American Institute of Mathematical Sciences

Advanced
December  2017, 6(4): 535-557. doi: 10.3934/eect.2017027

Finite determining parameters feedback control for distributed nonlinear dissipative systems -a computational study

Evelyn Lunasin 1,, and  Edriss S. Titi 2,3,

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

* Corresponding author: Evelyn Lunasin

Received  March 2017 Revised  August 2017 Published  September 2017

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 [7]. This feedback control algorithm overcomes some of the major difficulties in control of multi-scale processes: It does not require the presence of separation of scales nor does it assume the existence of a finite-dimensional globally invariant inertial manifold. In this work we present a theoretical framework for a control algorithm which allows us to give a systematic stability analysis, and present the parameter regime where stabilization or control objective is attained. In addition, the number of observables and controllers that were derived analytically and implemented in our numerical studies is consistent with the finite number of determining modes that are relevant to the underlying physical system. We verify the results computationally in the context of the Chafee-Infante reaction-diffusion equation, the Kuramoto-Sivashinsky equation, and other applied control problems, and observe that the control strategy is robust and independent of the model equation describing the dissipative system.

Keywords: Globally stabilizing feedback control, Chafee-Infante, Kuramoto-Sivashinsky, reaction-diffusion, Navier-Stokes equations, feedback control, data assimilation, determining modes, determining nodes, determining volume elements.
Mathematics Subject Classification: Primary: 35K57, 37L25, 37L30, 37N35, 93B52, 93C20, 93D15.
Citation: Evelyn Lunasin, Edriss S. Titi. Finite determining parameters feedback control for distributed nonlinear dissipative systems -a computational study. Evolution Equations & Control Theory, 2017, 6 (4) : 535-557. doi: 10.3934/eect.2017027
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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[8]

A. V. Babin and M. Vishik, Attractors of Evolutionary Partial Differential Equations, North-Holland, Amsterdam, London, NewYork, Tokyo, 1992. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[19]

P. Constantin and C. Foias, Navier-Stokes Equations, University of Chicago Press, Chicago, 1988. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[33]

C. Foias, O. P. Manley, R. Rosa and R. Temam, Navier-Stokes Equations and Turbulence, Cambridge University Press, 2001. doi: 10.1017/CBO9780511546754. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[48]

J. Goodman, Stability of the Kuramoto-Sivashinky and related systems, Commun. Pure Appl. Math., 47 (1994), 293-306. doi: 10.1002/cpa.3160470304. Google Scholar

[49]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, Math. Survey and Monographs, 25 AMS, Providence, R. I., 1988. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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: 1706.00162Google 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[75]

G. R. Sell and Y. You, Dynamics of Evolutionary Equations, Springer, 2002. doi: 10.1007/978-1-4757-5037-9. Google Scholar

[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. Google Scholar

[78]

R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, New York: Springer, 1988. doi: 10.1007/978-1-4684-0313-8. Google Scholar

[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. Google Scholar

[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. Google Scholar

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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[8]

A. V. Babin and M. Vishik, Attractors of Evolutionary Partial Differential Equations, North-Holland, Amsterdam, London, NewYork, Tokyo, 1992. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[19]

P. Constantin and C. Foias, Navier-Stokes Equations, University of Chicago Press, Chicago, 1988. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[33]

C. Foias, O. P. Manley, R. Rosa and R. Temam, Navier-Stokes Equations and Turbulence, Cambridge University Press, 2001. doi: 10.1017/CBO9780511546754. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[48]

J. Goodman, Stability of the Kuramoto-Sivashinky and related systems, Commun. Pure Appl. Math., 47 (1994), 293-306. doi: 10.1002/cpa.3160470304. Google Scholar

[49]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, Math. Survey and Monographs, 25 AMS, Providence, R. I., 1988. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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: 1706.00162Google 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[75]

G. R. Sell and Y. You, Dynamics of Evolutionary Equations, Springer, 2002. doi: 10.1007/978-1-4757-5037-9. Google Scholar

[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. Google Scholar

[78]

R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, New York: Springer, 1988. doi: 10.1007/978-1-4684-0313-8. Google Scholar

[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. Google Scholar

[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. Google Scholar

Figure 1.  (a) Closed-loop profile showing stability of the $u(x,t)=~0$ steady state solution. (b) Top-view
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  (a) Open-loop profile showing stability of the $u(x,t)=~0$ steady state solution when $\nu = 1.1 > 1$ (b) Profile of $u(x, t=200)$
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  (a) Open-loop profile showing instability of the $u(x,t)=0$ steady state solution when $\nu = 4/15 < 1$. (b) Top view profile of $u(x, t)$
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4.  (a) Open-loop profile showing instability of the $u(x,t)=0$ steady state solution for $0<t<40$ for $\nu = 4/15 < 1$, then the feedback control with $\mu=20$ is turned on for $t>40$ which exponentially stabilizes the system. (b) Top view profile of $u(x, t)$
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 5.  (a) Closed-loop profile showing fast stabilization of the $u(x,t)=0$ steady state solution for $\nu = 4/20 < 1$, and with $\mu=20$. (b) Top view profile of $u(x, t)$
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 6.  (a) With $u_0 = 1e^{-10}\cos x (1 + \sin x)$, the film height starts to destabilize around $t = 32$ and then once feedback control is turned on at $t_c=40$, the solution stabilizes to zero again. (b) A top view of the controlled profile
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 7.  (a) Open-loop profile showing instability of the $u(x,t)=0$ steady state solution. (b) Top-view of $u(x,t)$
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 8.  (a) Closed-loop profile showing stabilization to $u(x,t)=0$ steady state solution. (b) Top-view
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 9.  (a) Closed-loop profile showing eventual stability. (b) Top-view
Figure Options

Download full-size image

Download as PowerPoint slide

Table 1.  Model parameters and type of interpolant operator for the controlled and uncontrolled 1D Chafee-Infante equations
Figure# Actuators $\mu$ $\nu$ $\alpha$Interpolant operator
1103001100 finite volume elements
Table Options

Download as excel

Figure# Actuators $\mu$ $\nu$ $\alpha$Interpolant operator
1103001100 finite volume elements
Table 2.  Model parameters and type of interpolant operator for the un-controlled and controlled 1D Kuramoto-Sivashinksy equations
Figure# Actuators $\mu$ $\nu$ $t_c$Interpolant Operator
2001.10
3004/150
44204/150 Fourier modes
54204/200 finite volume
64204/2040 nodal values
Table Options

Download as excel

Figure# Actuators $\mu$ $\nu$ $t_c$Interpolant Operator
2001.10
3004/150
44204/150 Fourier modes
54204/200 finite volume
64204/2040 nodal values
Table 3.  Model parameters and type of interpolant operator for the un-controlled and controlled catalytic rod problem
Figure# Actuators $\mu$ $\nu$ $\beta_T$$\beta_U$$\gamma$interpolant operator
7001502.04.0
81301502.04.0finite volume
91301varying2.04.0finite volume
similar to Fig 81301502.04.0nodal values
Table Options

Download as excel

Figure# Actuators $\mu$ $\nu$ $\beta_T$$\beta_U$$\gamma$interpolant operator
7001502.04.0
81301502.04.0finite volume
91301varying2.04.0finite volume
similar to Fig 81301502.04.0nodal values
[1]

Abderrahim Azouani, Edriss S. Titi. Feedback control of nonlinear dissipative systems by finite determining parameters - A reaction-diffusion paradigm. Evolution Equations & Control Theory, 2014, 3 (4) : 579-594. doi: 10.3934/eect.2014.3.579
[2]

Hans G. Kaper, Bixiang Wang, Shouhong Wang. Determining nodes for the Ginzburg-Landau equations of superconductivity. Discrete & Continuous Dynamical Systems - A, 1998, 4 (2) : 205-224. doi: 10.3934/dcds.1998.4.205
[3]

H. T. Banks, John E. Banks, R. A. Everett, John D. Stark. An adaptive feedback methodology for determining information content in stable population studies. Mathematical Biosciences & Engineering, 2016, 13 (4) : 653-671. doi: 10.3934/mbe.2016013
[4]

A. V. Fursikov. Stabilization for the 3D Navier-Stokes system by feedback boundary control. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 289-314. doi: 10.3934/dcds.2004.10.289
[5]

Yanli Han, Yan Gao. Determining the viability for hybrid control systems on a region with piecewise smooth boundary. Numerical Algebra, Control & Optimization, 2015, 5 (1) : 1-9. doi: 10.3934/naco.2015.5.1
[6]

Andrei Fursikov, Alexey V. Gorshkov. Certain questions of feedback stabilization for Navier-Stokes equations. Evolution Equations & Control Theory, 2012, 1 (1) : 109-140. doi: 10.3934/eect.2012.1.109
[7]

Luigi C. Berselli, Franco Flandoli. Remarks on determining projections for stochastic dissipative equations. Discrete & Continuous Dynamical Systems - A, 1999, 5 (1) : 197-214. doi: 10.3934/dcds.1999.5.197
[8]

Klemens Fellner, Stefanie Sonner, Bao Quoc Tang, Do Duc Thuan. Stabilisation by noise on the boundary for a Chafee-Infante equation with dynamical boundary conditions. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 4055-4078. doi: 10.3934/dcdsb.2019050
[9]

Amin Boumenir. Determining the shape of a solid of revolution. Mathematical Control & Related Fields, 2019, 9 (3) : 509-515. doi: 10.3934/mcrf.2019023
[10]

Qi Wang, Yanren Hou. Determining an obstacle by far-field data measured at a few spots. Inverse Problems & Imaging, 2015, 9 (2) : 591-600. doi: 10.3934/ipi.2015.9.591
[11]

Evrad M. D. Ngom, Abdou Sène, Daniel Y. Le Roux. Global stabilization of the Navier-Stokes equations around an unstable equilibrium state with a boundary feedback controller. Evolution Equations & Control Theory, 2015, 4 (1) : 89-106. doi: 10.3934/eect.2015.4.89
[12]

Evrad M. D. Ngom, Abdou Sène, Daniel Y. Le Roux. Boundary stabilization of the Navier-Stokes equations with feedback controller via a Galerkin method. Evolution Equations & Control Theory, 2014, 3 (1) : 147-166. doi: 10.3934/eect.2014.3.147
[13]

Jean-Pierre Raymond, Laetitia Thevenet. Boundary feedback stabilization of the two dimensional Navier-Stokes equations with finite dimensional controllers. Discrete & Continuous Dynamical Systems - A, 2010, 27 (3) : 1159-1187. doi: 10.3934/dcds.2010.27.1159
[14]

Enrique Fernández-Cara. Motivation, analysis and control of the variable density Navier-Stokes equations. Discrete & Continuous Dynamical Systems - S, 2012, 5 (6) : 1021-1090. doi: 10.3934/dcdss.2012.5.1021
[15]

Tomás Caraballo, Peter E. Kloeden, José Real. Invariant measures and Statistical solutions of the globally modified Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2008, 10 (4) : 761-781. doi: 10.3934/dcdsb.2008.10.761
[16]

Pedro Marín-Rubio, Antonio M. Márquez-Durán, José Real. Pullback attractors for globally modified Navier-Stokes equations with infinite delays. Discrete & Continuous Dynamical Systems - A, 2011, 31 (3) : 779-796. doi: 10.3934/dcds.2011.31.779
[17]

Rocio de la Torre, Amaia Lusa, Manuel Mateo, El-Houssaine Aghezzaf. Determining personnel promotion policies in HEI. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-25. doi: 10.3934/jimo.2019031
[18]

Elena Braverman, Alexandra Rodkina. Stabilization of difference equations with noisy proportional feedback control. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2067-2088. doi: 10.3934/dcdsb.2017085
[19]

Kiah Wah Ong. Dynamic transitions of generalized Kuramoto-Sivashinsky equation. Discrete & Continuous Dynamical Systems - B, 2016, 21 (4) : 1225-1236. doi: 10.3934/dcdsb.2016.21.1225
[20]

Hongyu Liu, Jun Zou. Uniqueness in determining multiple polygonal scatterers of mixed type. Discrete & Continuous Dynamical Systems - B, 2008, 9 (2) : 375-396. doi: 10.3934/dcdsb.2008.9.375

2018 Impact Factor: 1.048

Tools

Metrics

  • PDF downloads (27)
  • HTML views (108)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences