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Relaxation of regularity for the Westervelt equation by nonlinear damping with applications in acoustic-acoustic and elastic-acoustic coupling
Feedback control of nonlinear dissipative systems by finite determining parameters - A reaction-diffusion paradigm
1. | Mohammed First University, National School of Applied Sciences Al Hoceima, Ajdir, 32003, Al Hoceima, Morocco |
2. | Department of Computer Science and Applied Mathematics, Weizmann Institute of Science,Rehovot 76100, Israel |
References:
[1] |
A. Armaou and P. D. Christofides, Feedback control of the Kuramoto-Sivashinsky equation,, Physica D, 137 (2000), 49.
doi: 10.1016/S0167-2789(99)00175-X. |
[2] |
A. Azouani, E. Olson and E. S. Titi, Continuous data assimilation using general interpolant observables,, Journal of Nonlinear Analysis, 24 (2014), 277.
doi: 10.1007/s00332-013-9189-y. |
[3] |
A. V. Babin and M. Vishik, Attractors of Evolution Partial Differential Equations,, North-Holland, (1992).
|
[4] |
H. Bessaih, E. Olson and E. S. Titi, Continuous assimilation of data with stochastic noise, preprint,, , (). |
[5] |
P. G. Ciarlet, The Finite Element Method for Elliptic Problems,, Classics in Applied Mathematics, 40 (2002).
doi: 10.1137/1.9780898719208. |
[6] |
B. Cockburn, D. A. Jones and E. S. Titi, Degrés de liberté déterminants pour équations non linéaires dissipatives,, C.R. Acad. Sci.-Paris, 321 (1995), 563.
|
[7] |
B. Cockburn, D. A. Jones and E. S. Titi, Estimating the number of asymptotic degrees of freedom for nonlinear dissipative systems,, Math. Comput., 66 (1997), 1073.
doi: 10.1090/S0025-5718-97-00850-8. |
[8] |
P. Constantin, Ch. Doering and E. S. Titi, Rigorous estimates of small scales in turbulent flows,, Journal of Mathematical Physics, 37 (1996), 6152.
doi: 10.1063/1.531769. |
[9] |
P. Constantin and C. Foias, Navier-Stokes Equations,, University of Chicago Press, (1988).
|
[10] |
P. Constantin, C. Foias, B. Nicolaenko and R. Temam, Integral Manifolds and Inertial Manifolds for Dissipative Partial Differential Equations,, Springer-Verlag, 70 (1989).
doi: 10.1007/978-1-4612-3506-4. |
[11] |
N. H. El-Farra, A. Armaou and P. D. Christofides, Analysis and control of parabolic PDE systems with input constraints,, Automatica, 39 (2003), 715.
doi: 10.1016/S0005-1098(02)00304-7. |
[12] |
C. Foias, M. S. Jolly, I. G. Kevrekidis, G. R. Sell and E. S. Titi, On the computation of inertial manifolds,, Physics Letters A, 131 (1988), 433.
doi: 10.1016/0375-9601(88)90295-2. |
[13] |
C. Foias, M. Jolly and R. Karavchenko, Determining forms for the Kuramoto-Sivashinsky and Lorenz equations: Analysis and computations,, (in preparation)., (). |
[14] |
C. Foias, M. Jolly, R. Kravchenko and E. S. Titi, A determining form for the 2D Navier-Stokes equations - the Fourier modes case,, Journal of Mathematical Physics, 53 (2012).
|
[15] |
C. Foias, M. Jolly, R. Karavchenko and E. S. Titi, A unified approach to determining forms for the 2D Navier-Stokes equations - the general interpolants case,, Uspekhi Matematicheskikh Nauk, 69 (2014), 359.
doi: 10.1070/RM2014v069n02ABEH004891. |
[16] |
C. Foias, O. P. Manley, R. Rosa and R. Temam, Navier-Stokes Equations and Turbulence,, Cambridge University Press, (2001).
doi: 10.1017/CBO9780511546754. |
[17] |
C. Foias, O. P. Manley, R. Temam and Y. Treve, Asymptotic analysis of the Navier-Stokes equations,, Physica D, 9 (1983), 157.
doi: 10.1016/0167-2789(83)90297-X. |
[18] |
C. Foias and G. Prodi, Sur le comportement global des solutions non stationnaires des équations de Navier-Stokes en dimension deux,, Rend. Sem. Mat. Univ. Padova, 39 (1967), 1.
|
[19] |
C. Foias, G. R. Sell and R. Temam, Inertial manifolds for nonlinear evolutionary equations,, Journal of Differential Equations, 73 (1988), 309.
doi: 10.1016/0022-0396(88)90110-6. |
[20] |
C. Foias, G. R. Sell and E. S. Titi, Exponential tracking and approximation of inertial manifolds for dissipative nonlinear equations,, Journal of Dynamics and Differential Equations, 1 (1989), 199.
doi: 10.1007/BF01047831. |
[21] |
C. Foias and R. Temam, Determination of the solutions of the Navier-Stokes equations by a set of nodal values,, Math. Comput., 43 (1984), 117.
doi: 10.1090/S0025-5718-1984-0744927-9. |
[22] |
C. Foias and R. Temam, Asymptotic numerical analysis for the Navier-Stokes equations,, in Nonlinear Dynamics and Turbulence, (1983), 139.
|
[23] |
C. Foias and E. S. Titi, Determining nodes, finite difference schemes and inertial manifolds,, Nonlinearity, 4 (1991), 135.
doi: 10.1088/0951-7715/4/1/009. |
[24] |
J. K. Hale, Asymptotic Behavior of Dissipative Systems,, Math. Survey and Monographs, 25 (1988).
|
[25] |
M. S. Jolly, I. G. Kevrekidis and E. S. Titi, Approximate inertial manifolds for the Kuramoto-Sivashinsky equation: analysis and computations,, Physica D, 44 (1990), 38.
doi: 10.1016/0167-2789(90)90046-R. |
[26] |
D. Jones and E. S. Titi, On the number of determining nodes for the 2-D Navier-Stokes equations,, J. Math. Anal. Appl., 168 (1992), 72.
doi: 10.1016/0022-247X(92)90190-O. |
[27] |
D. Jones and E. S. Titi, Determining finite volume elements for the 2-D Navier-Stokes equations,, Physica D, 60 (1992), 165.
doi: 10.1016/0167-2789(92)90233-D. |
[28] |
D. Jones and E. S. Titi, Upper bounds on the number of determining modes, nodes, and volume elements for the Navier-Stokes equations,, Indiana University Mathematics Journal, 42 (1993), 875.
doi: 10.1512/iumj.1993.42.42039. |
[29] |
I. Kukavica, On the number of determining nodes for the Ginzburg-Landau equation,, Nonlinearity, 5 (1992), 997.
doi: 10.1088/0951-7715/5/5/001. |
[30] |
E. Lunasin and E. S. Titi, Finite determining parameters feedback control for distributed nonlinear dissipative systems - a computational study,, (in preparation)., (). |
[31] |
J. Robinson, Infinite-Dimensional Dynamical Systems: An Introduction to Dissipative Parabolic PDEs and the Theory of Global attractors,, Cambridge Texts in Applied Mathematics, (2001).
doi: 10.1007/978-94-010-0732-0. |
[32] |
R. Rosa, Exact finite-dimensional feedback control via inertial manifold theory with application to the Chafee-Infante equation,, J. Dynamics and Diff. Eqs, 15 (2003), 61.
doi: 10.1023/A:1026153311546. |
[33] |
R. Rosa and R. Temam, Finite-dimensional feedback control of a scalar reaction-diffusion equation via inertial manifold theory, in Foundations of Computational Mathematics, (1997), 382.
|
[34] |
G. R. Sell and Y. You, Dynamics of Evolutionary Equations,, Applied Mathematical Sciences, 15 (2002).
doi: 10.1007/978-1-4757-5037-9. |
[35] |
S. Y. Shvartsman, C. Theodoropoulos, R. Rico-Martinez, I. G. Kevrekidis, E. S. Titi and T. J. Mountziares, Order reduction of nonlinear dynamic models for distributed reacting systems,, Journal of Process Control, 10 (2000), 177.
doi: 10.1016/S0959-1524(99)00029-3. |
[36] |
R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics,, $2^{nd}$ edition, (1997).
doi: 10.1007/978-1-4612-0645-3. |
[37] |
R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis,, Reprint of the 1984 edition, (1984).
|
show all references
References:
[1] |
A. Armaou and P. D. Christofides, Feedback control of the Kuramoto-Sivashinsky equation,, Physica D, 137 (2000), 49.
doi: 10.1016/S0167-2789(99)00175-X. |
[2] |
A. Azouani, E. Olson and E. S. Titi, Continuous data assimilation using general interpolant observables,, Journal of Nonlinear Analysis, 24 (2014), 277.
doi: 10.1007/s00332-013-9189-y. |
[3] |
A. V. Babin and M. Vishik, Attractors of Evolution Partial Differential Equations,, North-Holland, (1992).
|
[4] |
H. Bessaih, E. Olson and E. S. Titi, Continuous assimilation of data with stochastic noise, preprint,, , (). |
[5] |
P. G. Ciarlet, The Finite Element Method for Elliptic Problems,, Classics in Applied Mathematics, 40 (2002).
doi: 10.1137/1.9780898719208. |
[6] |
B. Cockburn, D. A. Jones and E. S. Titi, Degrés de liberté déterminants pour équations non linéaires dissipatives,, C.R. Acad. Sci.-Paris, 321 (1995), 563.
|
[7] |
B. Cockburn, D. A. Jones and E. S. Titi, Estimating the number of asymptotic degrees of freedom for nonlinear dissipative systems,, Math. Comput., 66 (1997), 1073.
doi: 10.1090/S0025-5718-97-00850-8. |
[8] |
P. Constantin, Ch. Doering and E. S. Titi, Rigorous estimates of small scales in turbulent flows,, Journal of Mathematical Physics, 37 (1996), 6152.
doi: 10.1063/1.531769. |
[9] |
P. Constantin and C. Foias, Navier-Stokes Equations,, University of Chicago Press, (1988).
|
[10] |
P. Constantin, C. Foias, B. Nicolaenko and R. Temam, Integral Manifolds and Inertial Manifolds for Dissipative Partial Differential Equations,, Springer-Verlag, 70 (1989).
doi: 10.1007/978-1-4612-3506-4. |
[11] |
N. H. El-Farra, A. Armaou and P. D. Christofides, Analysis and control of parabolic PDE systems with input constraints,, Automatica, 39 (2003), 715.
doi: 10.1016/S0005-1098(02)00304-7. |
[12] |
C. Foias, M. S. Jolly, I. G. Kevrekidis, G. R. Sell and E. S. Titi, On the computation of inertial manifolds,, Physics Letters A, 131 (1988), 433.
doi: 10.1016/0375-9601(88)90295-2. |
[13] |
C. Foias, M. Jolly and R. Karavchenko, Determining forms for the Kuramoto-Sivashinsky and Lorenz equations: Analysis and computations,, (in preparation)., (). |
[14] |
C. Foias, M. Jolly, R. Kravchenko and E. S. Titi, A determining form for the 2D Navier-Stokes equations - the Fourier modes case,, Journal of Mathematical Physics, 53 (2012).
|
[15] |
C. Foias, M. Jolly, R. Karavchenko and E. S. Titi, A unified approach to determining forms for the 2D Navier-Stokes equations - the general interpolants case,, Uspekhi Matematicheskikh Nauk, 69 (2014), 359.
doi: 10.1070/RM2014v069n02ABEH004891. |
[16] |
C. Foias, O. P. Manley, R. Rosa and R. Temam, Navier-Stokes Equations and Turbulence,, Cambridge University Press, (2001).
doi: 10.1017/CBO9780511546754. |
[17] |
C. Foias, O. P. Manley, R. Temam and Y. Treve, Asymptotic analysis of the Navier-Stokes equations,, Physica D, 9 (1983), 157.
doi: 10.1016/0167-2789(83)90297-X. |
[18] |
C. Foias and G. Prodi, Sur le comportement global des solutions non stationnaires des équations de Navier-Stokes en dimension deux,, Rend. Sem. Mat. Univ. Padova, 39 (1967), 1.
|
[19] |
C. Foias, G. R. Sell and R. Temam, Inertial manifolds for nonlinear evolutionary equations,, Journal of Differential Equations, 73 (1988), 309.
doi: 10.1016/0022-0396(88)90110-6. |
[20] |
C. Foias, G. R. Sell and E. S. Titi, Exponential tracking and approximation of inertial manifolds for dissipative nonlinear equations,, Journal of Dynamics and Differential Equations, 1 (1989), 199.
doi: 10.1007/BF01047831. |
[21] |
C. Foias and R. Temam, Determination of the solutions of the Navier-Stokes equations by a set of nodal values,, Math. Comput., 43 (1984), 117.
doi: 10.1090/S0025-5718-1984-0744927-9. |
[22] |
C. Foias and R. Temam, Asymptotic numerical analysis for the Navier-Stokes equations,, in Nonlinear Dynamics and Turbulence, (1983), 139.
|
[23] |
C. Foias and E. S. Titi, Determining nodes, finite difference schemes and inertial manifolds,, Nonlinearity, 4 (1991), 135.
doi: 10.1088/0951-7715/4/1/009. |
[24] |
J. K. Hale, Asymptotic Behavior of Dissipative Systems,, Math. Survey and Monographs, 25 (1988).
|
[25] |
M. S. Jolly, I. G. Kevrekidis and E. S. Titi, Approximate inertial manifolds for the Kuramoto-Sivashinsky equation: analysis and computations,, Physica D, 44 (1990), 38.
doi: 10.1016/0167-2789(90)90046-R. |
[26] |
D. Jones and E. S. Titi, On the number of determining nodes for the 2-D Navier-Stokes equations,, J. Math. Anal. Appl., 168 (1992), 72.
doi: 10.1016/0022-247X(92)90190-O. |
[27] |
D. Jones and E. S. Titi, Determining finite volume elements for the 2-D Navier-Stokes equations,, Physica D, 60 (1992), 165.
doi: 10.1016/0167-2789(92)90233-D. |
[28] |
D. Jones and E. S. Titi, Upper bounds on the number of determining modes, nodes, and volume elements for the Navier-Stokes equations,, Indiana University Mathematics Journal, 42 (1993), 875.
doi: 10.1512/iumj.1993.42.42039. |
[29] |
I. Kukavica, On the number of determining nodes for the Ginzburg-Landau equation,, Nonlinearity, 5 (1992), 997.
doi: 10.1088/0951-7715/5/5/001. |
[30] |
E. Lunasin and E. S. Titi, Finite determining parameters feedback control for distributed nonlinear dissipative systems - a computational study,, (in preparation)., (). |
[31] |
J. Robinson, Infinite-Dimensional Dynamical Systems: An Introduction to Dissipative Parabolic PDEs and the Theory of Global attractors,, Cambridge Texts in Applied Mathematics, (2001).
doi: 10.1007/978-94-010-0732-0. |
[32] |
R. Rosa, Exact finite-dimensional feedback control via inertial manifold theory with application to the Chafee-Infante equation,, J. Dynamics and Diff. Eqs, 15 (2003), 61.
doi: 10.1023/A:1026153311546. |
[33] |
R. Rosa and R. Temam, Finite-dimensional feedback control of a scalar reaction-diffusion equation via inertial manifold theory, in Foundations of Computational Mathematics, (1997), 382.
|
[34] |
G. R. Sell and Y. You, Dynamics of Evolutionary Equations,, Applied Mathematical Sciences, 15 (2002).
doi: 10.1007/978-1-4757-5037-9. |
[35] |
S. Y. Shvartsman, C. Theodoropoulos, R. Rico-Martinez, I. G. Kevrekidis, E. S. Titi and T. J. Mountziares, Order reduction of nonlinear dynamic models for distributed reacting systems,, Journal of Process Control, 10 (2000), 177.
doi: 10.1016/S0959-1524(99)00029-3. |
[36] |
R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics,, $2^{nd}$ edition, (1997).
doi: 10.1007/978-1-4612-0645-3. |
[37] |
R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis,, Reprint of the 1984 edition, (1984).
|
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