具有非紧半群生成元的代数Riccati方程的逼近
Approximation of Algebraic Riccati Equations with Generators of Noncompact Semigroups
AI总结：
本文研究非紧半群生成元对应的代数Riccati方程（ARE），证明其Bochner积分表示无紧性假设仍适定，推导Galerkin逼近解收敛性，验证有限元逼近对弱阻尼波动过程的最优性，并通过数值算例证实泛函增益的最优收敛速率。
摘要：
本文中，我们证明了代数Riccati方程（ARE）的Bochner积分表示在系数和半群算子无任何紧性假设的情况下是适定的。基于该结果，我们进一步确定，在若干假设下，这些方程的Galerkin逼近解收敛于无穷维解。更进一步，我们应用这一一般性结果证明，对于弱阻尼波动半群过程，ARE的有限元逼近在$H^1(Ω) \ imes L^2(Ω)$范数下是最优的。数值算例验证了弱阻尼波动最优控制系统的泛函增益在$H^1(Ω) \ imes L^2(Ω)$和$L^2(Ω)\ imes L^2(Ω)$两种范数下均具有最优收敛速率。
英文摘要：
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 $H^1(Ω) \times L^2(Ω)$ norm. Optimal convergence rates of the functional gain for a weakly damped wave optimal control system in both the $H^1(Ω) \times L^2(Ω)$ and $L^2(Ω)\times L^2(Ω)$ norms are demonstrated in the numerical examples.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
