PSEUDOSPECTRA AND LYAPUNOV STABILITY

Abstract

Classical stability analysis of linear models is based upon eigenvalues. This is most notably true for self-adjoint matrices and operators, which possess a basis of orthogonal eigenvectors. In recent decades, recognition has grown that one must proceed with greater caution when a matrix or operator lacks an orthogonal basis of eigenvectors (non-normal operators). In this paper we use Lyapunov equations and functions to consider perturbed matrices. The basic question is: what choice of Lyapunov function V would allow the largest perturbation and still guarantee that dV/dt is negative definite? By using a sub-optimal strategy and pseudospectra we find that this is determined by testing for the existence of solutions to a related ‘quadratic’ matrix equation - algebraic Riccati equation (ARE).