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doi: 10.3934/eect.2021044
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Stackelberg-Nash null controllability of heat equation with general dynamic boundary conditions

1. 

Faculté Polydisciplinaire de Ouarzazate, Université Ibn Zohr, Ouarzazate, B.P. 638, 45000, Morocco

2. 

Cadi Ayyad University, Faculty of Sciences Semlalia, LMDP, UMMISCO (IRD-UPMC) B.P. 2390, Marrakesh, Morocco

* Corresponding author: Idriss Boutaayamou

Received  December 2020 Revised  June 2021 Early access August 2021

This paper deals with the hierarchical control of the anisotropic heat equation with dynamic boundary conditions and drift terms. We use the Stackelberg-Nash strategy with one leader and two followers. To each fixed leader, we find a Nash equilibrium corresponding to a bi-objective optimal control problem for the followers. Then, by some new Carleman estimates, we prove a null controllability result.

Citation: Idriss Boutaayamou, Lahcen Maniar, Omar Oukdach. Stackelberg-Nash null controllability of heat equation with general dynamic boundary conditions. Evolution Equations and Control Theory, doi: 10.3934/eect.2021044
References:
[1]

E. M. Ait Ben Hassi, S. E. Chorfi, L. Maniar and O. Oukdach, Lipschitz Stability for an Inverse Source Problem in Anisotropic Parabolic Equations with Dynamic Boundary Conditions, Evol. Equ. Control. Theory., 2020.

[2]

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.

[3]

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.

[4]

F. D. Araruna, B. S. V. Araújo and E. Fernández-Cara, Stackelberg-Nash null controllability for some linear and semilinear degenerate parabolic equations, Math. Control Signals Systems, 30 (2018). doi: 10.1007/s00498-018-0220-6.

[5]

F. D. ArarunaE. Fernández-CaraS. Guerrero and M. C. Santos, New results on the Stackelberg-Nash exact control of linear parabolic equations, Syst. Control. Lett., 104 (2017), 78-85.  doi: 10.1016/j.sysconle.2017.03.009.

[6]

F. D. ArarunaE. Fernández-Cara and M. C. Santos, Stackelberg-Nash exact controllability for linear and semilinear parabolic equations, ESAIM Control Optim. Calc. Var., 21 (2015), 835-856.  doi: 10.1051/cocv/2014052.

[7]

F. D. ArarunaE. Fernández-Cara and L. C. D. Silva, Hierarchic control for the wave equation, J. Optimiz. Theory. App., 178 (2018), 264-288.  doi: 10.1007/s10957-018-1277-6.

[8]

F. D. Araruna, E. Fernández-Cara and L. C. da Silva, Hierarchical exact controllability of semilinear parabolic equations with distributed and boundary controls, Commun. Contemp. Math., 22 (2019), 1950034. doi: 10.1142/S0219199719500342.

[9]

I. BoutaayamouS. E. ChorfiL. Maniar and O. Oukdach, The cost of approximate controllability of heat equation with general dynamical boundary conditions, Port. Math., 78 (2020), 65-99.  doi: 10.4171/PM/2061.

[10]

I. BoutaayamouG. Fragnelli and L. Maniar, Carleman estimates for parabolic equations with interior degeneracy and Neumann boundary conditions, J. Anal. Math., 135 (2018), 1-35.  doi: 10.1007/s11854-018-0030-2.

[11]

F. Boyer, On the penalised HUM approach and its applications to the numerical approximation of null-controls for parabolic problems, CANUM 2012, 41e Congrés National d'Analyse Numérique, ESAIM Proc., EDP Sci., Les Ulis, 41 (2013), 15-58. doi: 10.1051/proc/201341002.

[12]

P. CannarsaG. Fragnelli and D. Rocchetti, Null controllability of degenerate parabolic operators with drift, Netw. Heterog. Media, 2 (2007), 695-715.  doi: 10.3934/nhm.2007.2.695.

[13]

P. P. Carvalho and E. Fernández-Cara, On the computation of Nash and Pareto equilibria for some bi-objective control problems, J. Sci. Comput., 78 (2019), 246-273.  doi: 10.1007/s10915-018-0764-0.

[14]

L. Cherfils and A. Miranville, On the Caginalp system with dynamic boundary conditions and singular potentials, Appl. Math., 54 (2009), 89-115.  doi: 10.1007/s10492-009-0008-6.

[15]

I. P. de JesusJ. Límaco and M. R. Clark, Hierarchical control for the one-dimensional plate equation with a moving boundary, J. Dyn. Control Syst., 24 (2018), 635-655.  doi: 10.1007/s10883-018-9413-z.

[16]

J. I. Díaz, On the von Neumann problem and the approximate controllability of Stackelberg-Nash strategy for some environmental problems, RACSAM. Rev. R. Acad. Cien. Exactas. Nat. Ser. A Mat., 96 (2002), 343-356. 

[17]

J. I. Díaz and J.-L. Lions, On the approximate controllability of Stackelberg-Nash strategies, Ocean Circulation and Pollution Control: A Mathematical and Numerical Investigation, Madrid, Springer, Berlin, (1997) 17-27.

[18]

I. Ekeland and R. Teman, Analyse Convexe et Problemes Variationnels, Dunod; Gauthier-Villars, Paris-Brussels-Montreal, Que., 1974.

[19]

C. FabreJ-P. Puel and E. Zuazua, Approximate controllability of the semilinear heat equation, Proc. Roy. Soc. Edinburgh Sect. A, 125 (1995), 31-61.  doi: 10.1017/S0308210500030742.

[20]

E. Fernández-Cara and P. de Carvalho, Numerical Stackelberg-Nash Control for the Heat Equation, SIAM Journal on Scientific Computing, 42 (2020), 26-78. 

[21]

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

[22]

C. G. Gal and L. Tebou, Carleman inequalities for wave equations with oscillatory boundary conditions and application, SIAM J. Control. Optim., 55 (2017), 324-364.  doi: 10.1137/15M1032211.

[23] R. Glowinski and J. Lions, Exact and Approximate Controllability for Distributed Parameter Systems, Cambridge University Press, Cambridge, 2008.  doi: 10.1017/CBO9780511721595.
[24]

G. R. Goldstein, Derivation and physical interpretation of general boundary conditions, Adv. Differential. Equ., 11 (2006), 457-480. 

[25]

M. GrasselliA. Miranville and G. Schimperna, The Caginalp phase-field system with coupled dynamic boundary conditions and singular potentials, Discrete Contin. Dyn. Syst., 28 (2010), 67-98.  doi: 10.3934/dcds.2010.28.67.

[26]

F. Guillen-GonzalezF. Marques-Lopes and M. Rojas-Medar, On the approximate controllability of Stackelberg-Nash strategy for Stokes equations, Proc. Amer. Math. Soc., 141 (2013), 1759-1773.  doi: 10.1090/S0002-9939-2012-11459-5.

[27]

V. Hernández-Santamaría and L. de Teresa, Some remarks on the hierarchic control for coupled parabolic PDEs, Recent Advances in PDEs: Analysis, Numerics and Control., Springer, Cham, 17 (2018), 117-137. 

[28]

V. Hernández-Santamaría and L. de Teresa, Robust Stackelberg controllability for linear and semilinear heat equations, Evol. Equ. Control. Theory., 7 (2018), 247-273.  doi: 10.3934/eect.2018012.

[29]

V. Hernández-SantamaríaL. de Teresa and A. Poznyak, Hierarchic control for a coupled parabolic system, Port. Math., 73 (2016), 115-137.  doi: 10.4171/PM/1979.

[30]

J. Jahn, Introduction to the Theory of Nonlinear Optimization, 3$^{nd}$ edition, Springer, Berlin, 2007.

[31]

A. Khoutaibi and L. Maniar, Null controllability for a heat equation with dynamic boundary conditions and drift terms, Evol. Equ. Control. Theory., 9 (2020), 535-559.  doi: 10.3934/eect.2020023.

[32]

A. Khoutaibi, L. Maniar, D. Mugnolo and A. Rhandi, Parabolic equations with dynamic boundary conditions and drift terms, preprint, arXiv: 1909.02377.

[33]

I. LasieckaR. Triggiani and P. F. Yao, Inverse/observability estimates for second-order hyperbolic equations with variable coefficients, J. Math. Anal. Appl., 235 (1999), 13-57.  doi: 10.1006/jmaa.1999.6348.

[34]

J. LimacoH. Clark and L. Medeiros, Remarks on hierarchic control, J. Math. Anal. Appl., 359 (2009), 368-383.  doi: 10.1016/j.jmaa.2009.05.040.

[35]

J.-L. Lions, Some remarks on Stackelberg's optimization, Math. Models. Methods. Appl. Sci., 4 (1994), 477-487.  doi: 10.1142/S0218202594000273.

[36]

J.-L. Lions, Hierarchic control, Proc. Indian Acad. Sci. Math. Sci., 104 (1994), 295-304.  doi: 10.1007/BF02830893.

[37]

L. ManiarM. Meyries and R. Schnaubelt, Null controllability for parabolic equations with dynamic boundary conditions of reactive-diffusion type, Evol. Equ. Control. Theory., 6 (2017), 381-407.  doi: 10.3934/eect.2017020.

[38]

A. Miranville and S. Zelik, Exponential attractors for the Cahn-Hilliard equation with dynamic boundary conditions, Math. Methods Appl.Sci., 28 (2005), 709-735.  doi: 10.1002/mma.590.

[39]

G. MophouK. Moumini and L. Djomegne Njoukoue, Robust hierarchic control for a population dynamics model with missing birth rate, Math. Control. Signals Syst., 32 (2020), 209-239.  doi: 10.1007/s00498-020-00260-0.

[40]

D. Mugnolo and S. Romanelli, Dirichlet form for general Wentzell boundary condition, analytic semigroup and cosinus operator function, Electron. J. Differ. Eq., 18 (2006), 1-20. 

[41]

J. Nash, Non-cooperative games, Ann. Math., 54 (1951), 286-295.  doi: 10.2307/1969529.

[42]

V. Pareto, Cours d'économie Politique, Switzerland, 1896. doi: 10.3917/droz.paret.1964.01.

[43]

K. D. PhungG. Wang and X. Zhang, On the existence of time optimal controls for linear evolution equations, Discrete Contin. Dyn. Syst. Ser. B, 8 (2007), 925-941.  doi: 10.3934/dcdsb.2007.8.925.

[44]

J. P. Puel, Applications of Global Carleman Inequalities to Controllability and Inverse Problems, Textos de Metodos Matematicos de l'Instituto de Matematica de l'UFRJ, 2008.

[45]

A. M. RamosR. Glowinski and J. Periaux, Nash equilibria for the multiobjective control of linear partial differential equations, J. Optim. Theory Appl., 112 (2002), 457-498.  doi: 10.1023/A:1017981514093.

[46]

A. M. RamosR. Glowinski and J. Periaux, Pointwise control of the Burgers equation and related Nash equilibria problems: Computational approach, J. Optim. Theory Appl., 112 (2001), 499-516.  doi: 10.1023/A:1017907930931.

[47]

M. E. Taylor, Partial Differential Equations I: Basic Theory, Applied Mathematical Sciences, 2$^{nd}$ edition, Applied Mathematical Sciences, 115. Springer, New York, 2011. doi: 10.1007/978-1-4419-7055-8.

[48]

J. L. Vázquez and E. Vitillaro, Heat equation with dynamical boundary conditions of reactive type, Commun. Part. Diff. Eq., 33 (2008), 561-612.  doi: 10.1080/03605300801970960.

[49]

H. von Stackelberg, Marktform und Gleichgewicht, Springer, 1934.

[50]

G. Wang, L. Wang, Y. Xu and Y. Zhang, Time Optimal Control of Evolution Equations, Cham: Birkhaüser, 2018. doi: 10.1007/978-3-319-95363-2.

show all references

References:
[1]

E. M. Ait Ben Hassi, S. E. Chorfi, L. Maniar and O. Oukdach, Lipschitz Stability for an Inverse Source Problem in Anisotropic Parabolic Equations with Dynamic Boundary Conditions, Evol. Equ. Control. Theory., 2020.

[2]

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.

[3]

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.

[4]

F. D. Araruna, B. S. V. Araújo and E. Fernández-Cara, Stackelberg-Nash null controllability for some linear and semilinear degenerate parabolic equations, Math. Control Signals Systems, 30 (2018). doi: 10.1007/s00498-018-0220-6.

[5]

F. D. ArarunaE. Fernández-CaraS. Guerrero and M. C. Santos, New results on the Stackelberg-Nash exact control of linear parabolic equations, Syst. Control. Lett., 104 (2017), 78-85.  doi: 10.1016/j.sysconle.2017.03.009.

[6]

F. D. ArarunaE. Fernández-Cara and M. C. Santos, Stackelberg-Nash exact controllability for linear and semilinear parabolic equations, ESAIM Control Optim. Calc. Var., 21 (2015), 835-856.  doi: 10.1051/cocv/2014052.

[7]

F. D. ArarunaE. Fernández-Cara and L. C. D. Silva, Hierarchic control for the wave equation, J. Optimiz. Theory. App., 178 (2018), 264-288.  doi: 10.1007/s10957-018-1277-6.

[8]

F. D. Araruna, E. Fernández-Cara and L. C. da Silva, Hierarchical exact controllability of semilinear parabolic equations with distributed and boundary controls, Commun. Contemp. Math., 22 (2019), 1950034. doi: 10.1142/S0219199719500342.

[9]

I. BoutaayamouS. E. ChorfiL. Maniar and O. Oukdach, The cost of approximate controllability of heat equation with general dynamical boundary conditions, Port. Math., 78 (2020), 65-99.  doi: 10.4171/PM/2061.

[10]

I. BoutaayamouG. Fragnelli and L. Maniar, Carleman estimates for parabolic equations with interior degeneracy and Neumann boundary conditions, J. Anal. Math., 135 (2018), 1-35.  doi: 10.1007/s11854-018-0030-2.

[11]

F. Boyer, On the penalised HUM approach and its applications to the numerical approximation of null-controls for parabolic problems, CANUM 2012, 41e Congrés National d'Analyse Numérique, ESAIM Proc., EDP Sci., Les Ulis, 41 (2013), 15-58. doi: 10.1051/proc/201341002.

[12]

P. CannarsaG. Fragnelli and D. Rocchetti, Null controllability of degenerate parabolic operators with drift, Netw. Heterog. Media, 2 (2007), 695-715.  doi: 10.3934/nhm.2007.2.695.

[13]

P. P. Carvalho and E. Fernández-Cara, On the computation of Nash and Pareto equilibria for some bi-objective control problems, J. Sci. Comput., 78 (2019), 246-273.  doi: 10.1007/s10915-018-0764-0.

[14]

L. Cherfils and A. Miranville, On the Caginalp system with dynamic boundary conditions and singular potentials, Appl. Math., 54 (2009), 89-115.  doi: 10.1007/s10492-009-0008-6.

[15]

I. P. de JesusJ. Límaco and M. R. Clark, Hierarchical control for the one-dimensional plate equation with a moving boundary, J. Dyn. Control Syst., 24 (2018), 635-655.  doi: 10.1007/s10883-018-9413-z.

[16]

J. I. Díaz, On the von Neumann problem and the approximate controllability of Stackelberg-Nash strategy for some environmental problems, RACSAM. Rev. R. Acad. Cien. Exactas. Nat. Ser. A Mat., 96 (2002), 343-356. 

[17]

J. I. Díaz and J.-L. Lions, On the approximate controllability of Stackelberg-Nash strategies, Ocean Circulation and Pollution Control: A Mathematical and Numerical Investigation, Madrid, Springer, Berlin, (1997) 17-27.

[18]

I. Ekeland and R. Teman, Analyse Convexe et Problemes Variationnels, Dunod; Gauthier-Villars, Paris-Brussels-Montreal, Que., 1974.

[19]

C. FabreJ-P. Puel and E. Zuazua, Approximate controllability of the semilinear heat equation, Proc. Roy. Soc. Edinburgh Sect. A, 125 (1995), 31-61.  doi: 10.1017/S0308210500030742.

[20]

E. Fernández-Cara and P. de Carvalho, Numerical Stackelberg-Nash Control for the Heat Equation, SIAM Journal on Scientific Computing, 42 (2020), 26-78. 

[21]

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

[22]

C. G. Gal and L. Tebou, Carleman inequalities for wave equations with oscillatory boundary conditions and application, SIAM J. Control. Optim., 55 (2017), 324-364.  doi: 10.1137/15M1032211.

[23] R. Glowinski and J. Lions, Exact and Approximate Controllability for Distributed Parameter Systems, Cambridge University Press, Cambridge, 2008.  doi: 10.1017/CBO9780511721595.
[24]

G. R. Goldstein, Derivation and physical interpretation of general boundary conditions, Adv. Differential. Equ., 11 (2006), 457-480. 

[25]

M. GrasselliA. Miranville and G. Schimperna, The Caginalp phase-field system with coupled dynamic boundary conditions and singular potentials, Discrete Contin. Dyn. Syst., 28 (2010), 67-98.  doi: 10.3934/dcds.2010.28.67.

[26]

F. Guillen-GonzalezF. Marques-Lopes and M. Rojas-Medar, On the approximate controllability of Stackelberg-Nash strategy for Stokes equations, Proc. Amer. Math. Soc., 141 (2013), 1759-1773.  doi: 10.1090/S0002-9939-2012-11459-5.

[27]

V. Hernández-Santamaría and L. de Teresa, Some remarks on the hierarchic control for coupled parabolic PDEs, Recent Advances in PDEs: Analysis, Numerics and Control., Springer, Cham, 17 (2018), 117-137. 

[28]

V. Hernández-Santamaría and L. de Teresa, Robust Stackelberg controllability for linear and semilinear heat equations, Evol. Equ. Control. Theory., 7 (2018), 247-273.  doi: 10.3934/eect.2018012.

[29]

V. Hernández-SantamaríaL. de Teresa and A. Poznyak, Hierarchic control for a coupled parabolic system, Port. Math., 73 (2016), 115-137.  doi: 10.4171/PM/1979.

[30]

J. Jahn, Introduction to the Theory of Nonlinear Optimization, 3$^{nd}$ edition, Springer, Berlin, 2007.

[31]

A. Khoutaibi and L. Maniar, Null controllability for a heat equation with dynamic boundary conditions and drift terms, Evol. Equ. Control. Theory., 9 (2020), 535-559.  doi: 10.3934/eect.2020023.

[32]

A. Khoutaibi, L. Maniar, D. Mugnolo and A. Rhandi, Parabolic equations with dynamic boundary conditions and drift terms, preprint, arXiv: 1909.02377.

[33]

I. LasieckaR. Triggiani and P. F. Yao, Inverse/observability estimates for second-order hyperbolic equations with variable coefficients, J. Math. Anal. Appl., 235 (1999), 13-57.  doi: 10.1006/jmaa.1999.6348.

[34]

J. LimacoH. Clark and L. Medeiros, Remarks on hierarchic control, J. Math. Anal. Appl., 359 (2009), 368-383.  doi: 10.1016/j.jmaa.2009.05.040.

[35]

J.-L. Lions, Some remarks on Stackelberg's optimization, Math. Models. Methods. Appl. Sci., 4 (1994), 477-487.  doi: 10.1142/S0218202594000273.

[36]

J.-L. Lions, Hierarchic control, Proc. Indian Acad. Sci. Math. Sci., 104 (1994), 295-304.  doi: 10.1007/BF02830893.

[37]

L. ManiarM. Meyries and R. Schnaubelt, Null controllability for parabolic equations with dynamic boundary conditions of reactive-diffusion type, Evol. Equ. Control. Theory., 6 (2017), 381-407.  doi: 10.3934/eect.2017020.

[38]

A. Miranville and S. Zelik, Exponential attractors for the Cahn-Hilliard equation with dynamic boundary conditions, Math. Methods Appl.Sci., 28 (2005), 709-735.  doi: 10.1002/mma.590.

[39]

G. MophouK. Moumini and L. Djomegne Njoukoue, Robust hierarchic control for a population dynamics model with missing birth rate, Math. Control. Signals Syst., 32 (2020), 209-239.  doi: 10.1007/s00498-020-00260-0.

[40]

D. Mugnolo and S. Romanelli, Dirichlet form for general Wentzell boundary condition, analytic semigroup and cosinus operator function, Electron. J. Differ. Eq., 18 (2006), 1-20. 

[41]

J. Nash, Non-cooperative games, Ann. Math., 54 (1951), 286-295.  doi: 10.2307/1969529.

[42]

V. Pareto, Cours d'économie Politique, Switzerland, 1896. doi: 10.3917/droz.paret.1964.01.

[43]

K. D. PhungG. Wang and X. Zhang, On the existence of time optimal controls for linear evolution equations, Discrete Contin. Dyn. Syst. Ser. B, 8 (2007), 925-941.  doi: 10.3934/dcdsb.2007.8.925.

[44]

J. P. Puel, Applications of Global Carleman Inequalities to Controllability and Inverse Problems, Textos de Metodos Matematicos de l'Instituto de Matematica de l'UFRJ, 2008.

[45]

A. M. RamosR. Glowinski and J. Periaux, Nash equilibria for the multiobjective control of linear partial differential equations, J. Optim. Theory Appl., 112 (2002), 457-498.  doi: 10.1023/A:1017981514093.

[46]

A. M. RamosR. Glowinski and J. Periaux, Pointwise control of the Burgers equation and related Nash equilibria problems: Computational approach, J. Optim. Theory Appl., 112 (2001), 499-516.  doi: 10.1023/A:1017907930931.

[47]

M. E. Taylor, Partial Differential Equations I: Basic Theory, Applied Mathematical Sciences, 2$^{nd}$ edition, Applied Mathematical Sciences, 115. Springer, New York, 2011. doi: 10.1007/978-1-4419-7055-8.

[48]

J. L. Vázquez and E. Vitillaro, Heat equation with dynamical boundary conditions of reactive type, Commun. Part. Diff. Eq., 33 (2008), 561-612.  doi: 10.1080/03605300801970960.

[49]

H. von Stackelberg, Marktform und Gleichgewicht, Springer, 1934.

[50]

G. Wang, L. Wang, Y. Xu and Y. Zhang, Time Optimal Control of Evolution Equations, Cham: Birkhaüser, 2018. doi: 10.1007/978-3-319-95363-2.

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