发表机构
Wentworth Institute of Technology; Florida State University（文沃斯理工学院; 佛罗里达州立大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对非均质多孔介质中带Radon测度追踪和逐点控制约束的Darcy最优控制问题，提出一种单片间断Galerkin框架，采用原始-对偶积极集策略求解，并验证了收敛性和局部质量守恒。
AI 中文摘要
我们研究了一个由非均质多孔介质中的Darcy方程控制的椭圆最优控制问题，其中控制具有逐点盒约束。目标泛函通过Radon测度表述，这使得可以在单一公式中追踪不同维度的观测集（包括点、曲线和子域）上的期望压力状态。状态方程和伴随方程采用对称内罚间断Galerkin方法离散，得到局部质量守恒的近似，该近似对强渗透率间断具有鲁棒性，而控制则用分片常数近似。状态、伴随和控制作为主要未知量保留在单个耦合最优性系统中，并通过原始-对偶积极集策略进行单片求解。我们建立了离散格式的稳定性和适定性，并推导了两个变量的先验$L^2$误差估计。主要困难在于由测度值追踪数据引起的伴随状态正则性降低。分析通过由连续最优状态驱动的中间伴随来控制由此产生的伴随-控制耦合。数值实验证实了点、曲线和子域观测的预测收敛速率，展示了网格无关的原始-对偶积极集迭代次数，并证明了局部质量守恒。
英文摘要
We study an elliptic optimal control problem governed by Darcy's equation in heterogeneous porous media, with pointwise box constraints on the control. The objective functional is formulated via a Radon measure, which allows the desired pressure state to be tracked on observation sets of varying dimension, including points, curves, and subdomains, within a single formulation. The state and adjoint equations are discretized by a symmetric interior penalty discontinuous Galerkin method, yielding a locally mass-conservative approximation that is robust across strong permeability discontinuities, while the control is approximated by piecewise constants. The state, adjoint, and control are retained as primary unknowns in a single coupled optimality system and solved monolithically by a primal-dual active set strategy. We establish stability and well-posedness of the discretization and derive a priori $L^2$ error estimates for both variables. The principal difficulty is the reduced regularity of the adjoint state induced by the measure-valued tracking data. The analysis controls the resulting adjoint-control coupling through an intermediate adjoint driven by the continuous optimal state. Numerical experiments confirm the predicted convergence rates for point, curve, and subdomain observations, exhibit mesh-independent primal-dual active set iteration counts, and demonstrate local mass conservation.
