Mesh-independent convergence of penalty methods applied to optimal control with partial differential equations

Abstract

In this paper, optimal control problems with elliptic state equations and constraints on controls are considered. Also state constraints are briefly discussed. Barrier-penalty methods are applied to treat the occurring restrictions. In the case of finite-dimensional optimization problems, the considered methods have a linear rate of convergence in dependence of the penalty parameter. However, in the case of infinite-dimensional problems, as studied in this article, the direct application of finite-dimensional theory, as given in Grossmann and Zadlo [A general class of penalty/barrier path-following Newton methods for nonlinear programming, Optimization 54 (2005), pp. 161–190], would lead to mesh-dependent order one estimates that deteriorate if the discretization is refined. In this article a first rigorous proof is given for inequality constrained problems that in the case of quadratic penalties a mesh-independence principle holds, i.e. the first-order convergence estimate holds for the continuous problem as well as for discretized problems independently of the discretization step size. The penalty techniques rest upon the control approximate reduction as discussed, e.g. in Grossmann et al. [C. Grossmann, H. Kunz, and R. Meischner, Elliptic control by penalty techniques with control reduction, in System Modeling and Optimization, IFIP Advances in Information and Communication Technology, Vol. 312, Springer, Berlin, 2009, pp. 251–267; M. Hinze, A variational discretization concept in control constrained optimization: the linear-quadratic case, Comput. Optim. Appl. 30 (2005), pp. 45–61]. For the discretization conforming linear element discretization is applied. Some numerical examples illustrate and confirm the theoretical results.