Almost periodic solutions for a weakly dissipated hybrid system

Previous Article
Algebraic characterization of autonomy and controllability of behaviours of spatially invariant systems
MCRF Home
This Issue
Next Article
Control of a Korteweg-de Vries equation: A tutorial

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

Almost periodic solutions for a weakly dissipated hybrid system

Sorin Micu ^1, and Ademir F. Pazoto ^2,

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
Related Papers

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 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.

show all references

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.

[1]	Hartmut Pecher. Almost optimal local well-posedness for the Maxwell-Klein-Gordon system with data in Fourier-Lebesgue spaces. Communications on Pure and Applied Analysis, 2020, 19 (6) : 3303-3321. doi: 10.3934/cpaa.2020146
[2]	Lukáš Poul. Existence of weak solutions to the Navier-Stokes-Fourier system on Lipschitz domains. Conference Publications, 2007, 2007 (Special) : 834-843. doi: 10.3934/proc.2007.2007.834
[3]	Genni Fragnelli, Paolo Nistri, Duccio Papini. Non-trivial non-negative periodic solutions of a system of doubly degenerate parabolic equations with nonlocal terms. Discrete and Continuous Dynamical Systems, 2011, 31 (1) : 35-64. doi: 10.3934/dcds.2011.31.35
[4]	V. Barbu. Periodic solutions to unbounded Hamiltonian system. Discrete and Continuous Dynamical Systems, 1995, 1 (2) : 277-283. doi: 10.3934/dcds.1995.1.277
[5]	Yinsong Bai, Lin He, Huijiang Zhao. Nonlinear stability of rarefaction waves for a hyperbolic system with Cattaneo's law. Communications on Pure and Applied Analysis, 2021, 20 (7&8) : 2441-2474. doi: 10.3934/cpaa.2021049
[6]	Thomas I. Seidman, Olaf Klein. Periodic solutions of isotone hybrid systems. Discrete and Continuous Dynamical Systems - B, 2013, 18 (2) : 483-493. doi: 10.3934/dcdsb.2013.18.483
[7]	Genni Fragnelli, Paolo Nistri, Duccio Papini. Corrigendum: Nnon-trivial non-negative periodic solutions of a system of doubly degenerate parabolic equations with nonlocal terms. Discrete and Continuous Dynamical Systems, 2013, 33 (8) : 3831-3834. doi: 10.3934/dcds.2013.33.3831
[8]	Xiao Wang, Zhaohui Yang, Xiongwei Liu. Periodic and almost periodic oscillations in a delay differential equation system with time-varying coefficients. Discrete and Continuous Dynamical Systems, 2017, 37 (12) : 6123-6138. doi: 10.3934/dcds.2017263
[9]	Bum Ja Jin, Kyungkeun Kang. Caccioppoli type inequality for non-Newtonian Stokes system and a local energy inequality of non-Newtonian Navier-Stokes equations without pressure. Discrete and Continuous Dynamical Systems, 2017, 37 (9) : 4815-4834. doi: 10.3934/dcds.2017207
[10]	Shin-Ichiro Ei, Kei Nishi, Yasumasa Nishiura, Takashi Teramoto. Annihilation of two interfaces in a hybrid system. Discrete and Continuous Dynamical Systems - S, 2015, 8 (5) : 857-869. doi: 10.3934/dcdss.2015.8.857
[11]	P.E. Kloeden. Pitchfork and transcritical bifurcations in systems with homogeneous nonlinearities and an almost periodic time coefficient. Communications on Pure and Applied Analysis, 2004, 3 (2) : 161-173. doi: 10.3934/cpaa.2004.3.161
[12]	Güher Çamliyurt, Igor Kukavica. A local asymptotic expansion for a solution of the Stokes system. Evolution Equations and Control Theory, 2016, 5 (4) : 647-659. doi: 10.3934/eect.2016023
[13]	Wen Si, Fenfen Wang, Jianguo Si. Almost-periodic perturbations of non-hyperbolic equilibrium points via Pöschel-Rüssmann KAM method. Communications on Pure and Applied Analysis, 2020, 19 (1) : 541-585. doi: 10.3934/cpaa.2020027
[14]	Tong Li, Nitesh Mathur. Riemann problem for a non-strictly hyperbolic system in chemotaxis. Discrete and Continuous Dynamical Systems - B, 2022, 27 (4) : 2173-2187. doi: 10.3934/dcdsb.2021128
[15]	Zhujun Jing, K.Y. Chan, Dashun Xu, Hongjun Cao. Bifurcations of periodic solutions and chaos in Josephson system. Discrete and Continuous Dynamical Systems, 2001, 7 (3) : 573-592. doi: 10.3934/dcds.2001.7.573
[16]	Tomás Caraballo, David Cheban. Almost periodic and asymptotically almost periodic solutions of Liénard equations. Discrete and Continuous Dynamical Systems - B, 2011, 16 (3) : 703-717. doi: 10.3934/dcdsb.2011.16.703
[17]	Mengyu Cheng, Zhenxin Liu. Periodic, almost periodic and almost automorphic solutions for SPDEs with monotone coefficients. Discrete and Continuous Dynamical Systems - B, 2021, 26 (12) : 6425-6462. doi: 10.3934/dcdsb.2021026
[18]	Xiang Li, Zhixiang Li. Kernel sections and (almost) periodic solutions of a non-autonomous parabolic PDE with a discrete state-dependent delay. Communications on Pure and Applied Analysis, 2011, 10 (2) : 687-700. doi: 10.3934/cpaa.2011.10.687
[19]	Demou Luo, Qiru Wang. Dynamic analysis on an almost periodic predator-prey system with impulsive effects and time delays. Discrete and Continuous Dynamical Systems - B, 2021, 26 (6) : 3427-3453. doi: 10.3934/dcdsb.2020238
[20]	Chun Liu, Jan-Eric Sulzbach. The Brinkman-Fourier system with ideal gas equilibrium. Discrete and Continuous Dynamical Systems, 2022, 42 (1) : 425-462. doi: 10.3934/dcds.2021123

2020 Impact Factor: 1.284

Tools

Metrics

Other articles
by authors

on AIMS
Sorin Micu Ademir F. Pazoto
on Google Scholar
Sorin Micu Ademir F. Pazoto

[Back to Top]