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:在这项工作中,我们证明方程式的波克纳整体代表方程式(ARE)在对系数和半组运算商没有任何紧凑性假设的情况下被充分定位。通过这一结果,我们可以确定,在某些假设下,加列金近似值的解决方案与无限的多元解决方案相融合。我们进一步应用这一总体结果来证明,ARE的有限元素近似值对于在 $H ⁇ 1 (\ Omega)\timedled wave 半组工艺中,对于在 $H ⁇ 1 (\ Omega)\time L ⁇ 2(\\ omega)$规范中微弱的波峰值最佳控制系统的功能增益最佳趋同率在数字示例中均有说明。