Optimal control for an elliptic system with convex polygonal control constraints (Q2841058)

The authors analyze the semismooth Newton method for constraints of polygonal type. It is supposed that \(\Omega\) is a bounded domain in \(\mathbb R^d\), \(d=2,3\), which is either convex or has a \(C^{1,1}\) boundary, and that states and controls vectors map from \(\Omega\) to \(\mathbb R^l\) \((l=2)\).NEWLINENEWLINEThe authors verify that the problem \(\Lambda y = B u\) in \(\Omega\) can be solved by a superlinearly convergent semismooth Newton method. The work contains the existence and uniqueness of the optimal solution, the first-order necessary condition and its equivalent formulations. Moreover, the Newton differentiability for the nonlinear equation representing the optimality system is proved. A semismooth Newton algorithm, together with its convergence analysis is proposed. Finally, a numerical example is presented to depict the superlinear convergence property of the algorithm.NEWLINENEWLINEThe numerical realization is based on a finite difference discretization with respect to a uniform axis-parallel grid. To demonstrate the mesh independence of the algorithm, the authors compute the same example on a series of grids for mesh sizes refined by a factor of \(1/2\).NEWLINENEWLINEThroughout this paper, the standard notation for a Sobolev space is used. The polygonal set \(K\) is taken as the intersection of \(m\) half spaces.NEWLINENEWLINEThe authors present (by introducing Lagrange multipliers) a complementarity system that is equivalent to the pointwise projection.