The mathematical theory of optimal control has rapidly developed into an important and separate field of applied mathematics. One area of application of this theory lies in aviation and space technology: aspects of optimization come into play whenever the motion of an aircraft or a space vessel has to follow a trajectory that is "optimal" in a sense to be specified.

Optimal control problems consist of a cost functional to be minimized, an initial value problem (or boundary value problem) for a differential equation describing the motion in order to determine the state, a control function, and various constraints that have to be obeyed.

Optimal control allows for the determination of not only the best estimate of parameter values but also the identification of control strategies to optimize system performance. Whether it involves managing groundwater resources, enhancing subsurface energy systems, or mitigating groundwater contamination, optimal control techniques help achieve desired objectives while considering the uncertainties in data.

• J. and S. Lee. 'Optimal control for Darcy’s equation in a heterogeneous porous media.' Applied Numerical Mathematics (2024)

A $C^0$ weak Galerkin method combined with an additive Schwarz preconditioner for solving optimal control problems governed by partial differential equations with general tracking cost functionals and pointwise state constraints. These problems pose significant analytical and numerical challenges due to the presence of fourth-order variational inequalities and the reduced regularity of solutions. Our first contribution is the design of a $C^0$ weak Galerkin method based on globally continuous quadratic Lagrange elements, enabling efficient element-wise stiffness matrix assembly and parameter-free implementation while maintaining accuracy, as supported by a rigorous error analysis. As a second contribution, we develop an additive Schwarz preconditioner tailored to the $C^0$ weak Galerkin method to improve solver performance for the resulting ill-conditioned linear systems. Numerical experiments confirm the effectiveness and robustness of the proposed method and preconditioner for both biharmonic and optimal control problems.

• J., S. Lee and K. Wang. 'A $C^0$ weak Galerkin method with preconditioning for constrained optimal control problems with general tracking', Journal of Scientific Computing (2026)

We propose and analyze a monotone finite element method for an elliptic distributed optimal control problem constrained by a convection-diffusion-reaction equation in the convection-dominated regime. The method is based on the edge-averaged finite element (EAFE) scheme, which is known to preserve the discrete maximum principle for convection-diffusion problems. We show that the EAFE discretization inherits the monotonicity property of the continuous problem and consequently preserves the desired-state bounds at the discrete level, ensuring that the numerical optimal state remains stable and free of nonphysical oscillations. The discrete formulation is analyzed using a combination of the EAFE consistency result and a discrete inf-sup condition, which together guarantee well-posedness and yield the optimal convergence order. Comprehensive numerical experiments are presented to confirm the theoretical findings and to demonstrate the robustness of the proposed scheme in the convection-dominated regimes.

• J., S. Lee, and S. Liu. 'A monotone finite element method for an elliptic distributed optimal control problem with a convection-dominated state equation', ournal of Computational and Applied Mathematics (2026)