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Control of a Korteweg-de Vries equation: A tutorial
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 |
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. |
[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.
|
[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.
|
[4] |
H. Bohr, Almost Periodic Functions,, Chelsea Publishing Company, (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.
doi: 10.1137/S0363012997316378. |
[6] |
T. Cazenave and A. Haraux, An Introduction to Semilinear Evolution Equations,, Oxford Lecture Series in Mathematics and its Applications, (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. 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. |
[9] |
C. Corduneanu, Almost Periodic Oscillations and Waves,, Springer, (2009).
doi: 10.1007/978-0-387-09819-7. |
[10] |
A. M. Fink, Almost Periodic Differetial Equation,, Lecture Notes in Mathematics, (1974).
|
[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. |
[12] |
A. Haraux, Semi-linear hyperbolic problems in bounded domains,, Math. Rep., 3 (1987), 1.
|
[13] |
A. E. Ingham, Some trigonometric inequalities with applications to the theory of series,, Math. Zeits., 41 (1936), 367.
doi: 10.1007/BF01180426. |
[14] |
J. E. Lagnese, Modelling and controllability of plate-beam systems,, J. Math. Systems Estim. Control, 4 (1994).
|
[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).
|
[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. |
[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. |
[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. |
[22] |
B. Rao, Uniform stabilization of a hybrid system of elasticity,, SIAM J. Control Optim., 33 (1995), 440.
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.
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.
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.
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.
|
[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.
|
[4] |
H. Bohr, Almost Periodic Functions,, Chelsea Publishing Company, (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.
doi: 10.1137/S0363012997316378. |
[6] |
T. Cazenave and A. Haraux, An Introduction to Semilinear Evolution Equations,, Oxford Lecture Series in Mathematics and its Applications, (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. 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. |
[9] |
C. Corduneanu, Almost Periodic Oscillations and Waves,, Springer, (2009).
doi: 10.1007/978-0-387-09819-7. |
[10] |
A. M. Fink, Almost Periodic Differetial Equation,, Lecture Notes in Mathematics, (1974).
|
[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. |
[12] |
A. Haraux, Semi-linear hyperbolic problems in bounded domains,, Math. Rep., 3 (1987), 1.
|
[13] |
A. E. Ingham, Some trigonometric inequalities with applications to the theory of series,, Math. Zeits., 41 (1936), 367.
doi: 10.1007/BF01180426. |
[14] |
J. E. Lagnese, Modelling and controllability of plate-beam systems,, J. Math. Systems Estim. Control, 4 (1994).
|
[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).
|
[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. |
[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. |
[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. |
[22] |
B. Rao, Uniform stabilization of a hybrid system of elasticity,, SIAM J. Control Optim., 33 (1995), 440.
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.
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.
doi: 10.1051/cocv:2005006. |
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