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

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 f...

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Bibliographic Details
Published in:Abstract and applied analysis 2020, Vol.2020 (2020), p.1-15
Main Authors: Kogut, Peter I., Manzo, Rosanna, Kupenko, Olha P.
Format: Article
Language:English
Subjects:
Approximation
Boundary value problems
Differential equations, Nonlinear
Elliptic functions
Equality
Equations of state
Mathematical analysis
Mathematical optimization
Mathematical research
Nonlinear control
Nonlinear equations
Nonlinearity
Optimal control
Optimization
Regularization
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Summary: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.
ISSN:1085-3375
1687-0409
DOI:10.1155/2020/7418707