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Approximate controllability of semilinear non-autonomous evolution systems with state-dependent delay
Finite determining parameters feedback control for distributed nonlinear dissipative systems -a computational study
1. | Department of Mathematics, United States Naval Academy, Annapolis, MD 21402, USA |
2. | Departments of Mathematics, Texas A & M University, College Station, TX 77843-3368, USA |
3. | Department of Computer Science and Applied Mathematics, The Weizmann Institute of Science, Rehovot 76100, Israel |
We investigate the effectiveness of a simple finite-dimensional feedback control scheme for globally stabilizing solutions of infinite-dimensional dissipative evolution equations introduced by Azouani and Titi in [
References:
[1] |
S. Ahuja, Reduction Methods for Feedback Stabilization of Fluid Flows, Ph. D Thesis, Dept. of Mechanical and Aerospace Engineering, Princeton University, 2009. Google Scholar |
[2] |
M. U. Altaf, E. S. Titi, T. Gebrael, O. Knio, L. Zhao, M. F. McCabe and I. Hoteit,
Downscaling the 2D Bénard convection equations using continuous data assimilation, Computat. Geosci., 21 (2017), 393-410.
doi: 10.1007/s10596-017-9619-2. |
[3] |
A. Armaou and P. D. Christofides,
Feedback control of the Kuramoto-Sivashinsky equation, Physica D, 137 (2000), 49-61.
doi: 10.1016/S0167-2789(99)00175-X. |
[4] |
A. Armou and P. D. Christofides,
Wave suppression by nonlinear finite-dimensional control, Eng. Sci., 55 (2000), 2627-2640.
doi: 10.1016/S0009-2509(99)00544-8. |
[5] |
A. Armou and P. D. Christofides,
Global stabilization of the Kuramoto-Sivashinsky Equation via distributed output feedback control, Syst. & Contr. Lett., 39 (2000), 283-294.
doi: 10.1016/S0167-6911(99)00108-5. |
[6] |
A. Azouani, E. Olson and E. S. Titi,
Continuous data assimilation using general interpolant observables, J. Nonlinear Sci., 24 (2014), 277-304.
doi: 10.1007/s00332-013-9189-y. |
[7] |
A. Azouani and E. S. Titi,
Feedback control of nonlinear dissipative systems by finite determining parameters -A reaction-diffusion Paradigm, Evolution Equations and Control Theory, 3 (2014), 579-594.
doi: 10.3934/eect.2014.3.579. |
[8] |
A. V. Babin and M. Vishik,
Attractors of Evolutionary Partial Differential Equations, North-Holland, Amsterdam, London, NewYork, Tokyo, 1992. |
[9] |
H. Bessaih, E. Olson and E. S. Titi,
Continuous assimilation of data with stochastic noise, Nonlinearity, 28 (2015), 729-753.
doi: 10.1088/0951-7715/28/3/729. |
[10] |
J. Bronski and T. Gambill,
Uncertainty estimates and $L_2$ bounds for the Kuramoto-Sivashinsky equation, Nonlinearity, 19 (2006), 2023-2039.
doi: 10.1088/0951-7715/19/9/002. |
[11] |
C. Cao, I. Kevrekidis and E. S. Titi,
Numerical criterion for the stabilization of steady states of the Navier-Stokes equations, Indiana University Mathematics Journal, 50 (2001), 37-96.
doi: 10.1512/iumj.2001.50.2154. |
[12] |
J. Charney, J. Halem and M. Jastrow,
Use of incomplete historical data to infer the present state of the atmosphere, Journal of Atmospheric Science, 26 (1969), 1160-1163.
doi: 10.1175/1520-0469(1969)026<1160:UOIHDT>2.0.CO;2. |
[13] |
L. H. Chen and H. C. Chang, Nonlinear waves on liquid film surfaces-Ⅱ. Bifurcation analyzes of the long-wave equation, Chem. Eng. Sci., 41 (1986), 2477-2486. Google Scholar |
[14] |
P. D. Christofides,
Nonlinear and robust control of PDE systems: Methods and Applications to Transport-Reaction Processes, Springer Science + Business Media, New York, 2001.
doi: 10.1007/978-1-4612-0185-4. |
[15] |
B. Cockburn, D. A. Jones and E. S. Titi,
Degrés de liberté déterminants pour équations non linéaires dissipatives, C.R. Acad. Sci.-Paris, Sér. I, 321 (1995), 563-568.
|
[16] |
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), 1073-1087.
doi: 10.1090/S0025-5718-97-00850-8. |
[17] |
B. I. Cohen, J. A. Krommes, W. M. Tang and M. N. Rosenbluth,
Non-linear saturation of the dissipative trapped-ion mode by mode coupling, Nuclear Fusion, 16 (1976), 971-974.
doi: 10.1088/0029-5515/16/6/009. |
[18] |
P. Collet, J.-P. Eckmann, H. Epstein and J. Stubbe,
A global attracting set for the Kuramoto-Sivashinksy equation, Commun. Math. Phys., 152 (1993), 203-214.
doi: 10.1007/BF02097064. |
[19] |
P. Constantin and C. Foias,
Navier-Stokes Equations, University of Chicago Press, Chicago, 1988. |
[20] |
P. Constantin, C. Foias, B. Nicolaenko and R. Temam,
Integral Manifolds and Inertial Manifolds for Dissipative Partial Differential Equations, Springer-Verlag, Applies Mathematical Sciences Series, 70 1989.
doi: 10.1007/978-1-4612-3506-4. |
[21] |
S. M. Cox and P. C. Matthews,
Exponential time differencing for stiff systems, J. Comput. Phys., 176 (2002), 430-455.
doi: 10.1006/jcph.2002.6995. |
[22] |
S. Dubljevic, N. El-Farra and P. Christofides,
Predictive control of transport-reaction processes, Computers and Chemical Engineering, 29 (2005), 2335-2345.
doi: 10.1016/j.compchemeng.2005.05.008. |
[23] |
S. Dubljevic, N. El-Farra and P. Christofides,
Predictive control of parabolic pdes with state and control constraints, Cinter. J. Rob. & Non. Contr., 16 (2006), 749-772.
doi: 10.1002/rnc.1097. |
[24] |
N. H. El-Farra, A. Armaou and P. D. Christofides,
Analysis and control of parabolic PDE systems with input constraints, Automatica, 39 (2003), 715-725.
doi: 10.1016/S0005-1098(02)00304-7. |
[25] |
A. Farhat, M. S. Jolly and E. S. Titi,
Continuous data assimilation for the 2D Bénard convection through velocity measurements alone, Physica D, 303 (2015), 59-66.
doi: 10.1016/j.physd.2015.03.011. |
[26] |
A. Farhat, E. Lunasin and E. S. Titi,
Abridged continuous data assimilation for the 2D Navier-Stokes Equations utilizing measurements of only one component of the velocity field, J. Math. Fluid Mech., 18 (2016), 1-23.
doi: 10.1007/s00021-015-0225-6. |
[27] |
A. Farhat, E. Lunasin and E. S. Titi,
Continuous data assimilation for a 2D Bénard convection system through horizontal velocity measurements alone, J. Nonlinear Sci., 27 (2017), 1065-1087.
doi: 10.1007/s00332-017-9360-y. |
[28] |
A. Farhat, E. Lunasin and E. S. Titi,
On the Charney conjecture of data assimilation employing temperature measurements alone: the paradigm of 3D Planetary Geostrophic model, Math. Clim. Weather Forecast, 2 (2016), 61-74.
doi: 10.1515/mcwf-2016-0004. |
[29] |
A. Farhat, E. Lunasin and E. S. Titi,
Data assimilation algorithm for 3D Bénard convection in porous media employing only temperature measurements, Jour. Math. Anal. Appl., 438 (2016), 492-506.
doi: 10.1016/j.jmaa.2016.01.072. |
[30] |
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), 433-436.
doi: 10.1016/0375-9601(88)90295-2. |
[31] |
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), 115623, 30 pp.
doi: 10.1063/1.4766459. |
[32] |
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, Russian Mathematical Surveys, 69 (2014), 359-381.
doi: 10.1070/RM2014v069n02ABEH004891. |
[33] |
C. Foias, O. P. Manley, R. Rosa and R. Temam,
Navier-Stokes Equations and Turbulence, Cambridge University Press, 2001.
doi: 10.1017/CBO9780511546754. |
[34] |
C. Foias, O. P. Manley, R. Temam and Y. Treve,
Asymptotic analysis of the Navier-Stokes equations, Physica D, 9 (1983), 157-188.
doi: 10.1016/0167-2789(83)90297-X. |
[35] |
C. Foias, C. Mondaini and E. S. Titi,
A discrete data assimilation scheme for the solutions of the 2D Navier-Stokes equations and their statistics, SIAM Journal on Applied Dynamical Systems, 15 (2016), 2109-2142.
doi: 10.1137/16M1076526. |
[36] |
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), 1-34.
|
[37] |
C. Foias, G. R. Sell and R. Temam,
Inertial manifolds for nonlinear evolutionary equations, Journal of Differential Equations, 73 (1988), 309-353.
doi: 10.1016/0022-0396(88)90110-6. |
[38] |
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), 199-244.
doi: 10.1007/BF01047831. |
[39] |
C. Foias and R. Temam,
Determination of the solutions of the Navier-Stokes equations by a set of nodal values, Math. Comput., 43 (1984), 117-133.
doi: 10.1090/S0025-5718-1984-0744927-9. |
[40] |
C. Foias and R. Temam, Asymptotic numerical analysis for the Navier-Stokes equations, Nonlinear Dynamics and Turbulence, eds. Barenblatt, Iooss, Joseph, Boston: Pitman Advanced Pub. Prog., 1983,139-155. |
[41] |
C. Foias and E. S. Titi,
Determining nodes, finite difference schemes and inertial manifolds, Nonlinearity, 4 (1991), 135-153.
doi: 10.1088/0951-7715/4/1/009. |
[42] |
M. Gesho, E. Olson and E. S. Titi,
A computational study of a data assimilation algorithm for the two-dimensional Navier-Stokes equations, Commun. Phys, 19 (2016), 1094-1110.
|
[43] |
M. Ghil, B. Shkoller and V. Yangarber, A balanced diagnostic system compatible with a barotropic prognostic model., Mon. Wea. Rev., 105 (1977), 1223-1238. Google Scholar |
[44] |
M. Ghil, M. Halem and R. Atlas, Time-continuous assimilation of remote-sounding data and its effect on weather forecasting, Mon. Wea. Rev., 107 (1978), 140-171. Google Scholar |
[45] |
L. Giacomeli and F. Otto,
New bounds for the Kuramoto-Sivashinsky equation, Commun. Pure. Appl. Math., 58 (2005), 297-318.
doi: 10.1002/cpa.20031. |
[46] |
S. N. Gomes, D. T. Papageorgiou and G. A. Pavliotis,
Stabilizing non-trivial solutions of generalized Kuramoto-Sivashinsky equation using feedback and optimal control, IMA J. Applied Mathematics, 82 (2017), 158-194.
doi: 10.1093/imamat/hxw011. |
[47] |
S. N. Gomes, M. Pradas, S. Kalliadasis, D. T. Papageorgiou and G. A. Pavliotis,
Controlling roughening processes in the stochastic Kuramoto-Sivashinsky equation, Physica D: Nonl. Phenom., 348 (2017), 33-43.
doi: 10.1016/j.physd.2017.02.011. |
[48] |
J. Goodman,
Stability of the Kuramoto-Sivashinky and related systems, Commun. Pure Appl. Math., 47 (1994), 293-306.
doi: 10.1002/cpa.3160470304. |
[49] |
J. K. Hale,
Asymptotic Behavior of Dissipative Systems, Math. Survey and Monographs, 25 AMS, Providence, R. I., 1988. |
[50] |
L. Illing, D. J. Gauthier and R. Roy,
Controlling optical chaos, Spatio-Temporal Dynamics, and Patterns, Advances in Atomic, Molecular and Optical Physics, 54 (2007), 615-697.
doi: 10.1016/S1049-250X(06)54010-8. |
[51] |
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), 38-60.
doi: 10.1016/0167-2789(90)90046-R. |
[52] |
M. Jolly, V. Martinez and E. S. Titi,
A data assimilation algorithm for the subcritical surface quasi-geostrophic equation, Advanced Nonlinear Studies, 17 (2017), 167-192.
doi: 10.1515/ans-2016-6019. |
[53] |
M. S. Jolly, T. Sadigov and E. S. Titi,
A determining form for the damped driven nonlinear Schrödinger equation-Fourier modes case, J. Diff. Eqns., 258 (2015), 2711-2744.
doi: 10.1016/j.jde.2014.12.023. |
[54] |
M. S. Jolly, T. Sadigov and E. S. Titi,
Determining form and data assimilation algorithm for weakly damped and driven Korteweg-de Vries equaton-Fourier modes case, Nonlinear Analysis: Real World Applications, 36 (2017), 287-317.
doi: 10.1016/j.nonrwa.2017.01.010. |
[55] |
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), 72-88.
doi: 10.1016/0022-247X(92)90190-O. |
[56] |
D. Jones and E. S. Titi,
Determining finite volume elements for the 2-D Navier-Stokes equations, Physica D, 60 (1992), 165-174.
doi: 10.1016/0167-2789(92)90233-D. |
[57] |
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), 875-887.
doi: 10.1512/iumj.1993.42.42039. |
[58] |
V. Kalantarov and E. S. Titi,
Finite-parameters feedback control for stabilizing damped nonlinear wave equations, Contemporary Mathematics: Nonlinear Analysis and Optimization, AMS, 669 (2016), 115-133.
doi: 10.1090/conm/659/13193. |
[59] |
V. Kalantarov and E. S. Titi, Global stabilization of the Navier-Stokes-Voight and the damped nonlinear wave equations by finite number of feedback controllers, Discrete and Continuous Dynamical Systems -B, (2017), to appear, arXiv: 1706.00162 Google Scholar |
[60] |
A. Kazaam and L. Trefethen,
Fourth-order time stepping for stiff PDEs, J. Sci Comp., 26 (2005), 1214-1233.
doi: 10.1137/S1064827502410633. |
[61] |
I. Kukavica,
On the number of determining nodes for the Ginzburg-Landau equation, Nonlinearity, 5 (1992), 997-1006.
doi: 10.1088/0951-7715/5/5/001. |
[62] |
Y. Kuramoto and T. Tsusuki,
Reductive perturbation approach to chemical instabilities, Prog. Theor. Phys., 52 (1974), 1399-1401.
doi: 10.1143/PTP.52.1399. |
[63] |
R. E. LaQuey, S. M. Mahajan, P. H. Rutherford and W. M. Tang,
Nonlinear saturation of the trapped-ion mode, Phys. Rev. Let., 34 (1975), 391-394.
doi: 10.2172/4202869. |
[64] |
C. H. Lee and H. T. Tran,
Reduced-order-based feedback control of the Kuramoto-Sivashinsky equation, Journal of Computational and Applied Mathematics, 173 (2005), 1-19.
doi: 10.1016/j.cam.2004.02.021. |
[65] |
M. Li and P. D. Christofides,
Optimal control of diffusion-convection-reaction processes using reduced order models, Computers and Chemical Engineering, 32 (2008), 2123-2135.
doi: 10.1016/j.compchemeng.2007.10.018. |
[66] |
Y. Lou and P. D. Christofides, Optimal actuator/sensor placement for nonlinear control of the Kuramoto-Sivashinksky equation, IEEE Transactions on Control Systems Tech., 11 (2002), 737-745. Google Scholar |
[67] |
P. Markowich, E. S. Titi and S. Trabelsi,
Continuous data assimilation for the three-dimensional Brinkman-Forchheimer-Extended Darcy model, Nonlinearity, 29 (2016), 1292-1328.
doi: 10.1088/0951-7715/29/4/1292. |
[68] |
C. Mondaini and E. S. Titi, Uniform in time error estimates for the postprocessing Galerkin method applied to a data assimilation algorithm, SIAM Journal on Numerical Analysis, (2017), to appear, arXiv: 1612.06998. Google Scholar |
[69] |
B. Nicolaenko, B. Scheurer and R. Temam,
Some global dynamical properties of the Kuramoto-Sivashinsky equation: nonlinear stability and attractors, Physica D, 16 (1985), 155-183.
doi: 10.1016/0167-2789(85)90056-9. |
[70] |
M. Oliver and E. S. Titi,
On the domain of analyticity for solutions of second order analytic nonlinear differential equations, J. Differential Equations, 174 (2001), 55-74.
doi: 10.1006/jdeq.2000.3927. |
[71] |
F. Otto,
Optimal bounds on the Kuramoto-Sivashinsky equations, Journal of Functional Analysis, 257 (2009), 2188-2245.
doi: 10.1016/j.jfa.2009.01.034. |
[72] |
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. |
[73] |
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), 61-86.
doi: 10.1023/A:1026153311546. |
[74] |
R. Rosa and R. Temam, Finite-dimensional feedback control of a scalar reaction-diffusion equation via inertial manifold theory, Foundations of Computational Mathematics, Selected papers of a conference held at IMPA, Rio de Janeiro, RJ, Brazil, eds. F. Cucker and M. Shub, Springer-Verlag, Berlin, (1997), 382-391. |
[75] |
G. R. Sell and Y. You,
Dynamics of Evolutionary Equations, Springer, 2002.
doi: 10.1007/978-1-4757-5037-9. |
[76] |
S. 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), 177-184. Google Scholar |
[77] |
G. I. Sivashinsky,
Nonlinear analysis of hydrodynamic instability in laminar flames, Acta Astronautica, 4 (1977), 1177-1206.
doi: 10.1016/0094-5765(77)90096-0. |
[78] |
R. Temam,
Infinite Dimensional Dynamical Systems in Mechanics and Physics, New York: Springer, 1988.
doi: 10.1007/978-1-4684-0313-8. |
[79] |
R. Temam,
Navier-Stokes Equations: Theory and Numerical Analysis, AMS Chelsea Publishing, Providence, RI, Theory and numerical analysis, Reprint of the 1984 edition, 2001.
doi: 10.1090/chel/343. |
[80] |
A. Thompson, S. B. Gomes, G. A. Pavliotis and D. T. Papageorgio, Stabilising falling liquid film flows using feedback control Physics of Fluids, 28 (2016), 012107.
doi: 10.1063/1.4938761. |
show all references
References:
[1] |
S. Ahuja, Reduction Methods for Feedback Stabilization of Fluid Flows, Ph. D Thesis, Dept. of Mechanical and Aerospace Engineering, Princeton University, 2009. Google Scholar |
[2] |
M. U. Altaf, E. S. Titi, T. Gebrael, O. Knio, L. Zhao, M. F. McCabe and I. Hoteit,
Downscaling the 2D Bénard convection equations using continuous data assimilation, Computat. Geosci., 21 (2017), 393-410.
doi: 10.1007/s10596-017-9619-2. |
[3] |
A. Armaou and P. D. Christofides,
Feedback control of the Kuramoto-Sivashinsky equation, Physica D, 137 (2000), 49-61.
doi: 10.1016/S0167-2789(99)00175-X. |
[4] |
A. Armou and P. D. Christofides,
Wave suppression by nonlinear finite-dimensional control, Eng. Sci., 55 (2000), 2627-2640.
doi: 10.1016/S0009-2509(99)00544-8. |
[5] |
A. Armou and P. D. Christofides,
Global stabilization of the Kuramoto-Sivashinsky Equation via distributed output feedback control, Syst. & Contr. Lett., 39 (2000), 283-294.
doi: 10.1016/S0167-6911(99)00108-5. |
[6] |
A. Azouani, E. Olson and E. S. Titi,
Continuous data assimilation using general interpolant observables, J. Nonlinear Sci., 24 (2014), 277-304.
doi: 10.1007/s00332-013-9189-y. |
[7] |
A. Azouani and E. S. Titi,
Feedback control of nonlinear dissipative systems by finite determining parameters -A reaction-diffusion Paradigm, Evolution Equations and Control Theory, 3 (2014), 579-594.
doi: 10.3934/eect.2014.3.579. |
[8] |
A. V. Babin and M. Vishik,
Attractors of Evolutionary Partial Differential Equations, North-Holland, Amsterdam, London, NewYork, Tokyo, 1992. |
[9] |
H. Bessaih, E. Olson and E. S. Titi,
Continuous assimilation of data with stochastic noise, Nonlinearity, 28 (2015), 729-753.
doi: 10.1088/0951-7715/28/3/729. |
[10] |
J. Bronski and T. Gambill,
Uncertainty estimates and $L_2$ bounds for the Kuramoto-Sivashinsky equation, Nonlinearity, 19 (2006), 2023-2039.
doi: 10.1088/0951-7715/19/9/002. |
[11] |
C. Cao, I. Kevrekidis and E. S. Titi,
Numerical criterion for the stabilization of steady states of the Navier-Stokes equations, Indiana University Mathematics Journal, 50 (2001), 37-96.
doi: 10.1512/iumj.2001.50.2154. |
[12] |
J. Charney, J. Halem and M. Jastrow,
Use of incomplete historical data to infer the present state of the atmosphere, Journal of Atmospheric Science, 26 (1969), 1160-1163.
doi: 10.1175/1520-0469(1969)026<1160:UOIHDT>2.0.CO;2. |
[13] |
L. H. Chen and H. C. Chang, Nonlinear waves on liquid film surfaces-Ⅱ. Bifurcation analyzes of the long-wave equation, Chem. Eng. Sci., 41 (1986), 2477-2486. Google Scholar |
[14] |
P. D. Christofides,
Nonlinear and robust control of PDE systems: Methods and Applications to Transport-Reaction Processes, Springer Science + Business Media, New York, 2001.
doi: 10.1007/978-1-4612-0185-4. |
[15] |
B. Cockburn, D. A. Jones and E. S. Titi,
Degrés de liberté déterminants pour équations non linéaires dissipatives, C.R. Acad. Sci.-Paris, Sér. I, 321 (1995), 563-568.
|
[16] |
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), 1073-1087.
doi: 10.1090/S0025-5718-97-00850-8. |
[17] |
B. I. Cohen, J. A. Krommes, W. M. Tang and M. N. Rosenbluth,
Non-linear saturation of the dissipative trapped-ion mode by mode coupling, Nuclear Fusion, 16 (1976), 971-974.
doi: 10.1088/0029-5515/16/6/009. |
[18] |
P. Collet, J.-P. Eckmann, H. Epstein and J. Stubbe,
A global attracting set for the Kuramoto-Sivashinksy equation, Commun. Math. Phys., 152 (1993), 203-214.
doi: 10.1007/BF02097064. |
[19] |
P. Constantin and C. Foias,
Navier-Stokes Equations, University of Chicago Press, Chicago, 1988. |
[20] |
P. Constantin, C. Foias, B. Nicolaenko and R. Temam,
Integral Manifolds and Inertial Manifolds for Dissipative Partial Differential Equations, Springer-Verlag, Applies Mathematical Sciences Series, 70 1989.
doi: 10.1007/978-1-4612-3506-4. |
[21] |
S. M. Cox and P. C. Matthews,
Exponential time differencing for stiff systems, J. Comput. Phys., 176 (2002), 430-455.
doi: 10.1006/jcph.2002.6995. |
[22] |
S. Dubljevic, N. El-Farra and P. Christofides,
Predictive control of transport-reaction processes, Computers and Chemical Engineering, 29 (2005), 2335-2345.
doi: 10.1016/j.compchemeng.2005.05.008. |
[23] |
S. Dubljevic, N. El-Farra and P. Christofides,
Predictive control of parabolic pdes with state and control constraints, Cinter. J. Rob. & Non. Contr., 16 (2006), 749-772.
doi: 10.1002/rnc.1097. |
[24] |
N. H. El-Farra, A. Armaou and P. D. Christofides,
Analysis and control of parabolic PDE systems with input constraints, Automatica, 39 (2003), 715-725.
doi: 10.1016/S0005-1098(02)00304-7. |
[25] |
A. Farhat, M. S. Jolly and E. S. Titi,
Continuous data assimilation for the 2D Bénard convection through velocity measurements alone, Physica D, 303 (2015), 59-66.
doi: 10.1016/j.physd.2015.03.011. |
[26] |
A. Farhat, E. Lunasin and E. S. Titi,
Abridged continuous data assimilation for the 2D Navier-Stokes Equations utilizing measurements of only one component of the velocity field, J. Math. Fluid Mech., 18 (2016), 1-23.
doi: 10.1007/s00021-015-0225-6. |
[27] |
A. Farhat, E. Lunasin and E. S. Titi,
Continuous data assimilation for a 2D Bénard convection system through horizontal velocity measurements alone, J. Nonlinear Sci., 27 (2017), 1065-1087.
doi: 10.1007/s00332-017-9360-y. |
[28] |
A. Farhat, E. Lunasin and E. S. Titi,
On the Charney conjecture of data assimilation employing temperature measurements alone: the paradigm of 3D Planetary Geostrophic model, Math. Clim. Weather Forecast, 2 (2016), 61-74.
doi: 10.1515/mcwf-2016-0004. |
[29] |
A. Farhat, E. Lunasin and E. S. Titi,
Data assimilation algorithm for 3D Bénard convection in porous media employing only temperature measurements, Jour. Math. Anal. Appl., 438 (2016), 492-506.
doi: 10.1016/j.jmaa.2016.01.072. |
[30] |
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), 433-436.
doi: 10.1016/0375-9601(88)90295-2. |
[31] |
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), 115623, 30 pp.
doi: 10.1063/1.4766459. |
[32] |
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, Russian Mathematical Surveys, 69 (2014), 359-381.
doi: 10.1070/RM2014v069n02ABEH004891. |
[33] |
C. Foias, O. P. Manley, R. Rosa and R. Temam,
Navier-Stokes Equations and Turbulence, Cambridge University Press, 2001.
doi: 10.1017/CBO9780511546754. |
[34] |
C. Foias, O. P. Manley, R. Temam and Y. Treve,
Asymptotic analysis of the Navier-Stokes equations, Physica D, 9 (1983), 157-188.
doi: 10.1016/0167-2789(83)90297-X. |
[35] |
C. Foias, C. Mondaini and E. S. Titi,
A discrete data assimilation scheme for the solutions of the 2D Navier-Stokes equations and their statistics, SIAM Journal on Applied Dynamical Systems, 15 (2016), 2109-2142.
doi: 10.1137/16M1076526. |
[36] |
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), 1-34.
|
[37] |
C. Foias, G. R. Sell and R. Temam,
Inertial manifolds for nonlinear evolutionary equations, Journal of Differential Equations, 73 (1988), 309-353.
doi: 10.1016/0022-0396(88)90110-6. |
[38] |
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), 199-244.
doi: 10.1007/BF01047831. |
[39] |
C. Foias and R. Temam,
Determination of the solutions of the Navier-Stokes equations by a set of nodal values, Math. Comput., 43 (1984), 117-133.
doi: 10.1090/S0025-5718-1984-0744927-9. |
[40] |
C. Foias and R. Temam, Asymptotic numerical analysis for the Navier-Stokes equations, Nonlinear Dynamics and Turbulence, eds. Barenblatt, Iooss, Joseph, Boston: Pitman Advanced Pub. Prog., 1983,139-155. |
[41] |
C. Foias and E. S. Titi,
Determining nodes, finite difference schemes and inertial manifolds, Nonlinearity, 4 (1991), 135-153.
doi: 10.1088/0951-7715/4/1/009. |
[42] |
M. Gesho, E. Olson and E. S. Titi,
A computational study of a data assimilation algorithm for the two-dimensional Navier-Stokes equations, Commun. Phys, 19 (2016), 1094-1110.
|
[43] |
M. Ghil, B. Shkoller and V. Yangarber, A balanced diagnostic system compatible with a barotropic prognostic model., Mon. Wea. Rev., 105 (1977), 1223-1238. Google Scholar |
[44] |
M. Ghil, M. Halem and R. Atlas, Time-continuous assimilation of remote-sounding data and its effect on weather forecasting, Mon. Wea. Rev., 107 (1978), 140-171. Google Scholar |
[45] |
L. Giacomeli and F. Otto,
New bounds for the Kuramoto-Sivashinsky equation, Commun. Pure. Appl. Math., 58 (2005), 297-318.
doi: 10.1002/cpa.20031. |
[46] |
S. N. Gomes, D. T. Papageorgiou and G. A. Pavliotis,
Stabilizing non-trivial solutions of generalized Kuramoto-Sivashinsky equation using feedback and optimal control, IMA J. Applied Mathematics, 82 (2017), 158-194.
doi: 10.1093/imamat/hxw011. |
[47] |
S. N. Gomes, M. Pradas, S. Kalliadasis, D. T. Papageorgiou and G. A. Pavliotis,
Controlling roughening processes in the stochastic Kuramoto-Sivashinsky equation, Physica D: Nonl. Phenom., 348 (2017), 33-43.
doi: 10.1016/j.physd.2017.02.011. |
[48] |
J. Goodman,
Stability of the Kuramoto-Sivashinky and related systems, Commun. Pure Appl. Math., 47 (1994), 293-306.
doi: 10.1002/cpa.3160470304. |
[49] |
J. K. Hale,
Asymptotic Behavior of Dissipative Systems, Math. Survey and Monographs, 25 AMS, Providence, R. I., 1988. |
[50] |
L. Illing, D. J. Gauthier and R. Roy,
Controlling optical chaos, Spatio-Temporal Dynamics, and Patterns, Advances in Atomic, Molecular and Optical Physics, 54 (2007), 615-697.
doi: 10.1016/S1049-250X(06)54010-8. |
[51] |
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), 38-60.
doi: 10.1016/0167-2789(90)90046-R. |
[52] |
M. Jolly, V. Martinez and E. S. Titi,
A data assimilation algorithm for the subcritical surface quasi-geostrophic equation, Advanced Nonlinear Studies, 17 (2017), 167-192.
doi: 10.1515/ans-2016-6019. |
[53] |
M. S. Jolly, T. Sadigov and E. S. Titi,
A determining form for the damped driven nonlinear Schrödinger equation-Fourier modes case, J. Diff. Eqns., 258 (2015), 2711-2744.
doi: 10.1016/j.jde.2014.12.023. |
[54] |
M. S. Jolly, T. Sadigov and E. S. Titi,
Determining form and data assimilation algorithm for weakly damped and driven Korteweg-de Vries equaton-Fourier modes case, Nonlinear Analysis: Real World Applications, 36 (2017), 287-317.
doi: 10.1016/j.nonrwa.2017.01.010. |
[55] |
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), 72-88.
doi: 10.1016/0022-247X(92)90190-O. |
[56] |
D. Jones and E. S. Titi,
Determining finite volume elements for the 2-D Navier-Stokes equations, Physica D, 60 (1992), 165-174.
doi: 10.1016/0167-2789(92)90233-D. |
[57] |
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), 875-887.
doi: 10.1512/iumj.1993.42.42039. |
[58] |
V. Kalantarov and E. S. Titi,
Finite-parameters feedback control for stabilizing damped nonlinear wave equations, Contemporary Mathematics: Nonlinear Analysis and Optimization, AMS, 669 (2016), 115-133.
doi: 10.1090/conm/659/13193. |
[59] |
V. Kalantarov and E. S. Titi, Global stabilization of the Navier-Stokes-Voight and the damped nonlinear wave equations by finite number of feedback controllers, Discrete and Continuous Dynamical Systems -B, (2017), to appear, arXiv: 1706.00162 Google Scholar |
[60] |
A. Kazaam and L. Trefethen,
Fourth-order time stepping for stiff PDEs, J. Sci Comp., 26 (2005), 1214-1233.
doi: 10.1137/S1064827502410633. |
[61] |
I. Kukavica,
On the number of determining nodes for the Ginzburg-Landau equation, Nonlinearity, 5 (1992), 997-1006.
doi: 10.1088/0951-7715/5/5/001. |
[62] |
Y. Kuramoto and T. Tsusuki,
Reductive perturbation approach to chemical instabilities, Prog. Theor. Phys., 52 (1974), 1399-1401.
doi: 10.1143/PTP.52.1399. |
[63] |
R. E. LaQuey, S. M. Mahajan, P. H. Rutherford and W. M. Tang,
Nonlinear saturation of the trapped-ion mode, Phys. Rev. Let., 34 (1975), 391-394.
doi: 10.2172/4202869. |
[64] |
C. H. Lee and H. T. Tran,
Reduced-order-based feedback control of the Kuramoto-Sivashinsky equation, Journal of Computational and Applied Mathematics, 173 (2005), 1-19.
doi: 10.1016/j.cam.2004.02.021. |
[65] |
M. Li and P. D. Christofides,
Optimal control of diffusion-convection-reaction processes using reduced order models, Computers and Chemical Engineering, 32 (2008), 2123-2135.
doi: 10.1016/j.compchemeng.2007.10.018. |
[66] |
Y. Lou and P. D. Christofides, Optimal actuator/sensor placement for nonlinear control of the Kuramoto-Sivashinksky equation, IEEE Transactions on Control Systems Tech., 11 (2002), 737-745. Google Scholar |
[67] |
P. Markowich, E. S. Titi and S. Trabelsi,
Continuous data assimilation for the three-dimensional Brinkman-Forchheimer-Extended Darcy model, Nonlinearity, 29 (2016), 1292-1328.
doi: 10.1088/0951-7715/29/4/1292. |
[68] |
C. Mondaini and E. S. Titi, Uniform in time error estimates for the postprocessing Galerkin method applied to a data assimilation algorithm, SIAM Journal on Numerical Analysis, (2017), to appear, arXiv: 1612.06998. Google Scholar |
[69] |
B. Nicolaenko, B. Scheurer and R. Temam,
Some global dynamical properties of the Kuramoto-Sivashinsky equation: nonlinear stability and attractors, Physica D, 16 (1985), 155-183.
doi: 10.1016/0167-2789(85)90056-9. |
[70] |
M. Oliver and E. S. Titi,
On the domain of analyticity for solutions of second order analytic nonlinear differential equations, J. Differential Equations, 174 (2001), 55-74.
doi: 10.1006/jdeq.2000.3927. |
[71] |
F. Otto,
Optimal bounds on the Kuramoto-Sivashinsky equations, Journal of Functional Analysis, 257 (2009), 2188-2245.
doi: 10.1016/j.jfa.2009.01.034. |
[72] |
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. |
[73] |
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), 61-86.
doi: 10.1023/A:1026153311546. |
[74] |
R. Rosa and R. Temam, Finite-dimensional feedback control of a scalar reaction-diffusion equation via inertial manifold theory, Foundations of Computational Mathematics, Selected papers of a conference held at IMPA, Rio de Janeiro, RJ, Brazil, eds. F. Cucker and M. Shub, Springer-Verlag, Berlin, (1997), 382-391. |
[75] |
G. R. Sell and Y. You,
Dynamics of Evolutionary Equations, Springer, 2002.
doi: 10.1007/978-1-4757-5037-9. |
[76] |
S. 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), 177-184. Google Scholar |
[77] |
G. I. Sivashinsky,
Nonlinear analysis of hydrodynamic instability in laminar flames, Acta Astronautica, 4 (1977), 1177-1206.
doi: 10.1016/0094-5765(77)90096-0. |
[78] |
R. Temam,
Infinite Dimensional Dynamical Systems in Mechanics and Physics, New York: Springer, 1988.
doi: 10.1007/978-1-4684-0313-8. |
[79] |
R. Temam,
Navier-Stokes Equations: Theory and Numerical Analysis, AMS Chelsea Publishing, Providence, RI, Theory and numerical analysis, Reprint of the 1984 edition, 2001.
doi: 10.1090/chel/343. |
[80] |
A. Thompson, S. B. Gomes, G. A. Pavliotis and D. T. Papageorgio, Stabilising falling liquid film flows using feedback control Physics of Fluids, 28 (2016), 012107.
doi: 10.1063/1.4938761. |









Figure | # Actuators | | | | Interpolant operator |
1 | 10 | 300 | 1 | 100 | finite volume elements |
Figure | # Actuators | | | | Interpolant operator |
1 | 10 | 300 | 1 | 100 | finite volume elements |
Figure | # Actuators | | | | Interpolant Operator |
2 | 0 | 0 | 1.1 | 0 | |
3 | 0 | 0 | 4/15 | 0 | |
4 | 4 | 20 | 4/15 | 0 | Fourier modes |
5 | 4 | 20 | 4/20 | 0 | finite volume |
6 | 4 | 20 | 4/20 | 40 | nodal values |
Figure | # Actuators | | | | Interpolant Operator |
2 | 0 | 0 | 1.1 | 0 | |
3 | 0 | 0 | 4/15 | 0 | |
4 | 4 | 20 | 4/15 | 0 | Fourier modes |
5 | 4 | 20 | 4/20 | 0 | finite volume |
6 | 4 | 20 | 4/20 | 40 | nodal values |
Figure | # Actuators | | | | interpolant operator | ||
7 | 0 | 0 | 1 | 50 | 2.0 | 4.0 | |
8 | 1 | 30 | 1 | 50 | 2.0 | 4.0 | finite volume |
9 | 1 | 30 | 1 | varying | 2.0 | 4.0 | finite volume |
similar to Fig 8 | 1 | 30 | 1 | 50 | 2.0 | 4.0 | nodal values |
Figure | # Actuators | | | | interpolant operator | ||
7 | 0 | 0 | 1 | 50 | 2.0 | 4.0 | |
8 | 1 | 30 | 1 | 50 | 2.0 | 4.0 | finite volume |
9 | 1 | 30 | 1 | varying | 2.0 | 4.0 | finite volume |
similar to Fig 8 | 1 | 30 | 1 | 50 | 2.0 | 4.0 | nodal values |
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