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Optimal design of sensors for a damped wave equation. (English) Zbl 1381.93027

Summary: In this paper we model and solve the problem of shaping and placing in an optimal way sensors for a wave equation with constant damping in a bounded open connected subset \(\Omega\) of \(\mathbb{R}^n\). Sensors are modeled by subdomains of \(\Omega\) of a given measure \(L|\Omega|\), with \(0 < L < 1\). We prove that, if \(L\) is close enough to \(1\), then the optimal design problem has a unique solution, which is characterized by a finite number of low frequency modes. In particular the maximizing sequence built from spectral approximations is stationary.

MSC:

93B07 Observability
35L20 Initial-boundary value problems for second-order hyperbolic equations
35P20 Asymptotic distributions of eigenvalues in context of PDEs
49K45 Optimality conditions for problems involving randomness
93B51 Design techniques (robust design, computer-aided design, etc.)
93C20 Control/observation systems governed by partial differential equations
58J51 Relations between spectral theory and ergodic theory, e.g., quantum unique ergodicity
49K20 Optimality conditions for problems involving partial differential equations
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References:

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