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', Submitted, arXiv:2506.17619 (2025)