First order numerical algorithms for some optimal control problems with PDE constraints

Abstract

In this thesis, we focus on designing first-order numerical algorithms for some optimal control problems with partial differential equation (PDE) constraints. The first part is focused on some PDE-constrained optimal control problems with additional box or sparsity control constraints. We design some operator splitting type algorithms for these problems, and their common feature is that the PDE constraints and the additional box or sparsity control constraints are treated separately in numerical implementation. In particular, we develop an inexact Uzawa method and an inexact alternating direction method of multipliers for elliptic and parabolic optimal control problems with box control constraints, respectively; and a primal-dual hybrid gradient algorithm for a sparse optimal control problem with diffusion-advection equation constraint. The second part is focused on the bilinear optimal control of an advection-reaction-diffusion system, where the control variable arises as the velocity field in the advection term. For this problem, we prove the existence of optimal controls, derive the first-order optimality conditions in general settings, and design a nested conjugate gradient method. These new algorithms are designed in accordance with the structures of the problems under consideration, and they can be implemented easily. Their efficiency is promisingly validated by the results of some preliminary numerical experiments and convergence properties are also studied.