On Approximation of an Optimal Control Problem for Ill-Posed Strongly Nonlinear Elliptic Equation with p-Laplace Operator

Kogut, Peter I.; Kupenko, Olha P.

doi:10.1007/978-3-319-96755-4_23

Cited by 5 publications

(4 citation statements)

References 10 publications

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“…In particular, to specify the term ½y, y f , we have the following result (we refer to [20], Lemma 2.1) where this result was proven for a particular nonlinearity f ðyÞ = e y (see also [27,28,32] for the more general cases).…”

Section: Preliminariesmentioning

confidence: 99%

“…The key point of our consideration is that, in contrast to the well-known approaches (see, for instance, [20,27,28]), we do not assume here the fulfillment of the "standard" extra properties such that the domain Ω is an open subset of ℝ N with N > 2, this domain should be star-shaped and exists a weak solution y ∈ H 1 0 ðΩÞ of Dirichlet problem (2)-( 3) satisfying f ðyÞ ∈ L 2 ðΩÞ. Because of this, the existence of at least one optimal pair to the problem (1)-( 4) is an open question provided we restrict our consideration only by assumptions (a)-(c).…”

Section: Asymptotic Analysis Of Regularized Optimal Control Problemmentioning

confidence: 99%

“…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%

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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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Section: Preliminariesmentioning

confidence: 99%

Section: Asymptotic Analysis Of Regularized Optimal Control Problemmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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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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“…Analogous results for the case of more general nonlinear elliptic equations of the type div( a (∇·)) + f (·) with a (∇ y ) = |∇ y | p − 2 ∇ y and p ≥ 2 have been considered in Reference 4. Some related questions in this field can be found in the recent papers, 15,16 where some regularization and approximation issues for optimal control problems of ill‐posed elliptic equations with an exponential type of nonlinearity and distributed controls were considered.…”

Section: Introduction and Setting Of The Optimal Control Problemmentioning

confidence: 96%

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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On Tikhonov Regularization of Optimal Distributed Control Problem for an Ill-Posed Elliptic Equation with p-Laplace Operator and $$L^1$$-type of Non-linearity

Kogut

Kupenko

2020

Contemporary Approaches and Methods in Fundamental Mathematics and Mechanics

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On Approximation of an Optimal Control Problem for Ill-Posed Strongly Nonlinear Elliptic Equation with p-Laplace Operator

Cited by 5 publications

References 10 publications

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

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

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

On Tikhonov Regularization of Optimal Distributed Control Problem for an Ill-Posed Elliptic Equation with p-Laplace Operator and $$L^1$$-type of Non-linearity

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