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 $H^1(\Omega) \times L^2(\Omega)$ norm. Optimal convergence rates of the functional gain for a weakly damped wave optimal control system in both the $H^1(\Omega) \times L^2(\Omega)$ and $L^2(\Omega)\times L^2(\Omega)$ norms are demonstrated in the numerical examples.

翻译：本文证明了代数Riccati方程(ARE)的Bochner积分表示在不需对系数和半群算子施加任何紧致性假设的条件下是适定的。基于这一结果，我们进一步确定了在适当假设下，这些方程的Galerkin逼近解收敛到无穷维解。进而，我们应用该一般性结果证明了弱阻尼波动半群过程在$H^1(\Omega) \times L^2(\Omega)$范数下ARE的有限元逼近具有最优性。数值算例中展示了弱阻尼波动最优控制系统在$H^1(\Omega) \times L^2(\Omega)$和$L^2(\Omega)\times L^2(\Omega)$两种范数下功能增益的最优收敛速率。