Shape stability of optimal control problems in coefficients for coupled system of Hammerstein type

Olha P.  Kupenko; Rosanna  Manzo

doi:10.3934/dcdsb.2015.20.2967

Article Contents

2015, Volume 20, Issue 9: 2967-2992. Doi: 10.3934/dcdsb.2015.20.2967

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Shape stability of optimal control problems in coefficients for coupled system of Hammerstein type

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

1.
Dnipropetrovsk Mining University, Department of System Analysis and Control, Karl Marks av., 19, 49005 Dnipropetrovsk
2.
Università degli Studi di Salerno, Dipartimento di Ingegneria dell'Informazione, Ingegneria Elettrica e Matematica Applicata, Via Giovanni Paolo II, 132, 84084 Fisciano (SA)

Received: March 31, 2014

Revised: September 30, 2014

Published: October 2015

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Abstract

In this paper we consider an optimal control problem (OCP) for the coupled system of a nonlinear monotone Dirichlet problem with matrix-valued $L^\infty(\Omega;\mathbb{R}^{N\times N} )$-controls in coefficients and a nonlinear equation of Hammerstein type. Since problems of this type have no solutions in general, we make a special assumption on the coefficients of the state equation and introduce the class of so-called solenoidal admissible controls. Using the direct method in calculus of variations, we prove the existence of an optimal control. We also study the stability of the optimal control problem with respect to the domain perturbation. In particular, we derive the sufficient conditions of the Mosco-stability for the given class of OCPs.

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Mathematics Subject Classification: Primary: 49J20, 35J57; Secondary: 49J45, 93C73.

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