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Control of networks of Euler-Bernoulli beams

Published online by Cambridge University Press:  15 August 2002

Bertrand Dekoninck  and
Serge Nicaise
Bertrand Dekoninck
Affiliation:
Université de Valenciennes et du Hainaut Cambrésis, LIMAV, BP. 311, 59304 Valenciennes Cedex, France; [email protected].
Serge Nicaise
Affiliation:
Université de Valenciennes et du Hainaut Cambrésis, LIMAV, BP. 311, 59304 Valenciennes Cedex, France; [email protected].
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Abstract

We consider the exact controllability problem by boundary action of hyperbolic systems of networks of Euler-Bernoulli beams. Using the multiplier method and Ingham's inequality, we give sufficient conditions insuring the exact controllability for all time. These conditions are related to the spectral behaviour of the associated operator and are sufficiently concrete in order to be able to check them on particular networks as illustrated on simple examples.

Keywords

ControlEuler-Bernoulli beamsnetworksspectral analysis
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 4 , 1999 , pp. 57 - 81
Copyright
© EDP Sciences, SMAI, 1999

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References

Ali Mehmeti, F., A characterisation of generalized C ∞ notion on nets. Int. Eq. and Operator Theory 9 (1986) 753-766. CrossRef
Ali Mehmeti, F., Regular solutions of transmission and interaction problems for wave equations. Math. Meth. Appl. Sci. 11 (1989) 665-685. CrossRef
Ball, J.M. and Slemrod, M., Nonharmonic Fourier series and the stabilization of distributed semi-linear control systems. Comm. Pure Appl. Math. 32 (1979) 555-587. CrossRef
von Below, J., A characteristic equation associated to an eigenvalue problem on c2 -networks. Linear Alg. Appl. 71 (1985) 309-325. CrossRef
von Below, J., Classical solvability of linear parabolic equations on networks. J. Diff. Eq. 72 (1988) 316-337. CrossRef
von Below, J., Sturm-Liouville eigenvalue problems on networks. Math. Meth. Appl. Sci. 10 (1988) 383-395. CrossRef
J. von Below, Parabolic Network Equations. Habilitation Thesis, Eberhard-Karls-Universität Tübingen (1993).
von Below, J. and Nicaise, S., Dynamical interface transition with diffusion in ramified media. Comm. Partial Diff. Eq. 21 (1996) 255-279. CrossRef
Borovskikh, A., Mustafokulov, R., Lazarev, K. and Pokornyi, Yu., A class of fourth-order differential equations on a spatial net. Doklady Math. 52 (1995) 433-435.
Chen, G., Delfour, M., Krall, A. and Payre, G., Modelling, stabilization and control of serially connected beams. SIAM J. Control and Opt. 25 (1987) 526-546. CrossRef
Chen, G., Krantz, S., Russell, D., Wayne, C., West, H. and Coleman, M., Analysis, design, and behavior of dissipative joints for coupled beams. SIAM J. Appl. Math. 49 (1989) 1665-1693. CrossRef
Chen, G. and Zhou, J., The wave propagation method for the analysis of boudary stabilization in vibrating structures. SIAM J. Appl. Math. 50 (1990) 1254-1283. CrossRef
Ciarlet, P.G., Le Dret, H. and Nzengwa, R., Junctions between three-dimension and two-dimensional linearly elastic structures. J. Math. Pures Appl. 68 (1989) 261-295.
F. Conrad, Stabilization of vibrating beams by a specific feedback, A.V. Balakrishnan and J.P. Zolésio Eds., Stabilization of flexible structures, Opt. Software Inc. (1988) 36-51.
B. Dekoninck and S. Nicaise, The eigenvalue problem for networks of beams. Preprint LIMAV 96-9, University of Valenciennes, Linear Alg. Appl. (submitted).
P. Grisvard, Elliptic problems in nonsmooth domains. Monographs and Studies in Mathematics 21 (Pitman, Boston, 1985).
Grisvard, P., Contrôlabilité exacte des solutions de l'équation des ondes en présence de singularités. J. Math. Pures Appl. 68 (1989) 215-259.
Ingham, A.E., Some trigonometrical inequalities with applications in the theory of series. Math. Z. 41 (1936) 367-369. CrossRef
V. Komornik, Exact controllability and stabilization. The multiplier method. RMA 36 Masson, Paris (1994).
J.E. Lagnese, Modeling and controllability of plate-beam systems. J. Math. Systems, Estimation and Control. 5 (1995) 141-187.
Lagnese, J.E., Leugering, G. and Schmidt, E.J.P.G., Modeling of dynamic networks of thin thermoelastic beams. Math. Meth. Appl. Sci. 16 (1993) 327-358. CrossRef
Lagnese, J.E., Leugering, G. and Schmidt, E.J.P.G., Control of planar networks of Timoshenko beams. SIAM J. Cont. Opt. 31 (1993) 780-811. CrossRef
J.E. Lagnese, G. Leugering and E.J.P.G. Schmidt, Modeling, analysis and control of dynamic elastic multi-link structures, Birkhäuser, Boston (1994).
H. Le Dret, Problèmes variationnels dans les multi-domaines. Modélisation des jonctions et applications. RMA 19, Masson, Paris (1991).
G. Leugering and E.J.P.G. Schmidt, On the control of networks of vibrating strings and beams, in Proc. of the 28th IEEE Conference on Decision and Control, Vol. 3, IEEE (1989) 2287-2290.
J.-L. Lions, Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués. Tome 1, RMA 8, Masson, Paris (1988).
Nicaise, S., Exact controllability of a pluridimensional coupled problem. Rev. Math. Univ. Complutense Madrid 5 (1992) 91-135.
S. Nicaise, About the Lamé system in a polygonal or a polyhedral domain and a coupled problem between the Lamé system and the plate equation II: Exact controllability. Ann. Scuola Normale Sup. Pisa, Series IV 20 (1993) 163-191.
Nicaise, S., Boundary exact controllability of interface problems with singularities I: Addition of the coefficients of singularities. SIAM J. Contr. Opt. 34 (1996) 1512-1533. CrossRef
Nicaise, S., Boundary exact controllability of interface problems with singularities II: Addition of internal controls. SIAM J. Contr. Opt. 35 (1997) 585-603. CrossRef
J.P. Puel and E. Zuazua, Exact controllability for a model of multidimensional flexible structure. Proc. Royal Soc. Edinburgh 123 A (1993) 323-344.
Schmidt, E.J.P.G., On the modelling and exact controllability of networks of vibrating strings. SIAM J. Contr. Opt. 30 (1992) 229-245. CrossRef