: A Toolbox

Abstract— In this paper, we present an algorithmic approach to the construction of Lyapunov functions for infinitedimensional systems. This paper unifies and extends many previous results which have appeared in conference and journal format. The unifying principle is that a linear matrix parametrization of operators in Hilbert space inevitably leads to a linear parametrization of positive forms using positive semidefinite matrices via squared representations. For linear systems, these positive forms are defined by positive linear operators and define quadratic Lyapunov functions. For nonlinear systems, the forms are defined by nonlinear operators and will define non-quadratic Lyapunov functions. Special cases of these results include operators defined by multipliers and kernels which are: polynomial; piecewise-polynomial; or semiseparable and apply to systems with delay; multiple spatial domains; or mixed boundary conditions. We also introduce a set of efficient software tools for creating these functionals. Finally, we illustrate the approach with numerical examples.