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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 & 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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[18]

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

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

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

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

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

[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.  Google Scholar

[32]

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

[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.  Google Scholar

[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.  Google Scholar

[35]

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

[36]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[41]

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

[42]

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

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[49]

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

[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.  Google Scholar

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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[18]

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

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

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

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

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

[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.  Google Scholar

[32]

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

[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.  Google Scholar

[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.  Google Scholar

[35]

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

[36]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[41]

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

[42]

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

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[49]

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

[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.  Google Scholar

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