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Optimal observation of the one-dimensional wave equation. (English) Zbl 1302.93052

Summary: In this paper, we consider the homogeneous one-dimensional wave equation on \([0,\pi]\) with Dirichlet boundary conditions, and observe its solutions on a subset \(\omega\) of \([0,\pi]\). Let \(L\in(0,1)\). We investigate the problem of maximizing the observability constant, or its asymptotic average in time, over all possible subsets \(\omega\) of \([0,\pi]\) of Lebesgue measure \(L\pi\). We solve this problem by means of Fourier series considerations, give the precise optimal value and prove that there does not exist any optimal set except for \(L=1/2\). When \(L\neq 1/2\) we prove the existence of solutions of a relaxed minimization problem, proving a no gap result. Following P. Hébrard, A. Henrott [”Optimal shape and position of the actuators for the stabilization of a string”, Syst. Control Lett. 48, No. 3-4, 199-209 (2003; Zbl 1134.93399)] and P. Hébrard, A. Henrott [”A spillover phenomenon in the optimal location of actuators”, SIAM J. Control Optimization 44, No. 1, 349-366 (2005; Zbl 1083.49002)], we then provide and solve a modal approximation of this problem, show the oscillatory character of the optimal sets, the so-called spillover phenomenon, which explains the lack of existence of classical solutions for the original problem.

MSC:

93B07 Observability
35L05 Wave equation
49K20 Optimality conditions for problems involving partial differential equations
42B37 Harmonic analysis and PDEs
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