We deal with an optimal control problem for or one class of non-linear elliptic equations with an exponential type of non-linearity. The boundary control is a density of surface traction acting on a part of boundary of an open domain. The aim is to minimize the discrepancy between a given distribution and the current system state. A special family of regularized optimization problems is introduced using a variant of the classical Tikhonov regularization and it is proven that each of these problem is consistent, well-posed, and their solutions allow to attain (in the limit) 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.
Some Results on Optimal Control Problem for Ill-Posed Nonlinear Elliptic Equations
R. Manzo
2023-01-01
Abstract
We deal with an optimal control problem for or one class of non-linear elliptic equations with an exponential type of non-linearity. The boundary control is a density of surface traction acting on a part of boundary of an open domain. The aim is to minimize the discrepancy between a given distribution and the current system state. A special family of regularized optimization problems is introduced using a variant of the classical Tikhonov regularization and it is proven that each of these problem is consistent, well-posed, and their solutions allow to attain (in the limit) 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.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
