Optimal location of controllers for the one-dimensional wave equation |
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Authors: | Yannick Privat Emmanuel Trélat Enrique Zuazua |
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Affiliation: | 1. IRMAR, ENS Cachan Bretagne, Univ. Rennes 1, CNRS, UEB, av. Robert Schuman, 35170 Bruz, France;2. Université Pierre et Marie Curie (Univ. Paris 6) and Institut Universitaire de France, CNRS UMR 7598, Laboratoire Jacques-Louis Lions, F-75005 Paris, France;3. BCAM – Basque Center for Applied Mathematics, Mazarredo, 14 E-48009 Bilbao, Basque Country, Spain;4. Ikerbasque, Basque Foundation for Science, Alameda Urquijo 36-5, Plaza Bizkaia, E-48011 Bilbao, Basque Country, Spain |
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Abstract: | In this paper, we consider the homogeneous one-dimensional wave equation defined on . For every subset of positive measure, every , and all initial data, there exists a unique control of minimal norm in steering the system exactly to zero. In this article we consider two optimal design problems. Let . The first problem is to determine the optimal shape and position of ω in order to minimize the norm of the control for given initial data, over all possible measurable subsets ω of of Lebesgue measure Lπ. The second problem is to minimize the norm of the control operator, over all such subsets. Considering a relaxed version of these optimal design problems, we show and characterize the emergence of different phenomena for the first problem depending on the choice of the initial data: existence of optimal sets having a finite or an infinite number of connected components, or nonexistence of an optimal set (relaxation phenomenon). The second problem does not admit any optimal solution except for . Moreover, we provide an interpretation of these problems in terms of a classical optimal control problem for an infinite number of controlled ordinary differential equations. This new interpretation permits in turn to study modal approximations of the two problems and leads to new numerical algorithms. Their efficiency will be exhibited by several experiments and simulations. |
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Keywords: | Wave equation Exact controllability HUM method Shape optimization Relaxation Optimal control Pontryagin Maximum Principle |
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