线性热-孔隙弹性控制的最优控制问题的数值逼近
Numerical approximation of distributed optimal control problems governed by linear thermo-poroelasticity
- Baylor University（贝勒大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文针对线性热-孔隙弹性最优控制问题，提出三场对称公式，证明适定性与最优解存在唯一性，推导伴随系统，并给出全离散格式的误差估计与数值验证。
AI中文摘要：
本文提出并分析了一种新颖的三场对称公式，用于求解由线性热-孔隙弹性控制的最优控制问题。状态变量为固体位移$\boldsymbol{u}$、流体压力$p$和温度$\theta$，分布式流体源$m_p$和分布式热源$m_\theta$均作为控制变量。我们证明了对称微分公式的适定性，建立了最优控制对$(\bar{m}_p,\bar{m}_\theta)$的存在唯一性，并通过耦合的三场伴随系统推导出一阶必要最优性条件。对于基于dG(0)时间离散的全离散格式，我们对两个控制均采用变分离散方法，并推导出状态、伴随和控制误差的阶为$\mathcal{O}(h^s + \Delta t)$的先验误差估计。我们还提供了制造解实验，验证了全离散最优性系统，并展示了我们分析所涵盖的存储退化情形。
英文摘要：
In this paper, we propose and analyze a novel three-field symmetric formulation for optimal control problems governed by linear thermo-poroelasticity. The state variables are the solid displacement $\boldsymbol{u}$, the fluid pressure $p$, and the temperature $θ$, and both the distributed fluid source $m_p$ and the distributed heat source $m_θ$ serve as control variables. We prove well-posedness of the symmetric differentiated formulation, establish the existence and uniqueness of an optimal control pair $(\bar{m}_p,\bar{m}_θ)$, and derive the first-order necessary optimality conditions via a coupled three-field adjoint system. For the fully discrete scheme based on dG(0) time discretization, we employ a variational discretization approach for both controls and derive a priori error estimates of order $\mathcal{O}(h^s + Δt)$ for the state, adjoint, and control errors. We also present manufactured-solution experiments that validate the fully discrete optimality system and illustrate the storage-degenerate regime covered by our analysis.
