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Approximation of optimal control problems in the coefficient for the p-Laplace equation. I. Convergence result

Casas E, Kogut PI, Leugering G (2016)

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Publication Status: Published

Publication Type: Journal article, Original article

Publication year: 2016

Journal

Publisher: Society for Industrial and Applied Mathematics Publications

Book Volume: 54

Pages Range: 1406-1422

Journal Issue: 3

DOI: 10.1137/15M1028108

Abstract

We study a Dirichlet optimal control problem for a quasi-linear monotone elliptic equation, the so-called weighted p-Laplacian problem. The coefficient of the -Laplacian, the weight u, we take as a control in BV (Ω) ∩ L∞(Ω). In this article, we use box-type constraints for the control such that there is a strictly positive lower and some upper bound. In order to handle the inherent degeneracy of the -Laplacian, we use a regularization, sometimes referred to as the -Laplacian. We derive existence and uniqueness of solutions to the underlying boundary value problem and the optimal control problem. In fact, we introduce a two-parameter model for the weighted -Laplacian, where we approximate the nonlinearity by a bounded monotone function, parametrized by κ. Further, we discuss the asymptotic behavior of the solutions to the regularized problem on each (ϵ, κ)-level as the parameters tend to zero and infinity, respectively.

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APA:

Casas, E., Kogut, P.I., & Leugering, G. (2016). Approximation of optimal control problems in the coefficient for the p-Laplace equation. I. Convergence result. SIAM Journal on Control and Optimization, 54(3), 1406-1422. https://dx.doi.org/10.1137/15M1028108

MLA:

Casas, Eduardo, Peter I. Kogut, and Günter Leugering. "Approximation of optimal control problems in the coefficient for the p-Laplace equation. I. Convergence result." SIAM Journal on Control and Optimization 54.3 (2016): 1406-1422.

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