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September  2020, 9(3): 605-635. doi: 10.3934/eect.2020026

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

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  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.

Citation: R. Demarque, J. Límaco, L. Viana. Local null controllability of coupled degenerate systems with nonlocal terms and one control force. Evolution Equations & Control Theory, 2020, 9 (3) : 605-635. doi: 10.3934/eect.2020026
References:
[1]

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

[2]

E. M. Ait Ben HassiF. Ammar KhodjaA. 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.  Google Scholar

[3]

E. M. Ait Ben HassiM. 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.  Google Scholar

[4]

F. Alabau-BoussouiraP. 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.  Google Scholar

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[6]

F. Ammar-KhodjaA. BenabdallahM. 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.  Google Scholar

[7]

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

[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.  Google Scholar

[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.  Google Scholar

[10]

H. ClarkE. Fernández-CaraJ. 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.  Google Scholar

[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.  Google Scholar

[12]

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

[13]

R. DemarqueJ. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

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

[18]

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

[19]

E. Fernández-CaraS. GuerreroO. 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.  Google Scholar

[20]

E. Fernández-CaraS. GuerreroO. 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.  Google Scholar

[21]

E. Fernández-CaraJ. 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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

L. MedeirosJ. 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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

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

[2]

E. M. Ait Ben HassiF. Ammar KhodjaA. 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.  Google Scholar

[3]

E. M. Ait Ben HassiM. 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.  Google Scholar

[4]

F. Alabau-BoussouiraP. 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.  Google Scholar

[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.  Google Scholar

[6]

F. Ammar-KhodjaA. BenabdallahM. 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.  Google Scholar

[7]

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

[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.  Google Scholar

[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.  Google Scholar

[10]

H. ClarkE. Fernández-CaraJ. 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.  Google Scholar

[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.  Google Scholar

[12]

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

[13]

R. DemarqueJ. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

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

[18]

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

[19]

E. Fernández-CaraS. GuerreroO. 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.  Google Scholar

[20]

E. Fernández-CaraS. GuerreroO. 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.  Google Scholar

[21]

E. Fernández-CaraJ. 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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

L. MedeirosJ. 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.  Google Scholar

[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.  Google Scholar

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