The Sensitivity of the Stable Lyapunov Equation

Abstract

We present an analysis of the sensitivity of the solution of the Lyapunov equation $A^ * X + XA = - W$, where A is stable. This analysis leads to a spectral norm bound on the relative perturbation of the solution which is optimal for a certain class of estimates and which is essentially equivalent to the Frobenius norm bound obtained from the associated Kronecker product system. The latter bound can be expressed in terms of ${\operatorname{sep}}(A^ * , - A)$ and is known to accurately reflect the sensitivity of the Lyapunov problem, but it is hard to interpret in terms of the original matrix A. In contrast, the spectral norm bound which we derive is directly related to the minimal $L_2$ damping of the dynamical system $\dot z = Az$. Moreover, this dynamical link with the sensitivity problem leads to a new method of systematically investigating the norm behavior of $e^{At} $ as well as providing a wealth of information about control theoretic aspects of $\dot z = Az$, when A is the closed loop state matrix.