Weak optimal controls in coefficients for linear elliptic problems

Buttazzo, Giuseppe; Kogut, Peter I.

doi:10.1007/s13163-010-0030-y

Cited by 29 publications

(35 citation statements)

References 16 publications

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“…As follows from convergence properties (15) and (17), the equality (85) implies the strong convergence ! in the variable space L 2 .0, T; L 2 R N , Q n dx/ Á .…”

Section: Proofmentioning

confidence: 91%

“…However, these classes are essentially wider than the space

C_{0}^{\infty} {(normalΩ)}^{\frac{N (N + 1)}{2}}

in the definition of the weak convergence in variable space

L^{2} {(normalΩ, ζ_{n} dx)}^{\frac{N (N + 1)}{2}}

(see ). Therefore, in order to pass to the limit in that integral identities as n → ∞ , we make use the following trick (see Buttazzo and Kogut ). For every fixed

n \in double-struckN

, we denote by

((), {\overset{ζ}{true}}_{n}, {\overset{boldu}{true}}_{n}) \in L^{2} ((), 0, T; H_{loc}^{1} ({double-struckR}^{N})) \times L^{1} ((), 0, T; {scriptW}_{loc}^{1, 1} ({double-struckR}^{N}; S))

an arbitrary extension of the functions ( ζ n , u n ) to the whole of space

{double-struckR}^{N}

such that the sequence

{({}, ((), {\overset{ζ}{true}}_{n}, {\overset{boldu}{true}}_{n}))}_{n \in double-struckN}

satisfies the properties:

{\overset{ζ}{true}}_{n} \in L^{2} (0, T; H^{1} (Q)), {\overset{ζ}{true}}_{n}^{'} \in L^{2} (0, T; {(H^{1} (Q))}^{'})

…”

Section: Setting Of the Optimal Control Problem And Existence Theoremmentioning

confidence: 99%

“…To the authors' best knowledge, the existence of optimal solutions for the aforementioned problem is an open question. Moreover, only few papers deal with optimal control problems for the degenerate partial differential equations (see for example ).…”

Section: Introductionmentioning

confidence: 99%

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Optimal and approximate boundary controls of an elastic body with quasistatic evolution of damage

Kogut

Leugering

2014

Math Methods in App Sciences

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Communicated by M. BrokateIn this paper, we study an optimal control problem for the mixed boundary value problem for an elastic body with quasistatic evolution of an internal damage variable. We suppose that the evolution of microscopic cracks and cavities responsible for the damage is described by a nonlinear parabolic equation. A density of surface traction p acting on a part of boundary of an elastic body is taken as a boundary control. Because the initial boundary value problem of this type can exhibit the Lavrentieff phenomenon and non-uniqueness of weak solutions, we deal with the solvability of this problem in the class of weak variational solutions. Using the convergence concept in variable spaces and following the direct method in calculus of variations, we prove the existence of optimal and approximate solutions to the optimal control problem under rather general assumptions on the quasistatic evolution of damage.

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“…As follows from convergence properties (15) and (17), the equality (85) implies the strong convergence ! in the variable space L 2 .0, T; L 2 R N , Q n dx/ Á .…”

Section: Proofmentioning

confidence: 91%

“…However, these classes are essentially wider than the space

C_{0}^{\infty} {(normalΩ)}^{\frac{N (N + 1)}{2}}

in the definition of the weak convergence in variable space

L^{2} {(normalΩ, ζ_{n} dx)}^{\frac{N (N + 1)}{2}}

(see ). Therefore, in order to pass to the limit in that integral identities as n → ∞ , we make use the following trick (see Buttazzo and Kogut ). For every fixed

n \in double-struckN

, we denote by

((), {\overset{ζ}{true}}_{n}, {\overset{boldu}{true}}_{n}) \in L^{2} ((), 0, T; H_{loc}^{1} ({double-struckR}^{N})) \times L^{1} ((), 0, T; {scriptW}_{loc}^{1, 1} ({double-struckR}^{N}; S))

an arbitrary extension of the functions ( ζ n , u n ) to the whole of space

{double-struckR}^{N}

such that the sequence

{({}, ((), {\overset{ζ}{true}}_{n}, {\overset{boldu}{true}}_{n}))}_{n \in double-struckN}

satisfies the properties:

{\overset{ζ}{true}}_{n} \in L^{2} (0, T; H^{1} (Q)), {\overset{ζ}{true}}_{n}^{'} \in L^{2} (0, T; {(H^{1} (Q))}^{'})

…”

Section: Setting Of the Optimal Control Problem And Existence Theoremmentioning

confidence: 99%

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Optimal and approximate boundary controls of an elastic body with quasistatic evolution of damage

Kogut

Leugering

2014

Math Methods in App Sciences

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“…In order to avoid degeneracy with respect to the control A(x), we assume that matrix A(x) has a uniformly bounded spectrum away from zero. As for the optimal control problems in coefficients for degenerate elliptic equations and variational inequalities, we can refer to [5,8,9,10,16,17,19].…”

Section: Introductionmentioning

confidence: 99%

On attainability of optimal controls in coefficients for system of Hammerstein type with anisotropic p-Laplacian

2016

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In this paper we consider an optimal control\ud problem (OCP) for the coupled system of a nonlinear monotone Dirichlet problem with anisotropic p-Laplacian and matrix-valued L^infinity(Omega, R^(NxN))-controls in its coefficients\ud and a nonlinear equation of Hammerstein type. Using the direct\ud method in calculus of variations, we prove the existence of an optimal control in considered problem and provide sensitivity analysis for a specific case of considered problem with respect to two-parameter regularization

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“…However, this type does not exhaust all weak solutions to the above problem. There is another type of weak solutions called non-variational [20,22], singular [3,13,14,19], pathological [16,17] and others. As for the optimal control problem (1) we have the following result [9] (see [8] for comparison): for any approximation {A * k } k∈N of the matrix A * ∈ L 2 Ω; S N skew with properties {A * k } k∈N ⊂ L ∞ (Ω; S N skew ) and A * k → A * strongly in L 2 (Ω; S N skew ), optimal solutions to the corresponding regularized OCPs associated with matrices A * k always lead in the limit as k → ∞ to some admissible (but not optimal in general) solution ( A, y ) of the original OCP (1).…”

mentioning

confidence: 99%

Optimal $L^2$-control problem in coefficients for a linear elliptic equation. II. Approximation of solutions and optimality conditions

Horsin¹,

Kogut²,

Wilk³

2016

MCRF

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In this paper we study we study a Dirichlet optimal control problem associated with a linear elliptic equation the coefficients of which we take as controls in the class of integrable functions. The characteristic feature of this control object is the fact that the skew-symmetric part of matrix-valued control A(x) belongs to L 2 -space (rather than L ∞ ). In spite of the fact that the equations of this type can exhibit non-uniqueness of weak solutions, the corresponding OCP, under rather general assumptions on the class of admissible controls, is well-posed and admits a nonempty set of solutions [9]. However, the optimal solutions to such problem may have a singular character. We show that some of optimal solutions can be attainable by solutions of special optimal control problems in perforated domains with fictitious boundary controls on the holes.

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Weak optimal controls in coefficients for linear elliptic problems

Cited by 29 publications

References 16 publications

Optimal and approximate boundary controls of an elastic body with quasistatic evolution of damage

Optimal and approximate boundary controls of an elastic body with quasistatic evolution of damage

On attainability of optimal controls in coefficients for system of Hammerstein type with anisotropic p-Laplacian

Optimal $L^2$-control problem in coefficients for a linear elliptic equation. II. Approximation of solutions and optimality conditions

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