×

Optimal design of the damping set for the stabilization of the wave equation. (English) Zbl 1105.49005

Summary: We consider the problem of optimizing the shape and position of the damping set for the internal stabilization of the linear wave equation in \(\mathbb {R}^N\), \(N=1,2\). In a first theoretical part, we reformulate the problem into an equivalent non-convex vector variational one using a characterization of divergence-free vector fields. Then, by means of gradient Young measures, we obtain a relaxed formulation of the problem in which the original cost density is replaced by its constrained quasi-convexification. This implies that the new relaxed problem is well-posed in the sense that there exists a minimizer and, in addition, the infimum of the original problem coincides with the minimum of the relaxed one. In a second numerical part, we address the resolution of the relaxed problem using a first-order gradient descent method. We present some numerical experiments which highlight the influence of the over-damping phenomena and show that for large values of the damping potential the original problem has no minimizer. We then propose a penalization technique to recover the minimizing sequences of the original problem from the optimal solution of the relaxed one.

MSC:

49J20 Existence theories for optimal control problems involving partial differential equations
35B37 PDE in connection with control problems (MSC2000)
35L05 Wave equation
93C20 Control/observation systems governed by partial differential equations
49J45 Methods involving semicontinuity and convergence; relaxation
49M30 Other numerical methods in calculus of variations (MSC2010)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Allaire, G., Shape Optimization by the Homogenization Method, Appl. Math. Sci., vol. 146 (2002), Springer · Zbl 0990.35001
[2] Castro, C.; Cox, S. J., Achieving arbitrarily large decay in the damped wave equation, SIAM J. Control Optim., 6, 1748-1755 (2001) · Zbl 0983.35095
[3] Cox, S. J.; Zuazua, E., The rate at which energy decays in a damped string, Comm. Partial Differential Equations, 19, 213-243 (1994) · Zbl 0818.35072
[4] Cox, S. J., Designing for optimal energy absorption II, The damped wave equation, Internat. Ser. Numer. Math., 126, 103-109 (1998) · Zbl 0992.93037
[5] Fahroo, F.; Ito, K., Variational formulation of optimal damping designs, Contemp. Math., 209, 95-114 (1997) · Zbl 0931.49019
[6] Freitas, P., On some eigenvalues problems related to the wave equation with indefinite damping, J. Differential Equations, 127, 320-335 (1996) · Zbl 0857.35077
[7] López-Gómez, J., On the linear damped wave equation, J. Differential Equations, 134, 26-45 (1997) · Zbl 0959.35109
[8] Hebrard, P.; Henrot, A., Optimal shape and position of the actuators for the stabilization of a string, Systems Control Lett., 48, 199-209 (2003) · Zbl 1134.93399
[9] Kotiuga, P. R., Clebsch potentials and the visualization of three-dimensional solenoidal vector fields, IEEE Trans. Magnetics, 27, 5, 3986-3989 (1991)
[10] Lions, J.-L.; Magenes, E., Problèmes aux Limites Non Homogènes et Applications (1968), Dunod: Dunod Paris · Zbl 0165.10801
[11] A. Münch, F. Maestre, P. Pedregal, Optimal design under the one-dimensional wave equation, submitted for publication; A. Münch, F. Maestre, P. Pedregal, Optimal design under the one-dimensional wave equation, submitted for publication
[12] A. Münch, Optimal internal stabilization of a damped wave equation by a level set approach, submitted for publication; A. Münch, Optimal internal stabilization of a damped wave equation by a level set approach, submitted for publication
[13] Nigam, S. D.; Usha, R.; Swaminathan, K., Divergence-free vector fields, J. Math. Phys. Sci., 14, 5, 523-527 (1980) · Zbl 0475.53014
[14] Pedregal, P., Parametrized Measures and Variational Principles (1997), Birkhäuser · Zbl 0879.49017
[15] Pedregal, P., Vector variational problems and applications to optimal design, ESAIM Control Optim. Calc. Var., 11, 357-381 (2005) · Zbl 1089.49022
[16] Shivamoggi, B. K., Theoretical Fluid Dynamics (1998), Wiley: Wiley New York · Zbl 0897.76001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.