Adaptive Finite Element Methods for Optimal Control Problems
Abstract: In this thesis we study the numerical solution of optimal control problems. The problems considered consist of a system of differential equations, the state equations, which are governed by a control variable. The goal is to determine the states and controls which minimize a given cost functional. The numerical method in this work is based on an indirect approach, which means that necessary conditions for optimality are first derived and then solved numerically, in our case by a finite element method. The optimality conditions are derived using Lagrange's method in the calculus of variations resulting in a boundary value problem for a system of differential/algebraic equations. These equations are discretized by a finite element method. The advantage of the finite element method is the possibility to use functional analysis to derive error estimates and in this work this is used to prove computable a posteriori error estimates. The error estimates are derived in the framework of dual weighted residuals which is well suited for optimal control problems since it is formulated within the Lagrange framework. Using an indirect method combined with an a posteriori error estimate makes it possible to implement adaptive finite element methods where the refinement of the computational mesh is automated. We have implemented such adaptive finite element methods for quadratic/linear optimal control problems, fully nonlinear problems, and for problems with inequality constraints on controls and states.
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