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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), 1-22.
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. |
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