Numerical approximation of distributed optimal control problems governed by linear thermo-poroelasticity
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.