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On optimality conditions for optimal control problem in coefficients for Δp-Laplacian

Olha P Kupenko12 and Rosanna Manzo3*

Author Affiliations

1 Department of System Analysis and Control, National Mining University, Karl Marx av., 19, Dnipropetrovsk, 49005, Ukraine

2 Institute of Applied and System Analysis, National Technical University of Ukraine ‘Kiev Polytechnical Institute’, Peremogy av., 37, build. 35, Kiev, 03056, Ukraine

3 Dipartimento di Ingegneria dell’Informazione, Ingegneria Elettrica e Matematica Applicata, Università degli Studi di Salerno, Via Giovanni Paolo II, 132, Fisciano, 84084, Italy

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Boundary Value Problems 2014, 2014:72  doi:10.1186/1687-2770-2014-72

Published: 28 March 2014

Abstract

In this paper we study an optimal control problem for a nonlinear monotone Dirichlet problem where the control is taken as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M2">View MathML</a>-coefficient of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/72/mathml/M1">View MathML</a>-Laplacian. Given a cost function, the objective is to derive first-order optimality conditions and provide their substantiation. We propose some ideas and new results concerning the differentiability properties of the Lagrange functional associated with the considered control problem. The obtained adjoint boundary value problem is not coercive and, hence, it may admit infinitely many solutions. That is why we concentrate not only on deriving the adjoint system, but also, following the well-known Hardy-Poincaré Inequality, on a formulation of sufficient conditions which would guarantee the uniqueness of the adjoint state to the optimal pair.

MSC: 35J70, 49J20, 49J45, 93C73.

Keywords:
nonlinear Dirichlet problem; control in coefficients; optimality conditions