Second Order Optimality Conditions for Semilinear Elliptic Control Problems with Finitely Many State Constraints

Casas, Eduardo; Mateos, Mariano

doi:10.1137/s0363012900382011

Cited by 89 publications

(77 citation statements)

References 13 publications

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“…By using the assumption (∂ 2 l/∂u 2 )(x, y, u) ≥ m l > 0, we deduce from Casas and Mateos [6,Theorem 4.4] that the following two conditions are equivalent:…”

Section: )mentioning

confidence: 98%

See 1 more Smart Citation

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

Casas

Mateos

Tröltzsch

2005

Comput Optim Applic

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121

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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“…By using the assumption (∂ 2 l/∂u 2 )(x, y, u) ≥ m l > 0, we deduce from Casas and Mateos [6,Theorem 4.4] that the following two conditions are equivalent:…”

Section: )mentioning

confidence: 98%

“…This theorem is now standard and can be proved by using the implicit function theorem; see Casas and Mateos [6].…”

Section: )mentioning

confidence: 99%

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

Casas

Mateos

Tröltzsch

2005

Comput Optim Applic

Self Cite

121

123

View full text Add to dashboard Cite

show abstract

“…Let us point out the related work on sufficient optimality conditions for control of state-constrained optimal control problems, in e.g. [5][6][7]17,18].…”

Section: Introduction Problem Statement Regularizationmentioning

confidence: 99%

Sufficient optimality conditions and semi-smooth newton methods for optimal control of stationary variational inequalities

Kunisch

Wachsmuth

2011

ESAIM: COCV

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Abstract.In this paper sufficient second order optimality conditions for optimal control problems subject to stationary variational inequalities of obstacle type are derived. Since optimality conditions for such problems always involve measures as Lagrange multipliers, which impede the use of efficient Newton type methods, a family of regularized problems is introduced. Second order sufficient optimality conditions are derived for the regularized problems as well. It is further shown that these conditions are also sufficient for superlinear convergence of the semi-smooth Newton algorithm to be well-defined and superlinearly convergent when applied to the first order optimality system associated with the regularized problems.Mathematics Subject Classification. 49K20, 47J20, 49M15.

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“…For the above results the reader is referred to [5] or [6], where similar cases were studied. Let us remark that in the case where the set of zeros ofφ has a zero Lebesgue measure, thenū(x) is either α or β for almost all points x ∈ Ω, i.e.ū is a bang-bang control.…”

Section: Assumption (D1)mentioning

confidence: 99%

Second Order Conditions for L 2 Local Optimality in PDE Control

Casas

2013

IFIP Advances in Information and Communication Technology

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Abstract. In the second order analysis of infinite dimension optimization problems, we have to deal with the so-called two-norm discrepancy. As a consequence of this fact, the second order optimality conditions usually imply local optimality in the L ∞ sense. However, we have observed that the L 2 local optimality can be proved for many control problems of partial differential equations. This can be deduced from the standard second order conditions. To this end, we make some quite realistic assumptions on the second derivative of the cost functional. These assumptions do not hold if the control does not appear explicitly in the cost functional. In this case, the optimal control is usually of bang-bang type. For this type of problems we also formulate some new second order optimality conditions that lead to the strict L 2 local optimality of the bang-bang controls.Keywords: optimal control of partial differential equations, semilinear partial differential equations, second order optimality conditions, bangbang controls.

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Second Order Optimality Conditions for Semilinear Elliptic Control Problems with Finitely Many State Constraints

Cited by 89 publications

References 13 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

Sufficient optimality conditions and semi-smooth newton methods for optimal control of stationary variational inequalities

Second Order Conditions for L 2 Local Optimality in PDE Control

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