Abstract
The present article is concerned with the numerical implementation of the Hilbert uniqueness method for solving exact and approximate boundary controllability problems for the heat equation. Using convex duality, we reduce the solution of the boundary control problems to the solution of identification problems for the initial data of an adjoint heat equation. To solve these identification problems, we use a combination of finite difference methods for the time discretization, finite element methods for the space discretization, and of conjugate gradient and operator splitting methods for the iterative solution of the discrete control problems. We apply then the above methodology to the solution of exact and approximate boundary controllability test problems in two space dimensions. The numerical results validate the methods discussed in this article and clearly show the computational advantage of using second-order accurate time discretization methods to approximate the control problems.
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Dedicated to J. Céa on his 60th birthday
This work was partially supported by the National Science Foundation Grant INT-8612680 the Texas Board of Higher Education Grant ARP-003652156, and the Faculty Development Program, University of Houston (special thanks are due to Glenn Aumann and Garret J. Etgen). The authors also thank J. A. Wilson for diligently processing the manuscript of this article.
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Carthel, C., Glowinski, R. & Lions, J.L. On exact and approximate boundary controllabilities for the heat equation: A numerical approach. J Optim Theory Appl 82, 429–484 (1994). https://doi.org/10.1007/BF02192213
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DOI: https://doi.org/10.1007/BF02192213