PDE-Constrained Optimal Control Problems

My research in this area focuses on PDE-constrained optimal control problems, where the inherent constraints demand specialized numerical treatment.  

The goal is to design methods that balance efficiency with structure-preserving stability, ensuring accurate solutions even under challenging conditions such as pointwise bounds and convection-dominated regimes.

C0 Weak Galerkin Methods for Optimal Control

In a recent work, we introduced a C0 weak Galerkin method for PDE-constrained optimal control problems with general tracking and pointwise state constraints. Designed for efficiency, this scheme allows elementwise stiffness matrix assembly, eliminates penalty parameters, and supports scalable parallel solvers through additive Schwarz preconditioning, all while maintaining optimal convergence order.

Monotonicity Property in Optimal Control

My current work develops a monotone finite element scheme based on edge-averaged techniques that ensures the discrete optimal state preserves the desired-state bounds of the continuous problem. By enforcing a discrete maximum principle, this scheme eliminates spurious oscillations in convection-dominated regimes, providing both stability and physical consistency.

Related Publications

• S. Jeong, S. Lee, and S. Liu. A monotone finite element scheme preserving desired-state bounds in convection-dominated optimal control problems. (manuscript nearly completed, to be submitted).

• S. Jeong, S. Lee, and K. Wang. A C0 weak Galerkin method with preconditioning for optimal control problems with general tracking and pointwise state constraints. Submitted (2025).