Boundary optimal control problems with inequality constraints
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In my work, I will introduce and analyze different approaches to overcome some issues in computing fractional Sobolev norms in the context of PDE-constrained boundary optimal control problems with Dirichlet control. Dirichlet boundary optimal control problems are perhaps the most interesting class of optimal control problems constrained by partial differential equations (PDEs). In fact, the possibility of controlling the behaviour of a physical system may often take place only by changing the values of certain quantities at the boundary of the domain, especially when the interior of the physical system is not accessible and when no physical mechanism can be triggered inside the domain from the outside. A first approach that is considered in this work keeps the control function on the boundary by replacing fractional norms with integer norms. This approach is a simple way to overcome the need of explicitly computing fractional norms. Then, two other approaches are proposed that make use of lifting functions of the boundary controls. These lifting functions become a new class of control functions in the statement of the optimal control problem. They can be chosen to be defi ned either inside the original domain of the state problem or outside of it. We compare these two lifting approaches with the first approach on the boundary. Another goal of this research is to add control inequality constraints to all the above approaches. In order to deal with these, the Primal-Dual Active Set method is used. The optimality systems arising from first-order necessary conditions are discretized using the finite element method. Numerical results are presented to compare the convergence rate, computational cost, and accuracy of each optimal control problem.