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September  2020, 9(3): 605-634. doi: 10.3934/eect.2020026

Local null controllability of coupled degenerate systems with nonlocal terms and one control force

R. Demarque 1,, , J. Límaco 2, and  L. Viana 3,

1. 

Departamento de Ciências da Natureza, Universidade Federal Fluminense, Rio das Ostras, RJ, 28895-532, Brazil
2. 

Departamento de Matemática Aplicada, Universidade Federal Fluminense, Niterói, RJ, 24020-140, Brazil
3. 

Departamento de Análise, Universidade Federal Fluminense, Niterói, RJ, 24020-140, Brazil

* Corresponding author: reginaldodr@id.uff.br

Received  March 2019 Revised  August 2019 Published  September 2020 Early access  December 2019

In this paper, we are concerned with the internal control of a class of one-dimensional nonlinear parabolic systems with nonlocal and weakly degenerate diffusion coefficients. Our main theorem establishes a local null controllability result with only one internal control for a system of two equations. The proof, based on the ideias developed by Fursikov and Imanuvilov, is obtained from the global null controllability of the linearized system provided by Lyusternik's Inverse Mapping Theorem. This work extends the results previously treated by the authors for just one equation. For the system, the main issue is to obtain similar results with just one internal control, which requires a new Carleman estimate with the local term just depending on one of the state function.

Keywords: Degenerate parabolic systems, controllability, nonlinear parabolic system, nonlocal term, Carleman inequalities.
Mathematics Subject Classification: Primary: 35K65, 93B05; Secondary: 35K55, 93C20.
Citation: R. Demarque, J. Límaco, L. Viana. Local null controllability of coupled degenerate systems with nonlocal terms and one control force. Evolution Equations and Control Theory, 2020, 9 (3) : 605-634. doi: 10.3934/eect.2020026
References:
[1]

E. M. Ait Ben Hassi, F. Ammar Khodja, A. Hajjaj and L. Maniar, Null controllability of degenerate parabolic cascade systems, Port. Math., 68 (2011), 345-367.  doi: 10.4171/PM/1895.

[2]

E. M. Ait Ben Hassi, F. Ammar Khodja, A. Hajjaj and L. Maniar, Carleman estimates and null controllability of coupled degenerate systems, Evol. Equ. Control Theory, 2 (2013), 441-459.  doi: 10.3934/eect.2013.2.441.

[3]

E. M. Ait Ben Hassi, M. Fadili and L. Maniar, On algebraic condition for null controllability of some coupled degenerate systems, Math. Control Relat. Fields, 9 (2019), 77-95.  doi: 10.3934/mcrf.2019004.

[4]

F. Alabau-Boussouira, P. Cannarsa and G. Fragnelli, Carleman estimates for degenerate parabolic operators with applications to null controllability, J. Evol. Equ., 6 (2006), 161-204.  doi: 10.1007/s00028-006-0222-6.

[5]

V. M. Alekseev, V. M. Tikhomirov and S. V. Fomin, Optimal Control, Contemporary Soviet Mathematics. Consultants Bureau, New York, 1987. doi: 10.1007/978-1-4615-7551-1.

[6]

F. Ammar-Khodja, A. Benabdallah, M. González Burgos and L. de Teresa, Recent results on the controllability of linear coupled parabolic problems: A survey, Math. Control Relat. Fields, 1 (2011), 267-306.  doi: 10.3934/mcrf.2011.1.267.

[7]

P. Cannarsa and L. de Teresa, Controllability of 1-d coupled degenerate parabolic equations, Electron. J. Differential Equations, 73 (2009), 21 pp.

[8]

N. Carreno and S. Guerrero, Local null controllability of the n-dimensional navier–stokes system with n- 1 scalar controls in an arbitrary control domain, J. Math. Fluid Mech., 15 (2013), 139-153.  doi: 10.1007/s00021-012-0093-2.

[9]

N. Chang and M. Chipot, On some model diffusion problems with a nonlocal lower order term, Chinese Ann. Math. Ser. B, 24 (2003), 147-166.  doi: 10.1142/S0252959903000153.

[10]

H. Clark, E. Fernández-Cara, J. Límaco and L. Medeiros, Theoretical and numerical local null controllability for a parabolic system with local and nonlocal nonlinearities, Appl. Math. Comput., 223 (2013), 483-505.  doi: 10.1016/j.amc.2013.08.035.

[11]

J.-M. Coron and S. Guerrero, Null controllability of the n-dimensional stokes system with n- 1 scalar controls, J. Differential Equations, 246 (2009), 2908-2921.  doi: 10.1016/j.jde.2008.10.019.

[12]

G. De Marco, G. Gorni and G. Zampieri, Global inversion of functions: An introduction., NoDEA Nonlinear Differential Equations Appl., 1 (1994), 229-248.  doi: 10.1007/BF01197748.

[13]

R. Demarque, J. Límaco and L. Viana, Local null controllability for degenerate parabolic equations with nonlocal term, Nonlinear Anal. Real World Appl., 43 (2018), 523-547.  doi: 10.1016/j.nonrwa.2018.04.001.

[14]

I. Ekeland and R. Temam, Convex Analysis and Variational Problems, Classics in Applied Mathematics, 28. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1999. doi: 10.1137/1.9781611971088.

[15]

M. Fadili and L. Maniar, Null controllability of n-coupled degenerate parabolic systems with m-controls, J. Evol. Equ., 17 (2017), 1311-1340.  doi: 10.1007/s00028-017-0385-3.

[16]

H. O. Fattorini and D. L. Russell, Exact controllability theorems for linear parabolic equations in one space dimension, Arch. Rational Mech. Anal., 43 (1971), 272-292.  doi: 10.1007/BF00250466.

[17]

W. Feller, The parabolic differential equations and the associated semi-groups of transformations, Ann. of Math., 55 (1952), 468-519. 

[18]

W. Feller, Diffusion processes in one dimension, Trans. Amer. Math. Soc., 77 (1954), 1-31.  doi: 10.1090/S0002-9947-1954-0063607-6.

[19]

E. Fernández-Cara, S. Guerrero, O. Y. Imanuvilov and J-P. Puel, Local exact controllability of the navier–stokes system, J. Math. Pures Appl., 83 (2004), 1501-1542.  doi: 10.1016/j.matpur.2004.02.010.

[20]

E. Fernández-Cara, S. Guerrero, O. Y. Imanuvilov and J.-P. Puel, Some controllability results forthe n-dimensional navier–stokes and boussinesq systems with n-1 scalar controls, SIAM J. Control Optim., 45 (2006), 146-173.  doi: 10.1137/04061965X.

[21]

E. Fernández-Cara, J. Límaco and S. B. De Menezes, Null controllability for a parabolic equation with nonlocal nonlinearities, Systems & Control Letters, 61 (2012), 107-111.  doi: 10.1016/j.sysconle.2011.09.017.

[22]

E. Fernández-Cara and E. Zuazua, The cost of approximate controllability for heat equations: The linear case, Adv. Differential Equations, 5 (2000), 465-514. 

[23]

G. Floridia, Approximate controllability for nonlinear degenerate parabolic problems with bilinear control, J. Differential Equations, 257 (2014), 3382-3422.  doi: 10.1016/j.jde.2014.06.016.

[24]

A. V. Fursikov and O. Y. Imanuvilov, Controllability of Evolution Equations, Lecture Notes Series, 34. Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul, 1996.

[25]

O. Y. Imanuvilov, Remarks on exact controllability for the navier-stokes equations, ESAIM Control Optim. Calc. Var., 6 (2001), 39-72.  doi: 10.1051/cocv:2001103.

[26]

O. Y. Imanuvilov and M. Yamamoto, Carleman inequalities for parabolic equations in sobolev spaces of negative order and exact controllability for semilinear parabolic equations, Publ. Res. Inst. Math. Sci., 39 (2003), 227-274.  doi: 10.2977/prims/1145476103.

[27]

G. Lebeau and L. Robbiano, Contrôle exact de léquation de la chaleur, Comm. Partial Differential Equations, 20 (1995), 335-356.  doi: 10.1080/03605309508821097.

[28]

L. Medeiros, J. Límaco and S. B. Menezes, Vibrations of elastic strings: Mathematical aspects, part one, J. Comput. Anal. Appl., 4 (2002), 91-127.  doi: 10.1023/A:1012934900316.

[29]

O. A. Oleinik and V. N. Samokhin, Mathematical Models in Boundary Layer Theory, Applied Mathematics and Mathematical Computation, 15. Chapman & Hall/CRC, Boca Raton, FL, 1999.

show all references

References:
[1]

E. M. Ait Ben Hassi, F. Ammar Khodja, A. Hajjaj and L. Maniar, Null controllability of degenerate parabolic cascade systems, Port. Math., 68 (2011), 345-367.  doi: 10.4171/PM/1895.

[2]

E. M. Ait Ben Hassi, F. Ammar Khodja, A. Hajjaj and L. Maniar, Carleman estimates and null controllability of coupled degenerate systems, Evol. Equ. Control Theory, 2 (2013), 441-459.  doi: 10.3934/eect.2013.2.441.

[3]

E. M. Ait Ben Hassi, M. Fadili and L. Maniar, On algebraic condition for null controllability of some coupled degenerate systems, Math. Control Relat. Fields, 9 (2019), 77-95.  doi: 10.3934/mcrf.2019004.

[4]

F. Alabau-Boussouira, P. Cannarsa and G. Fragnelli, Carleman estimates for degenerate parabolic operators with applications to null controllability, J. Evol. Equ., 6 (2006), 161-204.  doi: 10.1007/s00028-006-0222-6.

[5]

V. M. Alekseev, V. M. Tikhomirov and S. V. Fomin, Optimal Control, Contemporary Soviet Mathematics. Consultants Bureau, New York, 1987. doi: 10.1007/978-1-4615-7551-1.

[6]

F. Ammar-Khodja, A. Benabdallah, M. González Burgos and L. de Teresa, Recent results on the controllability of linear coupled parabolic problems: A survey, Math. Control Relat. Fields, 1 (2011), 267-306.  doi: 10.3934/mcrf.2011.1.267.

[7]

P. Cannarsa and L. de Teresa, Controllability of 1-d coupled degenerate parabolic equations, Electron. J. Differential Equations, 73 (2009), 21 pp.

[8]

N. Carreno and S. Guerrero, Local null controllability of the n-dimensional navier–stokes system with n- 1 scalar controls in an arbitrary control domain, J. Math. Fluid Mech., 15 (2013), 139-153.  doi: 10.1007/s00021-012-0093-2.

[9]

N. Chang and M. Chipot, On some model diffusion problems with a nonlocal lower order term, Chinese Ann. Math. Ser. B, 24 (2003), 147-166.  doi: 10.1142/S0252959903000153.

[10]

H. Clark, E. Fernández-Cara, J. Límaco and L. Medeiros, Theoretical and numerical local null controllability for a parabolic system with local and nonlocal nonlinearities, Appl. Math. Comput., 223 (2013), 483-505.  doi: 10.1016/j.amc.2013.08.035.

[11]

J.-M. Coron and S. Guerrero, Null controllability of the n-dimensional stokes system with n- 1 scalar controls, J. Differential Equations, 246 (2009), 2908-2921.  doi: 10.1016/j.jde.2008.10.019.

[12]

G. De Marco, G. Gorni and G. Zampieri, Global inversion of functions: An introduction., NoDEA Nonlinear Differential Equations Appl., 1 (1994), 229-248.  doi: 10.1007/BF01197748.

[13]

R. Demarque, J. Límaco and L. Viana, Local null controllability for degenerate parabolic equations with nonlocal term, Nonlinear Anal. Real World Appl., 43 (2018), 523-547.  doi: 10.1016/j.nonrwa.2018.04.001.

[14]

I. Ekeland and R. Temam, Convex Analysis and Variational Problems, Classics in Applied Mathematics, 28. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1999. doi: 10.1137/1.9781611971088.

[15]

M. Fadili and L. Maniar, Null controllability of n-coupled degenerate parabolic systems with m-controls, J. Evol. Equ., 17 (2017), 1311-1340.  doi: 10.1007/s00028-017-0385-3.

[16]

H. O. Fattorini and D. L. Russell, Exact controllability theorems for linear parabolic equations in one space dimension, Arch. Rational Mech. Anal., 43 (1971), 272-292.  doi: 10.1007/BF00250466.

[17]

W. Feller, The parabolic differential equations and the associated semi-groups of transformations, Ann. of Math., 55 (1952), 468-519. 

[18]

W. Feller, Diffusion processes in one dimension, Trans. Amer. Math. Soc., 77 (1954), 1-31.  doi: 10.1090/S0002-9947-1954-0063607-6.

[19]

E. Fernández-Cara, S. Guerrero, O. Y. Imanuvilov and J-P. Puel, Local exact controllability of the navier–stokes system, J. Math. Pures Appl., 83 (2004), 1501-1542.  doi: 10.1016/j.matpur.2004.02.010.

[20]

E. Fernández-Cara, S. Guerrero, O. Y. Imanuvilov and J.-P. Puel, Some controllability results forthe n-dimensional navier–stokes and boussinesq systems with n-1 scalar controls, SIAM J. Control Optim., 45 (2006), 146-173.  doi: 10.1137/04061965X.

[21]

E. Fernández-Cara, J. Límaco and S. B. De Menezes, Null controllability for a parabolic equation with nonlocal nonlinearities, Systems & Control Letters, 61 (2012), 107-111.  doi: 10.1016/j.sysconle.2011.09.017.

[22]

E. Fernández-Cara and E. Zuazua, The cost of approximate controllability for heat equations: The linear case, Adv. Differential Equations, 5 (2000), 465-514. 

[23]

G. Floridia, Approximate controllability for nonlinear degenerate parabolic problems with bilinear control, J. Differential Equations, 257 (2014), 3382-3422.  doi: 10.1016/j.jde.2014.06.016.

[24]

A. V. Fursikov and O. Y. Imanuvilov, Controllability of Evolution Equations, Lecture Notes Series, 34. Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul, 1996.

[25]

O. Y. Imanuvilov, Remarks on exact controllability for the navier-stokes equations, ESAIM Control Optim. Calc. Var., 6 (2001), 39-72.  doi: 10.1051/cocv:2001103.

[26]

O. Y. Imanuvilov and M. Yamamoto, Carleman inequalities for parabolic equations in sobolev spaces of negative order and exact controllability for semilinear parabolic equations, Publ. Res. Inst. Math. Sci., 39 (2003), 227-274.  doi: 10.2977/prims/1145476103.

[27]

G. Lebeau and L. Robbiano, Contrôle exact de léquation de la chaleur, Comm. Partial Differential Equations, 20 (1995), 335-356.  doi: 10.1080/03605309508821097.

[28]

L. Medeiros, J. Límaco and S. B. Menezes, Vibrations of elastic strings: Mathematical aspects, part one, J. Comput. Anal. Appl., 4 (2002), 91-127.  doi: 10.1023/A:1012934900316.

[29]

O. A. Oleinik and V. N. Samokhin, Mathematical Models in Boundary Layer Theory, Applied Mathematics and Mathematical Computation, 15. Chapman & Hall/CRC, Boca Raton, FL, 1999.

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