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Control and Stabilization of Waves on 1-d Networks

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Modelling and Optimisation of Flows on Networks

Part of the book series: Lecture Notes in Mathematics ((LNMCIME,volume 2062))

Abstract

We present some recent results on control and stabilization of waves on 1-d networks.The fine time-evolution of solutions of wave equations on networks and, consequently, their control theoretical properties, depend in a subtle manner on the topology of the network under consideration and also on the number theoretical properties of the lengths of the strings entering in it. Therefore, the overall picture is quite complex.In this paper we summarize some of the existing results on the problem of controllability that, by classical duality arguments in control theory, can be reduced to that of observability of the adjoint uncontrolled system. The problem of observability refers to that of recovering the total energy of solutions by means of measurements made on some internal or external nodes of the network. They lead, by duality, to controllability results guaranteeing that L 2-controls located on those nodes may drive sufficiently smooth solutions to equilibrium at a final time. Most of our results in this context, obtained in collaboration with R. Dáger, refer to the problem of controlling the network from one single external node. It is, to some extent, the most complex situation since, obviously, increasing the number of controllers enhances the controllability properties of the system. Our methods of proof combine sidewise energy estimates (that in the particular case under consideration can be derived by simply applying the classical d’Alembert’s formula), Fourier series representations, non-harmonic Fourier analysis, and number theoretical tools.These control results belong to the class of the so-called open-loop control systems.We then discuss the problem of closed-loop control or stabilization by feedback. We present a recent result, obtained in collaboration with J. Valein, showing that the observability results previously derived, regardless of the method of proof employed, can also be recast a posteriori in the context of stabilization, so to derive explicit decay rates (as \(t \rightarrow \infty \)) for the energy of smooth solutions. The decay rate depends in a very sensitive manner on the topology of the network and the number theoretical properties of the lengths of the strings entering in it.In the end of the article we also present some challenging open problems.

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Notes

  1. 1.

    As mentioned above, most of this article is devoted to the case in which the observation is only done on one external node of the network.

  2. 2.

    Once more, we shall focus in the case in which one single controller acts on one of the external nodes of the network.

  3. 3.

    We refer to [48, 50] for relatively complete and updated surveys on the state of the art of the observability and controllability of PDE’s.

  4. 4.

    More rigorously, for the adjoint system, the initial data should be given at time t = T, but the system under consideration being time-reversible, we may consider equally that the initial data are given at t = 0.

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Acknowledgements

The author has been partially supported by Grant MTM2008-03541 of the MICINN, Spain, the ERC Advanced Grant FP7-246775 NUMERIWAVES and the Grant PI2010-04 of the Basque Government.

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  1. BCAM – Basque Center for Applied Mathematics, Alameda Mazarredo, 14, E-48009, Bilbao, Basque Country, Spain

    Enrique Zuazua

  2. Ikerbasque, Basque Foundation for Science, Alameda Urquijo 36-5, Plaza Bizkaia, 48011, Bilbao, Basque Country, Spain

    Enrique Zuazua

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Zuazua, E. (2013). Control and Stabilization of Waves on 1-d Networks. In: Modelling and Optimisation of Flows on Networks. Lecture Notes in Mathematics(), vol 2062. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-32160-3_9

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