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December  2014, 3(4): 579-594. doi: 10.3934/eect.2014.3.579

Feedback control of nonlinear dissipative systems by finite determining parameters - A reaction-diffusion paradigm

1. 

Mohammed First University, National School of Applied Sciences Al Hoceima, Ajdir, 32003, Al Hoceima, Morocco

2. 

Department of Computer Science and Applied Mathematics, Weizmann Institute of Science,Rehovot 76100, Israel

Received  May 2014 Revised  September 2014 Published  October 2014

We introduce here a simple finite-dimensional feedback control scheme for stabilizing solutions of infinite-dimensional dissipative evolution equations, such as reaction-diffusion systems, the Navier-Stokes equations and the Kuramoto-Sivashinsky equation. The designed feedback control scheme takes advantage of the fact that such systems possess finite number of determining parameters (degrees of freedom), namely, finite number of determining Fourier modes, determining nodes, and determining interpolants and projections. In particular, the feedback control scheme uses finitely many of such observables and controllers. This observation is of a particular interest since it implies that our approach has far more reaching applications, in particular, in data assimilation. Moreover, we emphasize that our scheme treats all kinds of the determining projections, as well as, the various dissipative equations with one unified approach. However, for the sake of simplicity we demonstrate our approach in this paper to a one-dimensional reaction-diffusion equation paradigm.
Citation: Abderrahim Azouani, Edriss S. Titi. Feedback control of nonlinear dissipative systems by finite determining parameters - A reaction-diffusion paradigm. Evolution Equations and Control Theory, 2014, 3 (4) : 579-594. doi: 10.3934/eect.2014.3.579
References:
[1]

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.

[2]

A. Azouani, E. Olson and E. S. Titi, Continuous data assimilation using general interpolant observables, Journal of Nonlinear Analysis, 24 (2014), 277-304. doi: 10.1007/s00332-013-9189-y.

[3]

A. V. Babin and M. Vishik, Attractors of Evolution Partial Differential Equations, North-Holland, Amsterdam, London, New York, Tokyo, 1992.

[4]

H. Bessaih, E. Olson and E. S. Titi, Continuous assimilation of data with stochastic noise, preprint,, , (). 

[5]

P. G. Ciarlet, The Finite Element Method for Elliptic Problems, Classics in Applied Mathematics, 40, SIAM, 2002. doi: 10.1137/1.9780898719208.

[6]

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.

[7]

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.

[8]

P. Constantin, Ch. Doering and E. S. Titi, Rigorous estimates of small scales in turbulent flows, Journal of Mathematical Physics, 37 (1996), 6152-6156. doi: 10.1063/1.531769.

[9]

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

[10]

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, Vol. 70, New York, 1989. doi: 10.1007/978-1-4612-3506-4.

[11]

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.

[12]

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.

[13]

C. Foias, M. Jolly and R. Karavchenko, Determining forms for the Kuramoto-Sivashinsky and Lorenz equations: Analysis and computations,, (in preparation)., (). 

[14]

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.

[15]

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, Uspekhi Matematicheskikh Nauk, 69 (2014), 359-381. doi: 10.1070/RM2014v069n02ABEH004891.

[16]

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

[17]

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.

[18]

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.

[19]

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.

[20]

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.

[21]

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.

[22]

C. Foias and R. Temam, Asymptotic numerical analysis for the Navier-Stokes equations, in Nonlinear Dynamics and Turbulence, Interaction Mech. Math. Ser., Pitman, Boston, MA, (1983), 139-155.

[23]

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.

[24]

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

[25]

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.

[26]

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.

[27]

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.

[28]

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.

[29]

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.

[30]

E. Lunasin and E. S. Titi, Finite determining parameters feedback control for distributed nonlinear dissipative systems - a computational study,, (in preparation)., (). 

[31]

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.

[32]

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.

[33]

R. Rosa and R. Temam, Finite-dimensional feedback control of a scalar reaction-diffusion equation via inertial manifold theory in Foundations of Computational Mathematics, Selected papers of a conference held at IMPA, Rio de Janeiro, RJ, Brazil, January 1997 (Eds. F. Cucker and M. Shub), Springer-Verlag, Berlin, (1997), 382-391.

[34]

G. R. Sell and Y. You, Dynamics of Evolutionary Equations, Applied Mathematical Sciences, 15, Springer-Verlag, New York, 2002. doi: 10.1007/978-1-4757-5037-9.

[35]

S. Y. 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. doi: 10.1016/S0959-1524(99)00029-3.

[36]

R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, $2^{nd}$ edition, Springer, New York, 1997. doi: 10.1007/978-1-4612-0645-3.

[37]

R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, Reprint of the 1984 edition, Springer, AMS Chelsea Publishing, Providence, RI, Theory and numerical analysis, 2001.

show all references

References:
[1]

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.

[2]

A. Azouani, E. Olson and E. S. Titi, Continuous data assimilation using general interpolant observables, Journal of Nonlinear Analysis, 24 (2014), 277-304. doi: 10.1007/s00332-013-9189-y.

[3]

A. V. Babin and M. Vishik, Attractors of Evolution Partial Differential Equations, North-Holland, Amsterdam, London, New York, Tokyo, 1992.

[4]

H. Bessaih, E. Olson and E. S. Titi, Continuous assimilation of data with stochastic noise, preprint,, , (). 

[5]

P. G. Ciarlet, The Finite Element Method for Elliptic Problems, Classics in Applied Mathematics, 40, SIAM, 2002. doi: 10.1137/1.9780898719208.

[6]

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.

[7]

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.

[8]

P. Constantin, Ch. Doering and E. S. Titi, Rigorous estimates of small scales in turbulent flows, Journal of Mathematical Physics, 37 (1996), 6152-6156. doi: 10.1063/1.531769.

[9]

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

[10]

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, Vol. 70, New York, 1989. doi: 10.1007/978-1-4612-3506-4.

[11]

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.

[12]

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.

[13]

C. Foias, M. Jolly and R. Karavchenko, Determining forms for the Kuramoto-Sivashinsky and Lorenz equations: Analysis and computations,, (in preparation)., (). 

[14]

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.

[15]

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, Uspekhi Matematicheskikh Nauk, 69 (2014), 359-381. doi: 10.1070/RM2014v069n02ABEH004891.

[16]

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

[17]

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.

[18]

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.

[19]

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.

[20]

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.

[21]

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.

[22]

C. Foias and R. Temam, Asymptotic numerical analysis for the Navier-Stokes equations, in Nonlinear Dynamics and Turbulence, Interaction Mech. Math. Ser., Pitman, Boston, MA, (1983), 139-155.

[23]

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.

[24]

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

[25]

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.

[26]

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.

[27]

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.

[28]

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.

[29]

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.

[30]

E. Lunasin and E. S. Titi, Finite determining parameters feedback control for distributed nonlinear dissipative systems - a computational study,, (in preparation)., (). 

[31]

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.

[32]

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.

[33]

R. Rosa and R. Temam, Finite-dimensional feedback control of a scalar reaction-diffusion equation via inertial manifold theory in Foundations of Computational Mathematics, Selected papers of a conference held at IMPA, Rio de Janeiro, RJ, Brazil, January 1997 (Eds. F. Cucker and M. Shub), Springer-Verlag, Berlin, (1997), 382-391.

[34]

G. R. Sell and Y. You, Dynamics of Evolutionary Equations, Applied Mathematical Sciences, 15, Springer-Verlag, New York, 2002. doi: 10.1007/978-1-4757-5037-9.

[35]

S. Y. 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. doi: 10.1016/S0959-1524(99)00029-3.

[36]

R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, $2^{nd}$ edition, Springer, New York, 1997. doi: 10.1007/978-1-4612-0645-3.

[37]

R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, Reprint of the 1984 edition, Springer, AMS Chelsea Publishing, Providence, RI, Theory and numerical analysis, 2001.

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