Ana posteriorierror analysis for an optimal control problem with point sources

Abstract

We propose and analyze a reliable and efficient a posteriori error estimator for a controlconstrained linear-quadratic optimal control problem involving Dirac measures; the control variable corresponds to the amplitude of forces modeled as point sources. The proposed a posteriori error estimator is defined as the sum of two contributions, which are associated with the state and adjoint equations. The estimator associated with the state equation is based on Muckenhoupt weighted Sobolev spaces, while the one associated with the adjoint is in the maximum norm and allows for unbounded right hand sides. The analysis is valid for two and three-dimensional domains. On the basis of the devised a posteriori error estimator, we design a simple adaptive strategy that yields optimal rates of convergence for the numerical examples that we perform. AN OPTIMAL CONTROL PROBLEM WITH POINT SOURCES3 polytope, we prove the global reliability and local efficiency of our proposed error estimator. The analysis is delicate since it involves the interaction of L ∞ (Ω), R l and weighted Sobolev spaces, combined with having to deal with the first-order necessary and sufficient optimality condition that characterizes the optimal controlū. It is important to comment that this work exploits the ideas developed in [4] for the a posteriori error analysis of the so-called pointwise tracking optimal control problem. Although the mathematical techniques are similar, the a posteriori error analysis of our control problem does not follow directly from [4]; it requires its own analysis. This is mainly due to the following reasons:• The optimal control variableū belongs to R l , while the one of the problem studied in [4] belongs to L 2 (Ω).This in a sense simplifies the analysis. For instance, as opposed to [4,30], we can obtain local efficiency estimates that do not require convexity of Ω. Nevertheless, it comes with its own set of complications. In particular, the low regularity of the state equation. • The adjoint problem is a Poisson equation with a forcing term y − y d , which does not belong to L ∞ (Ω).Consequently, we must consider a pointwise error indicator that accounts for unbounded right hand sides.Since we were not able to locate one in the literature, in section 4, we propose such an error indicator and provide its analysis on the basis of [9,15]. Notice that, thanks to the structure of the control problem of [4], such an estimator was not needed there. The outline of this paper is as follows. In section 2, we introduce the notation and functional framework we shall work with. Section 3 contains the description of our control problem and reviews the a priori error analysis developed in [7]. In section 4, we propose and analyze a pointwise a posteriori error estimator for the Laplacian that allows for unbounded right hand sides. Combining this estimator and another one based on Muckenhoupt weighted Sobolev spaces, in section 5 we devise an a posteriori error estimator for our optimal control problem. We show in sections 5.2 and 5.3, its ...

An a posteriori error analysis of an elliptic optimal control problem in measure space

“…The local efficiency of the indicators (4.13) is as follows. For a proof we refer the reader to [1,Lemma 4.3] Lemma 4.4 (local efficiency). Let z ∈ H 1 0 (Ω) ∩ L ∞ (Ω) and z T ∈ V(T ) be the solutions to problems (4.11) and (4.12), respectively.…”

“…Proof. We follow closely the arguments elaborated in [1,Lemma 5.4]. Let w ∈ H 1 0 (Ω) be such that w| T ∈ C 2 (Ω) for T ∈ T .…”

“…E ∞ (z T ; T ) := h 2−d/2 T f L 2 (T ) + h T [[∇z T · ν]] L ∞ (∂T \∂Ω) , and the error estimator E ∞ (z T ; T ) := max T ∈T E ∞ (z T ; T ).In order to present the reliability of the global error estimator E ∞ , we define(4.14) ℓ T := log max T ∈T 1 h T .The proof of the next result can be found in[1, Lemma 4.2].…”

Optimal control of an eddy current problem with a dipole source

“…* Department of Mathematics, University of Trento, Via Sommarive 14, Povo (TN), 38123 Italy (gabriele.caselli@unitn.it). 1 The very first exhaustive characterization of the traces of H(curl) functions in rather general domains came out in the early 2000s with [6] by Buffa et al In more recent times, the lack of reflexivity, compactness and differentiability (regularity) properties of the L 1 -spaces and norms led to the study of optimal control problems in measure spaces like M(Ω), the space of regular Borel measures, or L 2 (I, M(Ω)), which both exhibit better functional properties as well as similar sparsity features of optimal solutions; see Casas et al [7], Clason and Kunish [9] and Trautmann et al [19], where all these issues are widely discussed.…”

“…For these reasons, we decided to work with a fixed number of deltas (i.e., one, without loss of generality) in a fixed location, say x = x 0 . A similar approach has been carried out rather recently by Allendes et al [1], but there the state equation takes the form of a Poisson problem and the focus is shifted on the a posteriori error analysis for a FEM approximation.…”