American Institute of Mathematical Sciences

Advanced
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

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

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.  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

[1]

M. Grasselli, V. Pata. Asymptotic behavior of a parabolic-hyperbolic system. Communications on Pure & Applied Analysis, 2004, 3 (4) : 849-881. doi: 10.3934/cpaa.2004.3.849
[2]

Petra Csomós, Hermann Mena. Fourier-splitting method for solving hyperbolic LQR problems. Numerical Algebra, Control & Optimization, 2018, 8 (1) : 17-46. doi: 10.3934/naco.2018002
[3]

Yanqin Fang, Jihui Zhang. Multiplicity of solutions for the nonlinear Schrödinger-Maxwell system. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1267-1279. doi: 10.3934/cpaa.2011.10.1267
[4]

Zaihong Wang, Jin Li, Tiantian Ma. An erratum note on the paper: Positive periodic solution for Brillouin electron beam focusing system. Discrete & Continuous Dynamical Systems - B, 2013, 18 (7) : 1995-1997. doi: 10.3934/dcdsb.2013.18.1995
[5]

Arunima Bhattacharya, Micah Warren. $ C^{2, \alpha} $ estimates for solutions to almost Linear elliptic equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021024
[6]

Ying Yang. Global classical solutions to two-dimensional chemotaxis-shallow water system. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2625-2643. doi: 10.3934/dcdsb.2020198
[7]

Zhiming Guo, Zhi-Chun Yang, Xingfu Zou. Existence and uniqueness of positive solution to a non-local differential equation with homogeneous Dirichlet boundary condition---A non-monotone case. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1825-1838. doi: 10.3934/cpaa.2012.11.1825
[8]

Giovanni Cimatti. Forced periodic solutions for piezoelectric crystals. Communications on Pure & Applied Analysis, 2005, 4 (2) : 475-485. doi: 10.3934/cpaa.2005.4.475
[9]

Jaume Llibre, Luci Any Roberto. On the periodic solutions of a class of Duffing differential equations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (1) : 277-282. doi: 10.3934/dcds.2013.33.277
[10]

Elena Bonetti, Pierluigi Colli, Gianni Gilardi. Singular limit of an integrodifferential system related to the entropy balance. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 1935-1953. doi: 10.3934/dcdsb.2014.19.1935
[11]

Dmitry Treschev. A locally integrable multi-dimensional billiard system. Discrete & Continuous Dynamical Systems - A, 2017, 37 (10) : 5271-5284. doi: 10.3934/dcds.2017228
[12]

Nizami A. Gasilov. Solving a system of linear differential equations with interval coefficients. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2739-2747. doi: 10.3934/dcdsb.2020203
[13]

Eduardo Casas, Christian Clason, Arnd Rösch. Preface special issue on system modeling and optimization. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021008
[14]

Wei-Jian Bo, Guo Lin, Shigui Ruan. Traveling wave solutions for time periodic reaction-diffusion systems. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4329-4351. doi: 10.3934/dcds.2018189
[15]

Luke Finlay, Vladimir Gaitsgory, Ivan Lebedev. Linear programming solutions of periodic optimization problems: approximation of the optimal control. Journal of Industrial & Management Optimization, 2007, 3 (2) : 399-413. doi: 10.3934/jimo.2007.3.399
[16]

Jianping Gao, Shangjiang Guo, Wenxian Shen. Persistence and time periodic positive solutions of doubly nonlocal Fisher-KPP equations in time periodic and space heterogeneous media. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2645-2676. doi: 10.3934/dcdsb.2020199
[17]

Dugan Nina, Ademir Fernando Pazoto, Lionel Rosier. Controllability of a 1-D tank containing a fluid modeled by a Boussinesq system. Evolution Equations & Control Theory, 2013, 2 (2) : 379-402. doi: 10.3934/eect.2013.2.379
[18]

Xu Zhang, Xiang Li. Modeling and identification of dynamical system with Genetic Regulation in batch fermentation of glycerol. Numerical Algebra, Control & Optimization, 2015, 5 (4) : 393-403. doi: 10.3934/naco.2015.5.393
[19]

Guo-Bao Zhang, Ruyun Ma, Xue-Shi Li. Traveling waves of a Lotka-Volterra strong competition system with nonlocal dispersal. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 587-608. doi: 10.3934/dcdsb.2018035
[20]

Dan Wei, Shangjiang Guo. Qualitative analysis of a Lotka-Volterra competition-diffusion-advection system. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2599-2623. doi: 10.3934/dcdsb.2020197

2019 Impact Factor: 0.857

Tools

Metrics

  • PDF downloads (54)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences