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Optimization of non-cylindrical domains for the exact null controllability of the 1D wave equation. (English) Zbl 1467.49031

Summary: This work is concerned with the null controllability of the one-dimensional wave equation over non-cylindrical distributed domains. The controllability in that case has been obtained by C. Castro et al. [SIAM J. Control Optim. 52, No. 6, 4027–4056 (2015; Zbl 1320.35185)] for domains satisfying the usual geometric optic condition. We analyze the problem of optimizing the non-cylindrical support \(q\) of the control of minimal \(L^2(q)\)-norm. In this respect, we prove a uniform observability inequality for a class of domains \(q\) satisfying the geometric optic condition. The proof based on the d’Alembert formula relies on arguments from graph theory. Numerical experiments are discussed and highlight the influence of the initial condition on the optimal domains.

MSC:

49Q10 Optimization of shapes other than minimal surfaces
93C20 Control/observation systems governed by partial differential equations
49M41 PDE constrained optimization (numerical aspects)

Citations:

Zbl 1320.35185

Software:

FreeFem++
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References:

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