. On-veut-la-continuité-en, pour la norme de 1f.; ce qui impose déjà que ayx E HI(O, 1), i.e. y E H 2 (0, 1), z E HI(O, 1), u = z(O)

A. Ensuite-(-<po and U. Implique, -li z( ayx)x dx -li ayxzx dx -u[ayx(O) +

. Vry, O, 1) X HI(O, 1) X JR.2, avec u = z(O), et v = z(I)

E. H. Vz and I. , avecu = z(O), et v = z(I), i.e.: llzxaYx-(JCz(O) = 0

. Donc, Il (1'? + aY;) dx + me + Mij2] + l' g(s)Y/(O,s) dsl :<::: c te

T. Cazenave and A. Haraux, Introduction aux problèmes d'évolution semi-linéaires, Ellipses, 1990.

G. Chen, M. C. Delfour, A. M. Krall, and G. , Modeling, Stabilization and Control of Serially Connected Beams, SIAM Journal on Control and Optimization, vol.25, issue.3, pp.526-546, 1987.
DOI : 10.1137/0325029

G. Chen, S. G. Krantz, D. W. Ma, C. E. Wayne, and H. H. West, The Euler-Bernoulli beam equation with boundary energy dissipation, Operator methods for optimal control problems, Lecture in Pure App!. Math. Series, Marcel Dekker, pp.67-96, 1987.

B. Chentouf, Contribution à la stabilité et à la stabilisation de systèmes à paramètres répartis, Thèse de l, 1998.

F. Conrad and A. Mifdal, Strong stability of a mode! of an overhead crane, Control and Cybernetics, pp.363-394, 1998.

F. Conrad and A. Mifdal, Uniform stabilization of a hybrid system with a class of nonlinear feedback laws, paraître dans Advances in Mathematical Sciences and Applications

F. Conrad and O. Morgül, On the Stabilization of a Flexible Beam with a Tip Mass, SIAM Journal on Control and Optimization, vol.36, issue.6, pp.1962-1986, 1998.
DOI : 10.1137/S0363012996302366

J. S. Gibson, A note on stabilization of infinite dimensionllinear oscilliators by compact linear feedback, SIAM Journal on Control and Optimization, issue.18, pp.311-316, 1980.

P. Grabowski, Spectral approch ta well-posedness and stability analysis of hybrid feedback systems, 1996. [12] A. H. Haraux, Systèmes dynamiques dissipatifs et applications, 1991.

F. L. Huang, Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces, Ann. Diff. Eqs.l(I), pp.43-53, 1985.

R. E. Langer, On the zeros of exponential sums and integrals, Bulletin of the American Mathematical Society, vol.37, issue.4, pp.213-239, 1931.
DOI : 10.1090/S0002-9904-1931-05133-8

H. Laousy, Sur quelques problèmes de stabilisation de systèmes à paramètres distribués, Thèse de j, 1997.

H. Laousy and B. Chentouf, On the boundary stabilization of a hybrid system, 1999.

W. Littman and L. Markus, Stabilization of a hybrid system of elasticity by feedback boundary damping, Annali di Matematica Pura ed Applicata, vol.152, issue.1, pp.281-330, 1988.
DOI : 10.1007/978-1-4684-9333-7

A. Pazy, Semi-groups of linear operators and applications to partial differential equations, 1983.

B. Rao, Uniform Stabilization of a Hybrid System of Elasticity, SIAM Journal on Control and Optimization, vol.33, issue.2, pp.440-454, 1995.
DOI : 10.1137/S0363012992239879

P. Rideau, Contrôle d'un assemblage de poutres flexibles par des capteurs-actionneurs ponctuels : étude du spectre du système, 1995.

D. L. Russell, Decay rates for weakly damped systems in Hilbert space obtained with control-theoretic methods, Journal of Differential Equations, vol.19, issue.2, pp.344-370, 1975.
DOI : 10.1016/0022-0396(75)90009-1

F. Z. Saouri, Stabilisation d'une poutre avec controle force. Etude du taux optimal de décroissance de l'énergie élastique, p.47, 1997.

A. Shkalikov, Boundary problems for ordinary differential equations with parameter in the boundary conditions, Journal of Soviet Mathematics, vol.160, issue.No. 2, pp.1311-1342, 1986.
DOI : 10.1007/BF01084754

Y. D. Tamarkin, General problems oftbe tbeory of ordinary differential equations and expansions of arbitrary functions in series, 1917.