Properties of the derivative of the solution to the parametric Lyapunov equation

This paper studies upper and lower bounds for the derivative of the positive definite solution to a parametric Lyapunov equation with respect to its design parameter. Explicit bounds are established in terms of the solution itself, and a main feature is that they hold on the entire admissible parameter domain without any additional restriction on the parameter. In the limiting regimes, the associated coefficients are further refined, and the asymptotically sharp quantities are shown to be determined by zeros of Laguerre polynomials arising from certain auxiliary matrices. Numerical examples illustrate the theoretical results.