Convergence of approximations vs. regularity of solutions for convex, control-constrained optimal-control problems

Malanowski, K.

doi:10.1007/bf01447752

Cited by 137 publications

(74 citation statements)

References 12 publications

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“…The case of linearquadratic elliptic control problems by finite elements was discussed in early papers by Falk [9], Geveci [10] and Malanowski [16], and Arnautu and Neittaanmäki [2], who already proved the optimal error estimate of order h in the L 2 -norm. In [16], also the case of piecewise linear control functions is addressed. …”

mentioning

confidence: 99%

Error Estimates for the Numerical Approximation of Boundary Semilinear Elliptic Control Problems

Casas

Mateos

Tröltzsch

2005

Comput Optim Applic

122

123

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We study the numerical approximation of boundary optimal control problems governed by semilinear elliptic partial differential equations with pointwise constraints on the control. The analysis of the approximate control problems is carried out. The uniform convergence of discretized controls to optimal controls is proven under natural assumptions by taking piecewise constant controls. Finally, error estimates are established. Abstract. We study the numerical approximation of boundary optimal control problems governed by semilinear elliptic partial differential equations with pointwise constraints on the control. The analysis of the approximate control problems is carried out. The uniform convergence of discretized controls to optimal controls is proven under natural assumptions by taking piecewise constant controls. Finally, error estimates are established.Key words. Boundary control, semilinear elliptic equation, numerical approximation, error estimates AMS subject classifications. 49J20, 49K20, 49M05, 65K101. Introduction. With this paper, we continue the discussion of error estimates for the numerical approximation of optimal control problems we have started for semilinear elliptic equations and distributed controls in [1]. The case of distributed control is the easiest one with respect to the mathematical analysis. In [1] it was shown that, roughly speaking, the distance between a locally optimal controlū and its numerical approximationū h has the order of the mesh size h in the L 2 -norm and in the L ∞ -norm. This estimate holds for a finite element approximation of the equation by standard piecewise linear elements and piecewise constant control functions.The analysis for boundary controls is more difficult, since the regularity of the state function is lower than that for distributed controls. Moreover, the internal approximation of the domain causes problems. In the general case, we have to approximate the boundary by a polygon. This requires the comparison of the original control that is located at the boundary Γ and the approximate control that is defined on the polygonal boundary Γ h . Moreover, the regularity of elliptic equations in domains with corners needs special care. To simplify the analysis, we assume here that Ω is a polygonal domain of R 2 . Though this makes the things easier, the lower regularity of states in polygonal domains complicates, together with the presence of nonlinearities, the analysis.Another novelty of our paper is the numerical confirmation of the predicted error estimates. We present two examples, where we know the exact solutions. The first one is of linear-quadratic type, while the second one is semilinear. We are able to verify our error estimates quite precisely.Let us mention some further papers related to this subject. The case of linearquadratic elliptic control problems by finite elements was discussed in early papers by Falk [9], Geveci [10] and Malanowski [16], and Arnautu and Neittaanmäki [2], who already proved the optimal error estimate of order h in the L 2 -...

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mentioning

confidence: 99%

Error Estimates for the Numerical Approximation of Boundary Semilinear Elliptic Control Problems

Casas

Mateos

Tröltzsch

2005

Comput Optim Applic

122

123

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“…Issues related to the analysis of numerical algorithms for optimal control problems constrained to time-dependent problems were studied in [2,8,10,17,19,22,[24][25][26][27][28]33,37,40,41,43,44]. In the recent works of [4,31,32,34,35] discontinuous Galerkin schemes were analyzed for distributed optimal control problems constrained to linear parabolic PDE's.…”

Section: Related Resultsmentioning

confidence: 99%

Convergence of discontinuous Galerkin approximations of an optimal control problem associated to semilinear parabolic PDE's

Chrysafinos

2009

ESAIM: M2AN

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Abstract.A discontinuous Galerkin finite element method for an optimal control problem related to semilinear parabolic PDE's is examined. The schemes under consideration are discontinuous in time but conforming in space. Convergence of discrete schemes of arbitrary order is proven. In addition, the convergence of discontinuous Galerkin approximations of the associated optimality system to the solutions of the continuous optimality system is shown. The proof is based on stability estimates at arbitrary time points under minimal regularity assumptions, and a discrete compactness argument for discontinuous Galerkin schemes (see Walkington [SINUM (June 2008) (submitted), preprint available at

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“…The proof technique is based on a combination of arguments in [15] and [9]. Throughout we assume y • , z Q , z Ω , α Q , and α Ω are sufficiently regular; see [15], Table 1.…”

Section: Mesh-independencementioning

confidence: 99%

“…We proceed as in [15], Section 3, and [8]. Let Ω h be a family of grids depending on the parameter h > 0.…”

Section: Mesh-independencementioning

confidence: 99%

See 1 more Smart Citation

Mesh-independence and preconditioning for solving parabolic control problems with mixed control-state constraints

Hintermüller¹,

Kopacka²,

Volkwein³

2008

ESAIM: COCV

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Abstract.Optimal control problems for the heat equation with pointwise bilateral control-state constraints are considered. A locally superlinearly convergent numerical solution algorithm is proposed and its mesh independence is established. Further, for the efficient numerical solution reduced space and Schur complement based preconditioners are proposed which take into account the active and inactive set structure of the problem. The paper ends by numerical tests illustrating our theoretical findings and comparing the efficiency of the proposed preconditioners.Mathematics Subject Classification. 49K20, 65K05.

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Convergence of approximations vs. regularity of solutions for convex, control-constrained optimal-control problems

Cited by 137 publications

References 12 publications

Error Estimates for the Numerical Approximation of Boundary Semilinear Elliptic Control Problems

Error Estimates for the Numerical Approximation of Boundary Semilinear Elliptic Control Problems

Convergence of discontinuous Galerkin approximations of an optimal control problem associated to semilinear parabolic PDE's

Mesh-independence and preconditioning for solving parabolic control problems with mixed control-state constraints

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