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

Evelyn Lunasin; Edriss S. Titi

doi:10.3934/eect.2017027

Article Contents

2017, Volume 6, Issue 4: 535-557. Doi: 10.3934/eect.2017027

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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
* Corresponding author: Evelyn Lunasin

Received: February 28, 2017

Revised: July 31, 2017

Published: November 2017

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Abstract

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.

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Mathematics Subject Classification: Primary: 35K57, 37L25, 37L30, 37N35, 93B52, 93C20, 93D15.

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

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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 $u(x, t=200)$

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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 $u(x, t)$

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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 $u(x, t)$

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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 $u(x, t)$

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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

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

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

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

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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

1 10 300 1 100 finite volume elements

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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

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

| Show Table

DownLoad: CSV

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

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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References

[1]	S. Ahuja, Reduction Methods for Feedback Stabilization of Fluid Flows, Ph. D Thesis, Dept. of Mechanical and Aerospace Engineering, Princeton University, 2009.
[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.
[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.
[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.
[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
[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.
[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.
[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.
[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.

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

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Figure	# Actuators	$\mu$	$\nu$	$\alpha$	Interpolant operator
1	10	300	1	100	finite volume elements

Figure	# Actuators	$\mu$	$\nu$	$t_c$	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	$\mu$	$\nu$	$\beta_T$	$\beta_U$	$\gamma$	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

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

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