In this work, we present an abstract theory for the approximation of operator-valued Riccati equations posed on Hilbert spaces. It is demonstrated here, under the assumption of compactness in the coefficient operators, that the error of the approximate solution to the operator-valued Riccati equation is bounded above by the approximation error of the governing semigroup. 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 prove optimal convergence rates for the finite element approximation of the functional gain associated with model one-dimensional weakly damped wave and thermal LQR control systems. These theoretical claims are then corroborated with computational evidence.

翻译：本文提出了希尔伯特空间中算子值Riccati方程逼近的抽象理论。研究表明，在系数算子具有紧性的假设下，算子值Riccati方程近似解的上界误差受控于主导半群的逼近误差。该结果的一个重要推论是：当主导半群涉及抛物型及双曲型过程时，可准确预测算子值Riccati方程有限元逼近的最优收敛性。本文第一部分推导了依赖于时间与不依赖于时间的算子值Riccati方程的抽象理论；第二部分则证明了与一维弱阻尼波动系统及热LQR控制系统模型相关的泛函增益的有限元逼近具有最优收敛速率。最后，通过数值计算验证了上述理论结论。