Optimal control for an elliptic system with convex polygonal control constraints

In recent years, optimal control problems governed by partial differential equations with pointwise constraints on the controls have received a considerable amount of attention. The challenge consists in finding efficient numerical methods in spite of the nonsmoothness introduced by pointwise inequality constraints. Most investigations so far have considered the case of scalar-valued control with simple constraints (e.g. unilateral or bilateral constraints). For vector-valued controls, which appear, for example, in the context of the Navier–Stokes equations, the constraints that were investigated were of a box type. For scalar-valued controls with unilateral or bilateral constraints, the numerical realization of optimal control problems can advantageously be performed with the semismooth Newton method or, equivalently, by the primal–dual active set method. We refer to Hintermuller et al. (2003), Ito & Kunisch (2004), De Los Reyes & Kunisch (2005), Hintermuller & Hinze (2006), Ito & Kunisch (2008) and Ulbrich (2011) and the references cited therein. An alternative approach is based on interior point methods. They were analysed, for example, in Schiela & Weiser (2008) and Weiser et al. (2008). The semismooth Newton method is different from penalty—or barrier—function methods which rely on penalty—or barrier— parameters, respectively. It was proven to be locally superlinearly convergent, and for unilaterally constrained problems to be globally convergent; cf. Hintermuller et al. (2003). For vector-valued controls, there are only a few papers that deal with the case of general convex control constraints. In Wachsmuth (2006a,b) and Wachsmuth (2007), the first-order necessary and the second-order sufficient optimality conditions for general convex constraints are obtained. In Wachsmuth (2006b), a numerical procedure is proposed which, however, does not concretely exploit the structure of