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

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

* 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.

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:
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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. AltafE. S. TitiT. GebraelO. KnioL. ZhaoM. 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]

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[5]

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[6]

A. AzouaniE. 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]

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[9]

H. BessaihE. 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. CaoI. 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. CharneyJ. 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. CockburnD. 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. CockburnD. 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. CohenJ. A. KrommesW. 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. ColletJ.-P. EckmannH. 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. DubljevicN. 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. DubljevicN. 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-FarraA. 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. FarhatM. 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]

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Figure 1.  (a) Closed-loop profile showing stability of the $u(x,t)=~0$ steady state solution. (b) Top-view
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 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 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 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 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 7.  (a) Open-loop profile showing instability of the $u(x,t)=0$ steady state solution. (b) Top-view of $u(x,t)$
Figure 8.  (a) Closed-loop profile showing stabilization to $u(x,t)=0$ steady state solution. (b) Top-view
Figure 9.  (a) Closed-loop profile showing eventual stability. (b) Top-view
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
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
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
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
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