Abstract
This paper concerns a system of nonlinear wave equations describing the vibrations of a 3-dimensional network of elastic strings. The authors derive the equations and appropriate nodal conditions, determine equilibrium solutions, and, by using the methods of quasilinear hyperbolic systems, prove that for tree networks the natural initial, boundary value problem has classical solutions existing in neighborhoods of the “stretched” equilibrium solutions. Then the local controllability of such networks near such equilibrium configurations in a certain specified time interval is proved. Finally, it is proved that, given two different equilibrium states satisfying certain conditions, it is possible to control the network from states in a small enough neighborhood of one equilibrium to any state in a suitable neighborhood of the second equilibrium over a sufficiently large time interval.
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References
Courant, R. and Hilbert, D., Methods of Mathematical Physics, Volume II, Interscience Publishers, New York, 1962.
Deimling, K., Nonlinear Functional Analysis, Springer-Verlag, Berlin, 1985.
Ekeland, I. and Teman, R., Analyse Convexe et Problmes Variationelles, Dunod, Gauthier-Villars, Paris, 1974.
Gu, Q. L. and Li, T. T., Exact boundary controllability for quasilinear wave equations in a tree-like planar network of strings, Ann. I. H. Poincare AN, 26, 2009, 2373–2384.
Lagnese, J. E., Leugering, G. and Schmidt, E. J. P. G., Modeling, Analysis and Control of Dynamic Elastic Multi-Link Structures, Birkhauser, Boston, Basel, Berlin, 1994.
Leugering, G. and Schmidt, E. J. P. G., On the control of networks of vibrating strings and beams, Proceedings of the 28th Conference on Decision and Control, Tampa, Florida, December, 1989, 2287–2290.
Li, T. T., Controllability and Observability for Quasilinear Hyperbolic Systems, AIMS Series Applied Mathematics, Vol. 3, American Institute of Mathematical Sciences & Higher Education Press, Springfield, Beijing, 2010.
Li, T. T. and Jin, Y., Semiglobal solutions to the mixed initial-boundary value problem for quasilinear hyperbolic systems, Chin. Ann. Math., 22B(3), 2001, 325–336.
Li, T. T., Serre, D. and Zhang, H., The generalized Riemann problem for the motion of elastic strings, SIAM J. Math. Anal., 23, 1992, 1189–1203.
Li, T. T. and Peng, Y. J., Problème de Riemann géneralisé pour uns sorte de système des cables, Portugaliae Mathematica, 50(40), 1993, 407–437.
Li, T. T., Global solutions to systems of the motion of elastic strings, Computational Science for the 21st Century, John Wiley & Sons, New York, 1997, 13–22.
Littman, W., Hyperbolic boundary control in one space dimension, Proceedings of the 27th Conference on Decision and Control, Austin, Texas, December, 1988, 1253–1254.
Schmidt, E. J. P. G., On the modelling and exact controllability of networks of vibrating strings, SIAM J. Cont. and Opt., 31, 1992, 230–245.
Schmidt, E. J. P. G., On a non-linear wave equation and the control of an elastic string from one equilibrium location to another, J. Math. Anal. Appl., 272, 2002, 536–554.
Schmidt, E. J. P. G. and Ming, W., On the modelling and analysis of networks of vibrating strings and masses, Report #91-13 from the Department of Mathematics and Statistics, McGill University, Montreal, 1991.
Wang, Z., Exact controllability for nonautonomous first order quasilinear hyperbolic systems, Chin. Ann. Math., 27B(6), 2006, 643–656.
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Project supported by the DFG SPP1253: Optimization with PDE-Constaints, and the DFG-CE315 Cluster of Excellence: Engineering of Advanced Materials.
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Leugering, G.R., Schmidt, E.J.P.G. On exact controllability of networks of nonlinear elastic strings in 3-dimensional space. Chin. Ann. Math. Ser. B 33, 33–60 (2012). https://doi.org/10.1007/s11401-011-0693-9
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DOI: https://doi.org/10.1007/s11401-011-0693-9