First- and Second-Order Optimality Conditions for a Class of Optimal Control Problems with Quasilinear Elliptic Equations

Casas, Eduardo; Tröltzsch, Fredi

doi:10.1137/080720048

Cited by 61 publications

(60 citation statements)

References 24 publications

(25 reference statements)

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“…Moreover, this solution has W 2,p (Ω)-regularity. Although the equation (2.12) is not monotone, the authors were able to prove the well posedness of the equation in [5]. In fact, for any v ∈ W −1,p (Ω), the boundary value problem…”

Section: Assumptions and Preliminary Resultsmentioning

confidence: 89%

“…For the proof of this theorem, we refer the reader to Casas and Tröltzsch [5]. Moreover, the solution y u depends continuously of u.…”

Section: Assumptions and Preliminary Resultsmentioning

confidence: 99%

“…This equation has been studied by the authors in [5], where the uniqueness and regularity of the solution was investigated. Here we prove that the discrete adjoint state equation has also a unique solution in spite of its non-monotone character; see Theorem 4.1.…”

Section: Introductionmentioning

confidence: 99%

“…The regularity of the solutions to these equations, which is required for this analysis, is obtained from the first order necessary optimality conditions. These optimality conditions were proved in [5] and are included here for convenience. Although the equation (1.1) is not of monotone type, it has a unique solution.…”

Section: Introductionmentioning

confidence: 99%

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Numerical analysis of some optimal control problems governed by a class of quasilinear elliptic equations

Casas¹,

Tröltzsch²

2010

ESAIM: COCV

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Abstract.In this paper, we carry out the numerical analysis of a distributed optimal control problem governed by a quasilinear elliptic equation of non-monotone type. The goal is to prove the strong convergence of the discretization of the problem by finite elements. The main issue is to get error estimates for the discretization of the state equation. One of the difficulties in this analysis is that, in spite of the partial differential equation has a unique solution for any given control, the uniqueness of a solution for the discrete equation is an open problem.Mathematics Subject Classification. 49M25, 35J60, 35B37, 65N30.

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Section: Assumptions and Preliminary Resultsmentioning

confidence: 89%

“…For the proof of this theorem, we refer the reader to Casas and Tröltzsch [5]. Moreover, the solution y u depends continuously of u.…”

Section: Assumptions and Preliminary Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Numerical analysis of some optimal control problems governed by a class of quasilinear elliptic equations

Casas¹,

Tröltzsch²

2010

ESAIM: COCV

Self Cite

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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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Adaptive finite element discretization in PDE‐based optimization

Rannacher

Vexler

2010

GAMM-Mitteilungen

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Key words PDE-based optimization, finite element method, goal-oriented adaptivity MSC (2000) 49K20, 49M15, 49M15, 65M50, 65N50This article surveys recent developments in the adaptive numerical solution of optimal control problems governed by partial differential equations (PDE). By the Euler-Lagrange formalism the optimization problem is reformulated as a saddle-point problem (KKT system) that is discretized by a Galerkin finite element method (FEM). Following the Dual Weighted Residual (DWR) approach the accuracy of the approximation is controlled by residual-based a posteriori error estimates. This opens the way toward systematic complexity reduction in the solution of PDE-based optimal control problems occurring in science and engineering.

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First- and Second-Order Optimality Conditions for a Class of Optimal Control Problems with Quasilinear Elliptic Equations

Cited by 61 publications

References 24 publications

Numerical analysis of some optimal control problems governed by a class of quasilinear elliptic equations

Numerical analysis of some optimal control problems governed by a class of quasilinear elliptic equations

Second Order Conditions for L 2 Local Optimality in PDE Control

Adaptive finite element discretization in PDE‐based optimization

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