2022/09/11 by James Chung‐Wai Cheung, Cheung, James
Computer Science · Engineering · Mathematics · #Advanced Mathematical Modeling in Engineering #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical methods for differential equations #Optimization and Control (math.OC) #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.2209.04769
openalex publication_date 2022/09/11 · openalex created_date 2022/09/14 · openalex updated_date 2026/07/28
In this work, we demonstrate that the Bochner integral representation of the Algebraic Riccati Equations (ARE) are well-posed without any compactness assumptions on the coefficient and semigroup operators. From this result, we then are able to determine that, under some assumptions, the solution to the Galerkin approximations to these equations are convergent to the infinite dimensional solution. Going further, we apply this general result to demonstrate that the finite element approximation to the ARE are optimal for weakly damped wave semigroup processes in the H1(Ω) × L2(Ω) norm. Optimal convergence rates of the functional gain for a weakly damped wave optimal control system in both the H1(Ω) × L2(Ω) and L2(Ω)× L2(Ω) norms are demonstrated in the numerical examples.