scholarly journals An optimal control problem for some nonlinear elliptic equations with unbounded coefficients

2019 ◽  
Vol 24 (3) ◽  
pp. 1393-1409
Author(s):  
Gabriella Zecca ◽  
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Elliptic Equations ◽  
Nonlinear Elliptic Equations ◽  
Unbounded Coefficients ◽  
Nonlinear Elliptic

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

2020 ◽  
Vol 2020 ◽  
pp. 1-15
Author(s):  
Peter I. Kogut ◽  
Olha P. Kupenko ◽  
Rosanna Manzo
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Elliptic Equations ◽  
Original Problem ◽  
Optimization Problems ◽  
Optimal Solution ◽  
Control Object ◽  
Nonlinear Elliptic Equations ◽  
Nonlinear Elliptic

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.

scholarly journals Existence of Solution for Nonlinear Elliptic Equations with Unbounded Coefficients and Data

2009 ◽  
Vol 2009 ◽  
pp. 1-18 ◽  
Author(s):  
Hicham Redwane
Keyword(s):  
Elliptic Equations ◽  
Existence Result ◽  
Nonlinear Elliptic Equations ◽  
Existence Of Solution ◽  
Unbounded Coefficients ◽  
Renormalized Solution ◽  
Nonlinear Elliptic

An existence result of a renormalized solution for a class of nonlinear elliptic equations is established. The diffusion functions may not be in for a finite value of the unknown and the data belong to .

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