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January  2020, 25(1): 161-190. doi: 10.3934/dcdsb.2019177

Some remarks on the Robust Stackelberg controllability for the heat equation with controls on the boundary

1. 

Institut de Mathématiques de Toulouse, CNRS, UPS IMT, F-31062 Toulouse Cedex 9, France

2. 

Centro de Investigación en Matemáticas, UAEH, Carretera Pachuca-Tulancingo km 4.5 Pachuca, Hidalgo 42184, Mexico

Received  November 2018 Revised  February 2019 Published  January 2020 Early access  July 2019

Fund Project: The first author is supported by project IN102116 of DGAPA, UNAM (Mexico).

In this paper, we present some controllability results for the heat equation in the framework of hierarchic control. We present a Stackelberg strategy combining the concept of controllability with robustness: the main control (the leader) is in charge of a null-controllability objective while a secondary control (the follower) solves a robust control problem, this is, we look for an optimal control in the presence of the worst disturbance. We improve previous results by considering that either the leader or follower control acts on a small part of the boundary. We also present a discussion about the possibility and limitations of placing all the involved controls on the boundary.

Citation: Víctor Hernández-Santamaría, Liliana Peralta. Some remarks on the Robust Stackelberg controllability for the heat equation with controls on the boundary. Discrete and Continuous Dynamical Systems - B, 2020, 25 (1) : 161-190. doi: 10.3934/dcdsb.2019177
References:
[1]

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.

[2]

F. Ammar-KhojdaA. BenabdallahM. González-Burgos and L. de Teresa, New phenomena for the null controllability of parabolic systems: Minimal time and geometrical dependence, J. Math. Anal. Appl., 444 (2016), 1071-1113.  doi: 10.1016/j.jmaa.2016.06.058.

[3]

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), Art. 14, 31 pp. doi: 10.1007/s00498-018-0220-6.

[4]

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.

[5]

F.D. ArarunaE. Fernández-CaraS. Guerrero and M. C. Santos, New results on the Stackelberg Nash exact controllability for parabolic equations, Systems & Control Letters, 104 (2017), 78-85.  doi: 10.1016/j.sysconle.2017.03.009.

[6]

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

[7]

A. Belmiloudi, On some robust control problems for nonlinear parabolic equations, Int. J. Pure Appl. Math., 11 (2004), 119-151. 

[8]

T. R. Bewley, R. Temam and M. Ziane, A generalized framework for robust control in fluid mechanics, Center for Turbulence Research Annual Briefs, (1997), 299–316.

[9]

T. R. BewleyR. Temam and M. Ziane, A general framework for robust control in fluid mechanics, Physica D, 138 (2000), 360-392.  doi: 10.1016/S0167-2789(99)00206-7.

[10]

F. Boyer, On the penalised HUM approach and its applications to the numerical approximation of null-controls for parabolic problems, ESAIM Proceedings, 41 (2013), 15-58.  doi: 10.1051/proc/201341002.

[11]

N. Carreño and M. C. Santos, Stackelberg-Nash exact controllability for the Kuramoto-Sivashinsky equation, J. Differential Equations, 266 (2019), 6068-6108.  doi: 10.1016/j.jde.2018.10.043.

[12]

M. Duprez and P. Lissy, Indirect controllability of some linear parabolic systems of $m$ equations with $m-1$ controls involving coupling terms of zero or first order, J. Math. Pures Appl., 9 (2016), 905-934.  doi: 10.1016/j.matpur.2016.03.016.

[13]

I. Ekeland and R. Temam, Convex Analysis and Variational Problems, North-Holland, 1976.

[14]

O. Yu. Èmanuilov, Controllability of parabolic equations, (Russian)Sb. Math., 186 (1995), 879–900. doi: 10.1070/SM1995v186n06ABEH000047.

[15]

L. C. Evans, Partial differential equations, Graduate studies in Mathematics, American Mathematical Society, Providence, RI, 1991.

[16]

E. Fernández-Cara and S. Guerrero, Global Carleman inequalities for parabolic systems and applications to controllability, SIAM J. Control Optim., 45 (2006), 1399-1446.  doi: 10.1137/S0363012904439696.

[17]

E. Fernández-CaraS. GuerreroO. Yu. 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.

[18]

H. P. Geering, Optimal Control with Engineering Applications, Springer-Verlag, Berlin, 2007.

[19]

A. Fursikov and O. Yu. Imanuvilov, Controllability of Evolution Equations, Lecture Notes, Research Institute of Mathematics, Seoul National University, Korea, 1996.

[20] R. GlowinskiJ.-L. Lions and J. He, Exact and Approximate Controllability for Distributed Parameter Systems, Encyclopedia of Mathematics and its Applications, vol. 117, Cambridge University Press, Cambridge, 2008.  doi: 10.1017/CBO9780511721595.
[21]

M. González-Burgos and L. de Teresa, Controllability results for cascade systems of $m$ coupled parabolic PDEs by one control force, Port. Math., 67 (2010), 91-113.  doi: 10.4171/PM/1859.

[22]

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

[23]

V. Hernández-SantamaríaL. de Teresa and A. Poznyak, Corrigendum and addendum to "Hierarchic control for a coupled parabolic system", Portugaliae Math., 73 (2016), 115–137., Port. Math., 74 (2017), 161-168.  doi: 10.4171/PM/1998.

[24]

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.

[25]

V. Hernández-Santamaría and L. de Teresa, Some Remarks on the Hierarchic Control for Coupled Parabolic PDEs, In: Recent Advances in PDEs: Analysis, Numerics and Control In Honor of Prof. Fernández-Cara's 60th Birthday, SEMA-SIMAI Springer Series, 17 (2018), 117–137.

[26]

O. Yu. ImanuvilovJ. P. Puel and M. Yamamoto, Carleman estimates for parabolic equations with nonhomogeneous boundary conditions, Chin. Ann. Math. Ser. B, 30 (2009), 333-378.  doi: 10.1007/s11401-008-0280-x.

[27]

O. Yu. Imanuvilov and M. Yamamoto, Carleman inequalities for parabolic equations in Sobolev spaces of negative order and exact controllability for semilinear parabolic equations, Publ. RIMS, Kyoto Univ., 39 (2003), 227-274.  doi: 10.2977/prims/1145476103.

[28]

I. P. de JesusJ. Limaco 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.

[29]

J.-L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer-Verlag, New York-Berlin, 1971.

[30]

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

[31]

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

[32]

J.-L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications, Vol. II. Springer-Verlag, New York-Heidelberg, 1972.

[33]

C. Montoya and L. de Teresa, Robust Stackelberg controllability for the Navier-Stokes equations, NoDEA Nonlinear Differential Equations Appl., 25 (2018), Art. 46, 33 pp. doi: 10.1007/s00030-018-0537-3.

[34]

D. Pighin and E. Zuazua, Controllability under positivity constraints of semilinear heat equations, Math. Control. Relat. F., 8 (2018), 935-964.  doi: 10.3934/mcrf.2018041.

[35]

J. Simon, Compact sets in the space Lp(0, T;B), Ann. Mat. Pura Appl., 146 (1987), 65-96.  doi: 10.1007/BF01762360.

[36]

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

[37]

L. de Teresa, Insensitizing controls for a semilinear heat equation, Comm. Partial Differential Equations, 25 (2000), 39-72.  doi: 10.1080/03605300008821507.

[38]

E. Trélat, Optimal control and applications to aerospace: Some results and challenges, J. Optim. Theory Appl., 154 (2012), 713-758.  doi: 10.1007/s10957-012-0050-5.

[39]

F. Tröltzsch, Optimal Control of Partial Differential Equations. Theory, Methods and Applications, American Mathematical Society, Providence, RI, 2010. doi: 10.1090/gsm/112.

[40]

J. Zabczyk, Mathematical Control Theory: An Introduction, Systems & control: Foundations & applications, Birkhäuser, Boston, 1992.

show all references

References:
[1]

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.

[2]

F. Ammar-KhojdaA. BenabdallahM. González-Burgos and L. de Teresa, New phenomena for the null controllability of parabolic systems: Minimal time and geometrical dependence, J. Math. Anal. Appl., 444 (2016), 1071-1113.  doi: 10.1016/j.jmaa.2016.06.058.

[3]

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), Art. 14, 31 pp. doi: 10.1007/s00498-018-0220-6.

[4]

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.

[5]

F.D. ArarunaE. Fernández-CaraS. Guerrero and M. C. Santos, New results on the Stackelberg Nash exact controllability for parabolic equations, Systems & Control Letters, 104 (2017), 78-85.  doi: 10.1016/j.sysconle.2017.03.009.

[6]

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

[7]

A. Belmiloudi, On some robust control problems for nonlinear parabolic equations, Int. J. Pure Appl. Math., 11 (2004), 119-151. 

[8]

T. R. Bewley, R. Temam and M. Ziane, A generalized framework for robust control in fluid mechanics, Center for Turbulence Research Annual Briefs, (1997), 299–316.

[9]

T. R. BewleyR. Temam and M. Ziane, A general framework for robust control in fluid mechanics, Physica D, 138 (2000), 360-392.  doi: 10.1016/S0167-2789(99)00206-7.

[10]

F. Boyer, On the penalised HUM approach and its applications to the numerical approximation of null-controls for parabolic problems, ESAIM Proceedings, 41 (2013), 15-58.  doi: 10.1051/proc/201341002.

[11]

N. Carreño and M. C. Santos, Stackelberg-Nash exact controllability for the Kuramoto-Sivashinsky equation, J. Differential Equations, 266 (2019), 6068-6108.  doi: 10.1016/j.jde.2018.10.043.

[12]

M. Duprez and P. Lissy, Indirect controllability of some linear parabolic systems of $m$ equations with $m-1$ controls involving coupling terms of zero or first order, J. Math. Pures Appl., 9 (2016), 905-934.  doi: 10.1016/j.matpur.2016.03.016.

[13]

I. Ekeland and R. Temam, Convex Analysis and Variational Problems, North-Holland, 1976.

[14]

O. Yu. Èmanuilov, Controllability of parabolic equations, (Russian)Sb. Math., 186 (1995), 879–900. doi: 10.1070/SM1995v186n06ABEH000047.

[15]

L. C. Evans, Partial differential equations, Graduate studies in Mathematics, American Mathematical Society, Providence, RI, 1991.

[16]

E. Fernández-Cara and S. Guerrero, Global Carleman inequalities for parabolic systems and applications to controllability, SIAM J. Control Optim., 45 (2006), 1399-1446.  doi: 10.1137/S0363012904439696.

[17]

E. Fernández-CaraS. GuerreroO. Yu. 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.

[18]

H. P. Geering, Optimal Control with Engineering Applications, Springer-Verlag, Berlin, 2007.

[19]

A. Fursikov and O. Yu. Imanuvilov, Controllability of Evolution Equations, Lecture Notes, Research Institute of Mathematics, Seoul National University, Korea, 1996.

[20] R. GlowinskiJ.-L. Lions and J. He, Exact and Approximate Controllability for Distributed Parameter Systems, Encyclopedia of Mathematics and its Applications, vol. 117, Cambridge University Press, Cambridge, 2008.  doi: 10.1017/CBO9780511721595.
[21]

M. González-Burgos and L. de Teresa, Controllability results for cascade systems of $m$ coupled parabolic PDEs by one control force, Port. Math., 67 (2010), 91-113.  doi: 10.4171/PM/1859.

[22]

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

[23]

V. Hernández-SantamaríaL. de Teresa and A. Poznyak, Corrigendum and addendum to "Hierarchic control for a coupled parabolic system", Portugaliae Math., 73 (2016), 115–137., Port. Math., 74 (2017), 161-168.  doi: 10.4171/PM/1998.

[24]

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.

[25]

V. Hernández-Santamaría and L. de Teresa, Some Remarks on the Hierarchic Control for Coupled Parabolic PDEs, In: Recent Advances in PDEs: Analysis, Numerics and Control In Honor of Prof. Fernández-Cara's 60th Birthday, SEMA-SIMAI Springer Series, 17 (2018), 117–137.

[26]

O. Yu. ImanuvilovJ. P. Puel and M. Yamamoto, Carleman estimates for parabolic equations with nonhomogeneous boundary conditions, Chin. Ann. Math. Ser. B, 30 (2009), 333-378.  doi: 10.1007/s11401-008-0280-x.

[27]

O. Yu. Imanuvilov and M. Yamamoto, Carleman inequalities for parabolic equations in Sobolev spaces of negative order and exact controllability for semilinear parabolic equations, Publ. RIMS, Kyoto Univ., 39 (2003), 227-274.  doi: 10.2977/prims/1145476103.

[28]

I. P. de JesusJ. Limaco 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.

[29]

J.-L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer-Verlag, New York-Berlin, 1971.

[30]

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

[31]

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

[32]

J.-L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications, Vol. II. Springer-Verlag, New York-Heidelberg, 1972.

[33]

C. Montoya and L. de Teresa, Robust Stackelberg controllability for the Navier-Stokes equations, NoDEA Nonlinear Differential Equations Appl., 25 (2018), Art. 46, 33 pp. doi: 10.1007/s00030-018-0537-3.

[34]

D. Pighin and E. Zuazua, Controllability under positivity constraints of semilinear heat equations, Math. Control. Relat. F., 8 (2018), 935-964.  doi: 10.3934/mcrf.2018041.

[35]

J. Simon, Compact sets in the space Lp(0, T;B), Ann. Mat. Pura Appl., 146 (1987), 65-96.  doi: 10.1007/BF01762360.

[36]

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

[37]

L. de Teresa, Insensitizing controls for a semilinear heat equation, Comm. Partial Differential Equations, 25 (2000), 39-72.  doi: 10.1080/03605300008821507.

[38]

E. Trélat, Optimal control and applications to aerospace: Some results and challenges, J. Optim. Theory Appl., 154 (2012), 713-758.  doi: 10.1007/s10957-012-0050-5.

[39]

F. Tröltzsch, Optimal Control of Partial Differential Equations. Theory, Methods and Applications, American Mathematical Society, Providence, RI, 2010. doi: 10.1090/gsm/112.

[40]

J. Zabczyk, Mathematical Control Theory: An Introduction, Systems & control: Foundations & applications, Birkhäuser, Boston, 1992.

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