On optimal controls in coefficients for ill-posed non-Linear elliptic Dirichlet boundary value problems

Olha P. Kupenko; Rosanna Manzo

doi:10.3934/dcdsb.2018155

Article Contents

2018, Volume 23, Issue 4: 1363-1393. Doi: 10.3934/dcdsb.2018155

This issue Previous Article Next Article Well-posedness in critical spaces for a multi-dimensional compressible viscous liquid-gas two-phase flow model

On optimal controls in coefficients for ill-posed non-Linear elliptic Dirichlet boundary value problems

Olha P. Kupenko^1,2, and
Rosanna Manzo^3,

1.
National Mining University, Department of System Analysis and Control, Yavornitskii av., 19, 49005 Dnipro, Ukraine
2.
Institute for Applied System Analysis, National Academy of Sciences and Ministry of Education and Science of Ukraine, Peremogy av., 37/35, IASA, 03056 Kyiv, Ukraine
3.
Università degli Studi di Salerno, Dipartimento di Ingegneria dell'Informazione, Ingegneria Elettrica e Matematica Applicata, Via Giovanni Paolo Ⅱ, 132, 84084 Fisciano (SA), Italy

Received: June 30, 2016

Revised: December 31, 2017

Early access: April 2018

Published: April 2018

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Abstract

We consider an optimal control problem associated to Dirichlet boundary value problem for non-linear elliptic equation on a bounded domain $Ω$ . We take the coefficient $u(x)∈ L^∞(Ω)\cap BV(Ω)$ in the main part of the non-linear differential operator as a control and in the linear part of differential operator we consider coefficients to be unbounded skew-symmetric matrix $A_{skew}∈ L^q(Ω;\mathbb{S}^N_{skew})$ . We show that, in spite of unboundedness of the non-linear differential operator, the considered Dirichlet problem admits at least one weak solution and the corresponding OCP is well-possed and solvable. At the same time, optimal solutions to such problem can inherit a singular character of the matrices $A^{skew}$ . We indicate two types of optimal solutions to the above problem and show that one of them can be attained by optimal solutions of regularized problems for coercive elliptic equations with bounded coefficients, using the two-parametric regularization of the initial OCP.

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Mathematics Subject Classification: Primary: 49J20, 35J92; Secondary: 49J45, 49K20.

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