Continuous dependence of solutions to the Lyapunov equation relative to an elliptic differential operator of order 2

Abstract

We study a Lyapunov-type equationXL -BX = C, whereL, B, and C are given linear operators acting in infinite-dimensional Hilbert spaces, andL is a general elliptic differential operator of order 2 in a bounded domain of a Euclidean space. The equation is derived from a specific parabolic boundary feedback control system. When the coefficients ofL, especially those in the principal part and in the boundary condition, are perturbed, it is shown that the solutionX is strongly continuous relative to the coefficients ofL. The result is applied to the robustness analysis of a feedback stabilization scheme against such perturbations.