Analysis and Simulation of Optimal Control for a Two-Time-Scale Fractional Advection-Diffusion-Reaction Equation with Space-Time-Dependent Order and Coefficients

Multiscale Modeling &Simulation, Volume 21, Issue 4, Page 1690-1716, December 2023.
Abstract. We investigate an optimal control model with pointwise constraints governed by a two-time-scale time-fractional advection-diffusion-reaction equation with space-time-dependent fractional order and coefficients, which describes, e.g., the contaminant in groundwater under various transport scales or miscible displacement of hydrocarbon by injected fluid through heterogeneous porous media. To accommodate for the effects of complex fractional order and coefficients, an auxiliary equation method is proposed, along with the Fredholm alternative for compact operators, to analyze the well-posedness of the state equation. Additionally, a bootstrapping argument is utilized to progressively improve the solution regularity through a carefully designed pathway, leading to the maximal regularity estimates. Subsequently, we analyze the adjoint equation derived from the first-order optimality condition, which requires more subtle treatments due to the presence of hidden-memory variable-order fractional operators. Based on these findings, we ultimately analyze the well-posedness, first-order optimality conditions and maximal regularity estimates for the optimal control problem, and we conduct numerical experiments to investigate its behavior in potential applications.