A linear-quadratic elliptic control problem with pointwise box constraints on the state is considered. The state-constraints are treated by a Lavrentiev type regularization. It is shown that the Lagrange multiplier associated with the regularized state-constraints are functions in L 2 . Moreover, the convergence of the regularized controls is proven for regularization parameter tending to zero. To solve the problem numerically, an interior point method and a primal-dual active set strategy are implemented and treated in function space.
Abstract. The paper addresses primal interior point method for state-constrained PDE optimal control problems. By a Lavrentiev regularization, the state constraint is transformed to a mixed control-state constraint with bounded Lagrange multiplier. Existence and convergence of the central path are established, and linear convergence of a short-step pathfollowing method is shown. The behaviour of the method is are demonstrated by numerical examples.Key words. interior point method, function space, optimal control, mixed control-state constraints, Lavrentiev regularization AMS subject classifications. 90C51, 49J20, 65M151. Introduction. The application of interior point methods to optimal control problems has received a good deal of interest in the past years. This parallels the fast development of numerical methods in large scale optimization where interior point methods play an important role. In the context of PDE control, their performance was carefully tested by Haddoux et al. [8] for discretized versions of elliptic control problems. Similarly, Grund and Rösch [7] considered different codes of interior point methods for elliptic control problems with pointwise state-constraints. Trust-region interior point techniques have been considered by M. Ulbrich, S. Ulbrich and M. Heinkenschloss in [15] for the optimal control of semilinear parabolic equations in a function space setting. Moreover, affine-scaling interior-point methods were presented for semilinear parabolic boundary control in [14].In [17,16] primal-dual interior point methods have been analysed for ODE problems in the infinite dimensional function space setting and their computational realization by inexact pathfollowing methods has been suggested. In [18] this method has been enhanced on the control of elliptic PDE problems with control constrains.A satisfactory convergence theory, however, had only been obtained for control constraints, whereas results for state constraints are scarce. The difficulty arises from the fact that Lagrange multipliers for state constraints are usually only measures, which hampers theoretical convergence analysis and affects the numerical solution.Concerning the regularity of Lagrange multipliers, the situation changes for mixed control-state constraints such as constraints of bottleneck type. Under natural assumptions, their multipliers can shown to be functions in certain L p -spaces, we only mention [12,4,3]. In [9], the idea came up to add a tiny fraction of the control to the state constraint such that a mixed control-state constraint results. The Lagrange multiplier to this mixed constraint is a bounded and measurable function. This Lavrentiev-regularization for state constraints has been analyzed in the context of primal-dual active set methods for elliptic control problems. Some results concerning the convergence of the solutions of the regularized problem to that of the origine state constrained can be found in [9].
We present a smooth, i.e. differentiable regularization of the projection formula that occurs in constrained parabolic optimal control problems. We summarize the optimality conditions in function spaces for unconstrained and control-constrained problems subject to a class of parabolic partial differential equations. The optimality conditions are then given by coupled systems of parabolic PDEs. For constrained problems, a non-smooth projection operator occurs in the optimality conditions. For this projection operator, we present in detail a regularization method based on smoothed sign, minimum and maximum functions. For all three cases, i.e (1) the unconstrained problem, (2) the constrained problem including the projection, and (3) the regularized projection, we verify that the optimality conditions can be equivalently expressed by an elliptic boundary value problem in the space-time domain. For this problem and all three cases we discuss existence and uniqueness issues. Motivated by this elliptic problem, we use a simultaneous space-time discretization for numerical tests. Here we show how a standard finite element software environment allows to solve the problem and thus to verify the applicability of this approach without much implementation effort. We present numerical results for an example problem.
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