Advances in Differential Equations

On shape stability of Dirichlet optimal control problems in coefficients for nonlinear elliptic equations

Ciro D'Apice, Umberto De Maio, and Ol'ga P. Kogut

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In this paper we study a classical Dirichlet optimal control problem for a nonlinear elliptic equation with the coefficients as controls in $L^\infty(\Omega)$. Since such problems have no solutions in general, we make an assumption on the coefficients of the state equation and introduce the class of so-called solenoidal controls. Using the direct method in the calculus of variations, we prove the existence of at least one optimal pair. We also study the stability of the above optimal control problem with respect to the domain perturbation. With this goal we introduce the concept of Mosco-stability for such problems and analyze the variational properties of Mosco-stable problems with respect to different types of domain perturbations.

Article information

Adv. Differential Equations Volume 15, Number 7/8 (2010), 689-720.

First available in Project Euclid: 18 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35B20: Perturbations 35B30: Dependence of solutions on initial and boundary data, parameters [See also 37Cxx] 35J50: Variational methods for elliptic systems 35J65: Nonlinear boundary value problems for linear elliptic equations 47H05: Monotone operators and generalizations 49J20: Optimal control problems involving partial differential equations


D'Apice, Ciro; De Maio, Umberto; Kogut, Ol'ga P. On shape stability of Dirichlet optimal control problems in coefficients for nonlinear elliptic equations. Adv. Differential Equations 15 (2010), no. 7/8, 689--720.

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