Optimal Control of PDEs with Regularized Pointwise State Constraints

Abstract

This paper addresses the regularization of pointwise state constraints in optimal control problems. By analyzing the associated dual problem, it is shown that the regularized problems admit Lagrange multipliers in L 2 -spaces. Under a certain boundedness assumption, the solution of the regularized problem converges to the one of the original state constrained problem. The results of our analysis are confirmed by numerical tests.

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

“…Our main goal is the investigation of Lavrentiev-and source type regularization methods for parabolic problems. Lavrentiev regularization was introduced for elliptic problems with distributed controls by Meyer et al in [20], and, in a form closer to this paper, by Meyer and Tröltzsch in [19]. An extension to the case of elliptic boundary control (and pointwise constraints in the domain) was recently suggested in [30].…”

“…Based on these observations, in [20] the constraints y ≥ y a , u ≥ 0 have been considered by their regularization y + λu ≥ y a , u ≥ 0. It was shown in [20], that the associated regularized optimal controlsū λ tend toū in L 2 (Ω) as λ → 0.…”

“…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.…”

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

“…Therefore, it is a natural idea to regularize state constrained problems by means of mixed control-state constrained ones, since with regard to numerical solution techniques the regularized problems can be formally treated as in the case of pure control constraints (cf. e.g., [2,11,33,36,37,38,39,40]). However, so far an a posteriori error analysis of adaptive finite element approximations has not been provided for mixed control-state constrained control problems.…”

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

“…With regard to numerical solution techniques the regularized problems can be formally treated as in the case of control constraints (cf. e.g., [2,9,29,32,33,34]). In this paper, we will develop, analyze and implement the goal oriented weighted dual approach to mixed control-state constrained distributed optimal control problems for linear second order elliptic boundary value problems.…”