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Optimal shape and location of sensors for parabolic equations with random initial data. (English) Zbl 1319.35272

The purpose of this article is to study observability of parabolic equations, in particular the heat equation and the anomalous diffusion equation. The method of proof is Fourier series for random variables. The proofs use Fubinis theorem, minimax arguments, Bessel functions, Cauchy-Schwartz inequality, Kapteyn inequality for Bessel functions, Lagrange multiplier, Hölder inequality.

MSC:

35Q90 PDEs in connection with mathematical programming
80A20 Heat and mass transfer, heat flow (MSC2010)
35K10 Second-order parabolic equations
93B07 Observability
90C15 Stochastic programming
49Q10 Optimization of shapes other than minimal surfaces

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Ipopt; AMPL
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