Degree

Doctor of Philosophy (PhD)

Department

Mathematics

Document Type

Dissertation

Abstract

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.

Date

4-4-2023

Committee Chair

Brenner, Susanne C.

DOI

10.31390/gradschool_dissertations.6096

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