On Neumann boundary control problem for ill-posed strongly nonlinear elliptic equation with p-Laplace operator and $$L^1$$L1-type of nonlinearity

Manzo, Rosanna

doi:10.1007/s11587-019-00439-x

Cited by 10 publications

(6 citation statements)

References 15 publications

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“…To this end, we make use of some results of the variational convergence of minimization problems [26,[29][30][31] and begin with some auxiliaries (see also Refs. [14,15,30,35,36] for other aspects of this concept).…”

Section: Theoremmentioning

confidence: 99%

Variational Model with Nonstandard Growth Condition in Image Restoration and Contrast Enhancement

D’Apice,

Kogut,

Manzo

et al. 2024

Commun. Appl. Math. Comput.

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We propose a new variational model in Sobolev-Orlicz spaces with non-standard growth conditions of the objective functional and discuss its applications to the simultaneous contrast enhancement and denoising of color images. The characteristic feature of the proposed model is that we deal with a constrained non-convex minimization problem that lives in variable Sobolev-Orlicz spaces where the variable exponent is unknown a priori and it depends on a particular function that belongs to the domain of the objective functional. In contrast to the standard approach, we do not apply any spatial regularization to the image gradient. We discuss the consistency of the variational model, give the scheme for its regularization, derive the corresponding optimality system, and propose an iterative algorithm for practical implementations.

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Section: Theoremmentioning

confidence: 99%

Variational Model with Nonstandard Growth Condition in Image Restoration and Contrast Enhancement

D’Apice,

Kogut,

Manzo

et al. 2024

Commun. Appl. Math. Comput.

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“…To specify the term

{false[y, y false]}_{f}

, we make use of the following result (for the proof we refer to Reference 7).…”

Section: Previous Analysis Of Optimal Control Problem (1)‐(5)mentioning

confidence: 99%

“…In particular, the next Proposition 3 can be considered as some specification of the well‐known Boccardo‐Murat Theorem (see Boccardo and Murat 25 (theorem 2.1) . For the proof, we refer to Reference [7, proposition 3.3].…”

Section: Previous Analysis Of Optimal Control Problem (1)‐(5)mentioning

confidence: 99%

“…It means that there is no reason to assert the existence of weak solutions to ()‐() for a given

u \in A_{a d} (Γ_{N, ε})

and

g \in G_{a d}

, or to suppose that such solution, even if it exists, is unique. As for existence of bounded solutions to Neumann boundary value problem for p ‐Laplace equation with exponential type of nonlinearity, this issue has been studied in Reference 6 (see also Reference 7).…”

Section: Introduction and Setting Of The Optimal Control Problemmentioning

confidence: 99%

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Optimal boundary control problem for ill‐posed elliptic equation in domains with rugous boundary. Existence result and optimality conditions

D’Apice

Maio

Kogut

2020

Optim Control Appl Methods

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SummaryThe main purpose is the study of optimal control problem in a domain with rough boundary for the mixed Dirichlet‐Neumann boundary value problem for the strongly nonlinear elliptic equation with exponential nonlinearity. A density of surface traction u acting on a part of rough boundary is taken as a control. The optimal control problem is to minimize the discrepancy between a given distribution and the current system state. We deal with such case of nonlinearity when we cannot expect to have a solution of the state equation for a given control. After having defined a suitable functional class in which we look for solutions, we prove the consistency of the original optimal control problem and show that it admits a unique optimal solution. Then we derive a first‐order optimality system assuming the optimal solution is slightly more regular.

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“…We show that the boundedness of these terms on the set Ξ of feasible solutions to the original problem plays a crucial role in the study of asymptotic behaviour of global solutions to regularized OCPs. Having introduced a special family of optimization problems, we also show that there exists an optimal solution to the original OCP that can be attained with a prescribed level of accuracy by the sequence of optimal solutions for the regularized minimization problems (for benefit of this approach and its comparison with other ones, we refer to the recent papers [24][25][26][27][28][29][30][31]).…”

Section: Introductionmentioning

confidence: 99%

On Regularization of an Optimal Control Problem for Ill-Posed Nonlinear Elliptic Equations

Kogut

Kupenko

Manzo

2020

Abstract and Applied Analysis

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We discuss the existence issue to an optimal control problem for one class of nonlinear elliptic equations with an exponential type of nonlinearity. We deal with the control object when we cannot expect to have a solution of the corresponding boundary value problem in the standard functional space for all admissible controls. To overcome this difficulty, we make use of a variant of the classical Tikhonov regularization scheme. In particular, we eliminate the PDE constraints between control and state and allow such pairs run freely by introducing an additional variable which plays the role of “compensator” that appears in the original state equation. We show that this fictitious variable can be determined in a unique way. In order to provide an approximation of the original optimal control problem, we define a special family of regularized optimization problems. We show that each of these problems is consistent, well-posed, and their solutions allow to attain an optimal solution of the original problem as the parameter of regularization tends to zero. As a consequence, we prove the existence of optimal solutions to the original problem and propose a way for their approximation.

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On Neumann boundary control problem for ill-posed strongly nonlinear elliptic equation with p-Laplace operator and $$L^1$$L1-type of nonlinearity

Cited by 10 publications

References 15 publications

Variational Model with Nonstandard Growth Condition in Image Restoration and Contrast Enhancement

Variational Model with Nonstandard Growth Condition in Image Restoration and Contrast Enhancement

Optimal boundary control problem for ill‐posed elliptic equation in domains with rugous boundary. Existence result and optimality conditions

On Regularization of an Optimal Control Problem for Ill-Posed Nonlinear Elliptic Equations

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