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

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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:

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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.
[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.
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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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Figure	# Actuators	$\mu$	$\nu$	$\alpha$	Interpolant operator
1	10	300	1	100	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
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

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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

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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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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