Differential and Integral Equations

On Tikhonov regularization of optimal Neumann boundary control problem for an ill-posed strongly nonlinear elliptic equation with an exponential type of non-linearity

Rosanna Manzo

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Abstract

We discuss the existence of solutions to an optimal control problem for the mixed Dirichlet-Neumann boundary value problem for strongly non-linear elliptic equations with an exponential type of nonlinearity. A density of surface traction $u$ acting on a part of boundary of open domain is taken as a boundary control. The optimal control problem is to minimize the discrepancy between a given distribution $y_d\in L^2(\Omega)$ and the current system state. We deal with such case of nonlinearity when we cannot expect to have a solution of the original boundary value problem for each admissible control. Instead of this, we make use of a variant of the classical Tikhonov regularization. We eliminate the differential constraints between control and state and allow such pairs run freely in their respective sets of feasibility by introducing some additional variable which plays the role of ``defect". We show that this special residual function can be determined in a unique way. We introduce a special family of regularized optimization problems and show 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 establish sufficient conditions of the existence of optimal solutions to the given class of nonlinear Dirichlet BVP and propose the way for their approximation.

Article information

Source
Differential Integral Equations, Volume 33, Number 3/4 (2020), 139-162.

Dates
First available in Project Euclid: 21 March 2020

Permanent link to this document
https://projecteuclid.org/euclid.die/1584756016

Mathematical Reviews number (MathSciNet)
MR4079786

Subjects
Primary: 49J20: Optimal control problems involving partial differential equations 49K20: Problems involving partial differential equations 58J37: Perturbations; asymptotics

Citation

Manzo, Rosanna. On Tikhonov regularization of optimal Neumann boundary control problem for an ill-posed strongly nonlinear elliptic equation with an exponential type of non-linearity. Differential Integral Equations 33 (2020), no. 3/4, 139--162. https://projecteuclid.org/euclid.die/1584756016


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