Variational Methods of Optimal Control Theory

Abstract

Since our main interest is related to the mathematical theory of the parameterized optimal control problems (OCP _ε ) for partial differential equations (PDEs), we discuss in this chapter general questions of optimal control theory, different settings of optimal control problems (OCPs) for distributed systems in variable spaces, the direct method of Calculus of Variation, the topological properties of solutions to ill-posed problems, optimality conditions for a wide class of extremal problems in the form of variational inequalities and also questions related to the construction of approximative solutions to different classes of OCPs. We refer to Alekseev, Tikchomirov and Fomin (1987), Egorov (1978), Fursikov (2000), Ivanenko and Mel’nik (1988), Lions (1981), Zgurovsky and Mel’nik (2004), Bonnans and Shapiro (2000), etc., or the textbooks by Tröltzsch (2010) and Mordukhovich (2006) for more results concerning this topic. A recent account on PDE-constrained OCP’s can also be found in Hinze, Pinnau, Ulbrich and Ulbrich (2009).