# American Institute of Mathematical Sciences

March  2014, 4(1): 101-113. doi: 10.3934/mcrf.2014.4.101

## Almost periodic solutions for a weakly dissipated hybrid system

 1 Universitatea din Craiova, Craiova 200585, Romania 2 Institute of Mathematics, Federal University of Rio de Janeiro, UFRJ, P.O. Box 68530, CEP 21941-909, Rio de Janeiro, RJ, Brazil

Received  May 2012 Revised  January 2013 Published  December 2013

We consider a hybrid system coupling an elastic string with a rigid body at one end and we study the existence of an almost periodic solution when an almost periodic force $f$ acts on the body. The weak dissipation of the system does not allow to show the relative compactness of the trajectories which generally implies the existence of such solutions. Instead, we use Fourier analysis to show that the existence or not of the almost periodic solutions depends on the regularity and the exponents of the almost periodic nonhomogeneous term $f$.
Citation: Sorin Micu, Ademir F. Pazoto. Almost periodic solutions for a weakly dissipated hybrid system. Mathematical Control and Related Fields, 2014, 4 (1) : 101-113. doi: 10.3934/mcrf.2014.4.101
##### References:
 [1] B. d'Andréa-Novel, F. Boustany, F. Conrad and B. P. Rao, Feedback stabilization of a hybrid PDE-ODE system: Application to an overhead crane, Math. Control Signals Systems, 7 (1994), 1-22. doi: 10.1007/BF01211483. [2] G. Avalos and I. Lasiecka, Uniform decay rates for solutions to a structural acoustics model with nonlinear dissipation, Appl. Math. Comput. Sci., 8 (1998), 287-312. [3] H. T. Banks and R. C. Smith, Feedback control of noise in a 2-D nonlinear structural acoustics model, Discrete Contin. Dynam. Systems, 1 (1995), 119-149. [4] H. Bohr, Almost Periodic Functions, Chelsea Publishing Company, New York, 1947. [5] C. Castro and E. Zuazua, Boundary controllability of a hybrid system consisting in two flexible beams connected by a point mass, SIAM J. Control Optim., 36 (1998), 1576-1595. doi: 10.1137/S0363012997316378. [6] T. Cazenave and A. Haraux, An Introduction to Semilinear Evolution Equations, Oxford Lecture Series in Mathematics and its Applications, 13, The Clarendon Press, Oxford University Press, New York, 1998. [7] N. Cîndea, S. Micu and A. F. Pazoto, Periodic solutions for a weakly dissipated hybrid system, J. Math. Anal. Appl., 385 (2012), 399-413. [8] L. Cot, J.-P. Raymond and J. Vancostenoble, Exact controllability of an aeroacoustic model with a Neumann and a Dirichlet boundary control, SIAM J. Control Optim., 48 (2009), 1489-1518. doi: 10.1137/070685609. [9] C. Corduneanu, Almost Periodic Oscillations and Waves, Springer, New York, 2009. doi: 10.1007/978-0-387-09819-7. [10] A. M. Fink, Almost Periodic Differetial Equation, Lecture Notes in Mathematics, Vol. 377, Springer-Verlag, Berlin-New York, 1974. [11] S. Hansen and E. Zuazua, Exact controllability and stabilization of strings with point masses, SIAM J. Cont. Optim., 33 (1995), 1357-1391. doi: 10.1137/S0363012993248347. [12] A. Haraux, Semi-linear hyperbolic problems in bounded domains, Math. Rep., 3 (1987), 1-281. [13] A. E. Ingham, Some trigonometric inequalities with applications to the theory of series, Math. Zeits., 41 (1936), 367-379. doi: 10.1007/BF01180426. [14] J. E. Lagnese, Modelling and controllability of plate-beam systems, J. Math. Systems Estim. Control, 4 (1994), 47 pp. [15] E. B. Lee and Y. C. You, Stabilization of a vibrating string system linked by point masses, in Control of Boundaries and Stabilization (Clermont-Ferrand, 1988), Lecture Notes in Control and Inform. Sci., 125, Springer, Berlin, 1989, 177-198. [16] E. B. Lee and Y. C. You, Stabilization of a hybrid (string/point mass) system, in Proc. Fifth Int. Conf. Syst. Eng. (Dayton, Ohio, EUA), 1987. [17] B. M. Levitan and V. V. Zhikov, Almost periodic functions and differential equations, Cambridge University Press, Cambridge-New York, 1982. [18] W. Littman and L. Markus, Some recent results on control and stabilization of flexible structures, in Proc. COMCON on Stabilization of Flexible Structures (Montpellier, France), 1987, 151-161. [19] W. Littman and L. Markus, Exact boundary controllability of a hybrid system of elasticity, Arch. Rational Mech. Anal., 103 (1988), 193-236. doi: 10.1007/BF00251758. [20] S. Micu and E. Zuazua, Asymptotics for the spectrum of a fluid/structure hybrid system arising in the control of noise, SIAM J. Math. Anal., 29 (1998), 967-1001. doi: 10.1137/S0036141096312349. [21] O. Morgul, B. Rao and F. Conrad, On the stabilization of a cable with a tip mass, IEEE Trans. Automat. Control, 39 (1994), 2140-2145. doi: 10.1109/9.328811. [22] B. Rao, Uniform stabilization of a hybrid system of elasticity, SIAM J. Control Optim., 33 (1995), 440-454. doi: 10.1137/S0363012992239879. [23] B. Rao, Decay estimates of solutions for a hybrid system of flexible structures, European J. Appl. Math., 4 (1993), 303-319. doi: 10.1017/S0956792500001133. [24] J.-P. Raymond and M. Vanninathan, Exact controllability in fluid-solid structure: The Helmholtz model, ESAIM Control Optim. Calc. Var., 11 (2005), 180-203. doi: 10.1051/cocv:2005006.

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##### References:
 [1] B. d'Andréa-Novel, F. Boustany, F. Conrad and B. P. Rao, Feedback stabilization of a hybrid PDE-ODE system: Application to an overhead crane, Math. Control Signals Systems, 7 (1994), 1-22. doi: 10.1007/BF01211483. [2] G. Avalos and I. Lasiecka, Uniform decay rates for solutions to a structural acoustics model with nonlinear dissipation, Appl. Math. Comput. Sci., 8 (1998), 287-312. [3] H. T. Banks and R. C. Smith, Feedback control of noise in a 2-D nonlinear structural acoustics model, Discrete Contin. Dynam. Systems, 1 (1995), 119-149. [4] H. Bohr, Almost Periodic Functions, Chelsea Publishing Company, New York, 1947. [5] C. Castro and E. Zuazua, Boundary controllability of a hybrid system consisting in two flexible beams connected by a point mass, SIAM J. Control Optim., 36 (1998), 1576-1595. doi: 10.1137/S0363012997316378. [6] T. Cazenave and A. Haraux, An Introduction to Semilinear Evolution Equations, Oxford Lecture Series in Mathematics and its Applications, 13, The Clarendon Press, Oxford University Press, New York, 1998. [7] N. Cîndea, S. Micu and A. F. Pazoto, Periodic solutions for a weakly dissipated hybrid system, J. Math. Anal. Appl., 385 (2012), 399-413. [8] L. Cot, J.-P. Raymond and J. Vancostenoble, Exact controllability of an aeroacoustic model with a Neumann and a Dirichlet boundary control, SIAM J. Control Optim., 48 (2009), 1489-1518. doi: 10.1137/070685609. [9] C. Corduneanu, Almost Periodic Oscillations and Waves, Springer, New York, 2009. doi: 10.1007/978-0-387-09819-7. [10] A. M. Fink, Almost Periodic Differetial Equation, Lecture Notes in Mathematics, Vol. 377, Springer-Verlag, Berlin-New York, 1974. [11] S. Hansen and E. Zuazua, Exact controllability and stabilization of strings with point masses, SIAM J. Cont. Optim., 33 (1995), 1357-1391. doi: 10.1137/S0363012993248347. [12] A. Haraux, Semi-linear hyperbolic problems in bounded domains, Math. Rep., 3 (1987), 1-281. [13] A. E. Ingham, Some trigonometric inequalities with applications to the theory of series, Math. Zeits., 41 (1936), 367-379. doi: 10.1007/BF01180426. [14] J. E. Lagnese, Modelling and controllability of plate-beam systems, J. Math. Systems Estim. Control, 4 (1994), 47 pp. [15] E. B. Lee and Y. C. You, Stabilization of a vibrating string system linked by point masses, in Control of Boundaries and Stabilization (Clermont-Ferrand, 1988), Lecture Notes in Control and Inform. Sci., 125, Springer, Berlin, 1989, 177-198. [16] E. B. Lee and Y. C. You, Stabilization of a hybrid (string/point mass) system, in Proc. Fifth Int. Conf. Syst. Eng. (Dayton, Ohio, EUA), 1987. [17] B. M. Levitan and V. V. Zhikov, Almost periodic functions and differential equations, Cambridge University Press, Cambridge-New York, 1982. [18] W. Littman and L. Markus, Some recent results on control and stabilization of flexible structures, in Proc. COMCON on Stabilization of Flexible Structures (Montpellier, France), 1987, 151-161. [19] W. Littman and L. Markus, Exact boundary controllability of a hybrid system of elasticity, Arch. Rational Mech. Anal., 103 (1988), 193-236. doi: 10.1007/BF00251758. [20] S. Micu and E. Zuazua, Asymptotics for the spectrum of a fluid/structure hybrid system arising in the control of noise, SIAM J. Math. Anal., 29 (1998), 967-1001. doi: 10.1137/S0036141096312349. [21] O. Morgul, B. Rao and F. Conrad, On the stabilization of a cable with a tip mass, IEEE Trans. Automat. Control, 39 (1994), 2140-2145. doi: 10.1109/9.328811. [22] B. Rao, Uniform stabilization of a hybrid system of elasticity, SIAM J. Control Optim., 33 (1995), 440-454. doi: 10.1137/S0363012992239879. [23] B. Rao, Decay estimates of solutions for a hybrid system of flexible structures, European J. Appl. Math., 4 (1993), 303-319. doi: 10.1017/S0956792500001133. [24] J.-P. Raymond and M. Vanninathan, Exact controllability in fluid-solid structure: The Helmholtz model, ESAIM Control Optim. Calc. Var., 11 (2005), 180-203. doi: 10.1051/cocv:2005006.
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