2023/08/20 by James Chung‐Wai Cheung, Cheung, James · 2 citations
Computer Science · Engineering · Mathematics · #Advanced Mathematical Modeling in Engineering #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical methods in inverse problems #Optimization and Control (math.OC) #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.2308.10130
openalex publication_date 2023/08/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this work, we present an abstract theory for the approximation of operator-valued Riccati equations posed on Hilbert spaces. It is demonstrated here that the error of the approximate solution to the operator-valued Riccati equation is bounded above by the approximation error of the governing semigroup, under the assumption of boundedness on the semigroup and compactness on the coefficient operators. One significant outcome of this result is the correct prediction of optimal convergence for finite element approximations of the operator-valued Riccati equations for when the governing semigroup involves parabolic, as well as hyperbolic processes. We derive the abstract theory for the time-dependent and time-independent operator-valued Riccati equations in the first part of this work. In the second part, we derive optimal error estimates for the finite element approximation of the functional gain associated with model weakly damped wave and thermal LQR control systems. These theoretical claims are then corroborated with computational evidence.