Almost periodic solutions for a weakly dissipated hybrid system

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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

Abstract
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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 $f$.

Keywords: hybrid system, Almost periodic solutions, Ingham's inequality., Fourier expansion, non homogeneous hyperbolic system.

Mathematics Subject Classification: Primary: 35B15; Secondary: 93D2.

Citation: Sorin Micu, Ademir F. Pazoto. Almost periodic solutions for a weakly dissipated hybrid system. Mathematical Control & 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. doi: 10.1007/BF01211483. Google Scholar
[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. Google Scholar
[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. Google Scholar
[4]	H. Bohr, Almost Periodic Functions,, Chelsea Publishing Company, (1947). Google Scholar
[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. doi: 10.1137/S0363012997316378. Google Scholar
[6]	T. Cazenave and A. Haraux, An Introduction to Semilinear Evolution Equations,, Oxford Lecture Series in Mathematics and its Applications, (1998). Google Scholar
[7]	N. Cîndea, S. Micu and A. F. Pazoto, Periodic solutions for a weakly dissipated hybrid system,, J. Math. Anal. Appl., 385 (2012), 399. Google Scholar
[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. doi: 10.1137/070685609. Google Scholar
[9]	C. Corduneanu, Almost Periodic Oscillations and Waves,, Springer, (2009). doi: 10.1007/978-0-387-09819-7. Google Scholar
[10]	A. M. Fink, Almost Periodic Differetial Equation,, Lecture Notes in Mathematics, (1974). Google Scholar
[11]	S. Hansen and E. Zuazua, Exact controllability and stabilization of strings with point masses,, SIAM J. Cont. Optim., 33 (1995), 1357. doi: 10.1137/S0363012993248347. Google Scholar
[12]	A. Haraux, Semi-linear hyperbolic problems in bounded domains,, Math. Rep., 3 (1987), 1. Google Scholar
[13]	A. E. Ingham, Some trigonometric inequalities with applications to the theory of series,, Math. Zeits., 41 (1936), 367. doi: 10.1007/BF01180426. Google Scholar
[14]	J. E. Lagnese, Modelling and controllability of plate-beam systems,, J. Math. Systems Estim. Control, 4 (1994). Google Scholar
[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), 177. Google Scholar
[16]	E. B. Lee and Y. C. You, Stabilization of a hybrid (string/point mass) system,, in Proc. Fifth Int. Conf. Syst. Eng.* (Dayton*, (1987). Google Scholar
[17]	B. M. Levitan and V. V. Zhikov, Almost periodic functions and differential equations,, Cambridge University Press, (1982). Google Scholar
[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*, (1987), 151. Google Scholar
[19]	W. Littman and L. Markus, Exact boundary controllability of a hybrid system of elasticity,, Arch. Rational Mech. Anal., 103 (1988), 193. doi: 10.1007/BF00251758. Google Scholar
[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. doi: 10.1137/S0036141096312349. Google Scholar
[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. doi: 10.1109/9.328811. Google Scholar
[22]	B. Rao, Uniform stabilization of a hybrid system of elasticity,, SIAM J. Control Optim., 33 (1995), 440. doi: 10.1137/S0363012992239879. Google Scholar
[23]	B. Rao, Decay estimates of solutions for a hybrid system of flexible structures,, European J. Appl. Math., 4 (1993), 303. doi: 10.1017/S0956792500001133. Google Scholar
[24]	J.-P. Raymond and M. Vanninathan, Exact controllability in fluid-solid structure: The Helmholtz model,, ESAIM Control Optim. Calc. Var., 11 (2005), 180. doi: 10.1051/cocv:2005006. Google Scholar

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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. doi: 10.1007/BF01211483. Google Scholar
[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. Google Scholar
[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. Google Scholar
[4]	H. Bohr, Almost Periodic Functions,, Chelsea Publishing Company, (1947). Google Scholar
[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. doi: 10.1137/S0363012997316378. Google Scholar
[6]	T. Cazenave and A. Haraux, An Introduction to Semilinear Evolution Equations,, Oxford Lecture Series in Mathematics and its Applications, (1998). Google Scholar
[7]	N. Cîndea, S. Micu and A. F. Pazoto, Periodic solutions for a weakly dissipated hybrid system,, J. Math. Anal. Appl., 385 (2012), 399. Google Scholar
[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. doi: 10.1137/070685609. Google Scholar
[9]	C. Corduneanu, Almost Periodic Oscillations and Waves,, Springer, (2009). doi: 10.1007/978-0-387-09819-7. Google Scholar
[10]	A. M. Fink, Almost Periodic Differetial Equation,, Lecture Notes in Mathematics, (1974). Google Scholar
[11]	S. Hansen and E. Zuazua, Exact controllability and stabilization of strings with point masses,, SIAM J. Cont. Optim., 33 (1995), 1357. doi: 10.1137/S0363012993248347. Google Scholar
[12]	A. Haraux, Semi-linear hyperbolic problems in bounded domains,, Math. Rep., 3 (1987), 1. Google Scholar
[13]	A. E. Ingham, Some trigonometric inequalities with applications to the theory of series,, Math. Zeits., 41 (1936), 367. doi: 10.1007/BF01180426. Google Scholar
[14]	J. E. Lagnese, Modelling and controllability of plate-beam systems,, J. Math. Systems Estim. Control, 4 (1994). Google Scholar
[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), 177. Google Scholar
[16]	E. B. Lee and Y. C. You, Stabilization of a hybrid (string/point mass) system,, in Proc. Fifth Int. Conf. Syst. Eng.* (Dayton*, (1987). Google Scholar
[17]	B. M. Levitan and V. V. Zhikov, Almost periodic functions and differential equations,, Cambridge University Press, (1982). Google Scholar
[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*, (1987), 151. Google Scholar
[19]	W. Littman and L. Markus, Exact boundary controllability of a hybrid system of elasticity,, Arch. Rational Mech. Anal., 103 (1988), 193. doi: 10.1007/BF00251758. Google Scholar
[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. doi: 10.1137/S0036141096312349. Google Scholar
[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. doi: 10.1109/9.328811. Google Scholar
[22]	B. Rao, Uniform stabilization of a hybrid system of elasticity,, SIAM J. Control Optim., 33 (1995), 440. doi: 10.1137/S0363012992239879. Google Scholar
[23]	B. Rao, Decay estimates of solutions for a hybrid system of flexible structures,, European J. Appl. Math., 4 (1993), 303. doi: 10.1017/S0956792500001133. Google Scholar
[24]	J.-P. Raymond and M. Vanninathan, Exact controllability in fluid-solid structure: The Helmholtz model,, ESAIM Control Optim. Calc. Var., 11 (2005), 180. doi: 10.1051/cocv:2005006. Google Scholar

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