On two numerical methods for state-constrained elliptic control problems

Abstract

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.

Adaptive finite element methods for mixed control-state constrained optimal control problems for elliptic boundary value problems

“…The first example features a solution y that strongly oscillates around the origin where the coincidence set is a connected subdomain with smooth boundary. In contrast to that, the second example, which is taken from [32], features a multiplier in M + (Ω) where the coincidence set degenerates to a single point. In both cases the adaptive process generates finite element meshes that are close to those created when one uses the adaptive strategy for state constrained problems as suggested in [26].…”

On regularization methods for the numerical solution of parabolic control problems with pointwise state constraints

“…Later, in [20,21], a Lavrentiev type regularization was introduced that numerically behaves like the method of [4,14], but, after regularization, preserves the structure of a state-constrained problem. This might be of interest for the convergence analysis of numerical methods in function space.…”

Goal Oriented Mesh Adaptivity for Mixed Control-State Constrained Elliptic Optimal Control Problems

“…We conclude this section with the results for an example which was chosen as a test case in [28]. The data of the problem are as follows The optimal solution in the pure state constrained case is given by: y(r) ≡ 4 , p(r) = 1 4π r 2 − 1 2π ln(r) , u(r) ≡ 4 , σ = δ 0 .…”