On bounds of the solution to the parametric Lyapunov equations with applications

The parametric Lyapunov equation (PLE) plays a fundamental role in various control problems for both linear and nonlinear systems, and it can be used to determine feedback gains and the corresponding Lyapunov function. This paper presents estimates for upper and lower bounds on the eigenvalues of the solution to the PLE. These estimates are applicable to both multiple-input and single-input linear systems, effectively circumventing cumbersome matrix integrals. The upper bound is expressed as a polynomial function of the parameter in the PLE, while the lower bound is represented by a rational function of the parameter in the PLE. Notably, as the parameter approaches the left critical value or tends toward infinity, these estimates align with existing results regarding the properties of the solution, thereby confirming the tightness of the bounds. The derived estimations are further illustrated through two numerical examples. Finally, the established upper and lower bounds are utilized to enhance the results related to overshoot estimation in pole placement.