Computation of Feasible Parametric Regions for Lyapunov Functions

A computational approach of feasible regions for parameter-dependent Lyapunov functions for linear systems based on real-root-classification (RRC) is proposed in this paper. We refer to such Lyapunov functions as feasible if there exists a parameter set that guarantees the existence of a Lyapunov function. In this sense, the stability of parameter-dependent linear ordinary differential equations (ODEs) and differential algebraic equations (DAEs) is considered. To this end, the existence condition of Lyapunov functions is given in form of an algebraic condition by means of a quantifier elimination (QE) formulation. We apply RRC to solve the QE problem to obtain algebraic necessary and sufficient existence conditions. Several numerical examples demonstrate that this approach produces non-conservative stability regions in the parameter space.