Differential and Integral Equations

On homogenization of a mixed boundary optimal control problem

Ciro D'Apice and Umberto De Maio

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Abstract

We study the asymptotic behaviour of an optimal control problem for the Ukawa equation in a thick multi-structure with different types and classes of admissible boundary controls. This thick multi-structure consists of a domain (the junction's body) and a large number of $\varepsilon$-periodically situated thin cylinders. We consider two types of boundary controls, namely, the Dirichlet $H^{1/2}$-controls on the bases $\Gamma_{\varepsilon}$ of thin cylinders, and the Neumann $L^2$-controls on their 'vertical' sides. We present some ideas and results concerning of the asymptotic analysis for such problems as ${\varepsilon}\to 0$ and derive conditions under which the homogenized problem can be recovered in the explicit form. We show that the mathematical description of the homogenized optimal boundary control problem is different from the original one. These differences appear not only in the control constraints, limit cost functional, state equations, and boundary conditions, but also in the type of admissible controls for the limit problem - one of them is the Dirichlet $L^2$-control, whereas the second one is appeared as the distributed $L^2$-control.

Article information

Source
Differential Integral Equations, Volume 21, Number 3-4 (2008), 201-234.

Dates
First available in Project Euclid: 20 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.die/1356038777

Mathematical Reviews number (MathSciNet)
MR2484006

Zentralblatt MATH identifier
1224.35019

Subjects
Primary: 35B27: Homogenization; equations in media with periodic structure [See also 74Qxx, 76M50]
Secondary: 49J20: Optimal control problems involving partial differential equations 49J45: Methods involving semicontinuity and convergence; relaxation

Citation

D'Apice, Ciro; De Maio, Umberto. On homogenization of a mixed boundary optimal control problem. Differential Integral Equations 21 (2008), no. 3-4, 201--234. https://projecteuclid.org/euclid.die/1356038777


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