论文标题
部分可观测时空混沌系统的无模型预测
Space-time finite element discretization of parabolic optimal control problems with energy regularization
论文作者
论文摘要
我们分析了分布式抛物线抛物线最佳控制问题的数值解决方案在bochner空间中$ l^2（0，t; h^{ - 1}（ω））$。通过二元性，可以通过解决椭圆形准平台边界值问题的解决方案来评估相关规范。消除控件时，我们最终将使用降低的最优系统，这不过是耦合的前后背部原始方程和伴随方程的变化公式。使用Babuška的定理，我们在连续的情况下证明了独特的解决性。此外，我们为任何符合的时空有限元离散化建立了离散的INF-SUP条件，从而产生准最佳离散误差估计。各种数值示例证实了理论发现。我们强调的是，能量正则化导致更局部的控制，并具有不连续目标功能的更清晰的轮廓，这可以通过与$ l^2 $正则化的比较和稀疏的最佳控制方法进行了比较。
We analyze space-time finite element methods for the numerical solution of distributed parabolic optimal control problems with energy regularization in the Bochner space $L^2(0,T;H^{-1}(Ω))$. By duality, the related norm can be evaluated by means of the solution of an elliptic quasi-stationary boundary value problem. When eliminating the control, we end up with the reduced optimality system that is nothing but the variational formulation of the coupled forward-backward primal and adjoint equations. Using Babuška's theorem, we prove unique solvability in the continuous case. Furthermore, we establish the discrete inf-sup condition for any conforming space-time finite element discretization yielding quasi-optimal discretization error estimates. Various numerical examples confirm the theoretical findings. We emphasize that the energy regularization results in a more localized control with sharper contours for discontinuous target functions, which is demonstrated by a comparison with an $L^2$ regularization and with a sparse optimal control approach.
