Privat, Yannick; Trélat, Emmanuel; Zuazua, Enrique Optimal location of controllers for the one-dimensional wave equation. (English) Zbl 1296.49004 Ann. Inst. Henri Poincaré, Anal. Non Linéaire 30, No. 6, 1097-1126 (2013). Based on authors’ abstract: The system described by the homogeneous one–dimensional wave equation defined on \((0, \pi)\) is considered. For every subset \(\omega \subset [0, \pi]\) of positive measure, \(T \geq 2\pi\), there exists a unique control of minimal norm in \(L^2(0,T; L^2(\omega))\) steering the system exactly to zero. In this paper two optimal design problems are investigated. The first problem is to determine the optimal shape and position of \(\omega\) in order to minimize the norm of the control for given initial data over all possible subsets \(\omega \subset [0, \pi]\) of Lebegue measure \(L\pi\), where \(L \in (0,1)\). The second problem is to minimize the norm of the control operator over all such subsets. For both problems different interesting phenomena have been found and their interpretation in terms of a classical optimal control problem for an infinite number of controlled ODEs have been provided. Several numerical experiments and simulations are given. Reviewer: Wiesław Kotarski (Sosnowiec) Cited in 1 ReviewCited in 32 Documents MSC: 49J20 Existence theories for optimal control problems involving partial differential equations 35L05 Wave equation 49J15 Existence theories for optimal control problems involving ordinary differential equations 49Q10 Optimization of shapes other than minimal surfaces 49K35 Optimality conditions for minimax problems Keywords:wave equation; exact controllability; HUM method; shape opimization; optimal control; Pontryagin maximum principle Software:Ipopt; AMPL PDFBibTeX XMLCite \textit{Y. Privat} et al., Ann. Inst. Henri Poincaré, Anal. Non Linéaire 30, No. 6, 1097--1126 (2013; Zbl 1296.49004) Full Text: DOI References: [1] Akhiezer, N. 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