Doctor of Philosophy (PhD)
This dissertation concerns a linear-quadratic elliptic distributed optimal control problem with pointwise state constraints in two spatial dimensions, where the cost function tracks the state at points, curves and regions of a domain.
First we explore the elliptic optimal control problem subject to pointwise control constraints. This problem is reduced into a problem that only involves the control. The solution of the reduced problem is characterized by a variational inequality. Then we introduce the elliptic optimal control problem with general tracking and pointwise state constraints. Here we reformulate the optimal control problem into a problem that only involves the state, which is equivalent to a fourth order variational inequality. We derive the Karush-Kuhn-Tucker conditions from the variational inequality and find the regularity result of the solution.
The reduced minimization problem is solved by a C0 interior penalty method. The C0 interior penalty methods are very effective for fourth order problems and much simpler than C1 finite element methods. The discrete problem is a quadratic program with simple box constraints which can be solved efficiently by the primal-dual active set algorithm. We provide a convergence analysis and demonstrate the performance of the method through several numerical experiments.
Jeong, SeongHee, "Finite Element Methods for Elliptic Optimal Control Problems with General Tracking" (2023). LSU Doctoral Dissertations. 6096.
Brenner, Susanne C.
Available for download on Wednesday, April 03, 2024