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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

Víctor Hernández-Santamaría 1, and  Liliana Peralta 2,

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  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.

Keywords: Hierarchic control, robust control, Carleman estimates, boundary controllability, heat equation.
Mathematics Subject Classification: Primary: 49J20, 93B05, 49K35.
Citation: Víctor Hernández-Santamaría, Liliana Peralta. Some remarks on the Robust Stackelberg controllability for the heat equation with controls on the boundary. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 161-190. doi: 10.3934/dcdsb.2019177
References:
[1]

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.  Google Scholar

[2]

F. Ammar-Khojda, A. Benabdallah, M. 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.  Google Scholar

[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.  Google Scholar

[4]

F. D. Araruna, E. 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

[5]

F.D. Araruna, E. Fernández-Cara, S. 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.  Google Scholar

[6]

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., (2019).  doi: 10.1142/S0219199719500342.  Google Scholar

[7]

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

[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. Google Scholar

[9]

T. R. Bewley, R. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

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

[14]

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

[15]

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

[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.  Google Scholar

[17]

E. Fernández-Cara, S. Guerrero, O. 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.  Google Scholar

[18]

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

[19]

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

[20] R. Glowinski, J.-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.  Google Scholar
[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.  Google Scholar

[22]

F. Guillén-González, F. 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.  Google Scholar

[23]

V. Hernández-Santamaría, L. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

O. Yu. Imanuvilov, J. 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.  Google Scholar

[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.  Google Scholar

[28]

I. P. de Jesus, J. 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.  Google Scholar

[29]

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

[30]

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

[31]

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

[32]

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

[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.  Google Scholar

[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.  Google Scholar

[35]

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

[36]

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

[37]

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

[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.  Google Scholar

[39]

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

[40]

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

show all references

References:
[1]

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.  Google Scholar

[2]

F. Ammar-Khojda, A. Benabdallah, M. 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.  Google Scholar

[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.  Google Scholar

[4]

F. D. Araruna, E. 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

[5]

F.D. Araruna, E. Fernández-Cara, S. 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.  Google Scholar

[6]

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., (2019).  doi: 10.1142/S0219199719500342.  Google Scholar

[7]

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

[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. Google Scholar

[9]

T. R. Bewley, R. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

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

[14]

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

[15]

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

[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.  Google Scholar

[17]

E. Fernández-Cara, S. Guerrero, O. 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.  Google Scholar

[18]

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

[19]

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

[20] R. Glowinski, J.-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.  Google Scholar
[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.  Google Scholar

[22]

F. Guillén-González, F. 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.  Google Scholar

[23]

V. Hernández-Santamaría, L. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

O. Yu. Imanuvilov, J. 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.  Google Scholar

[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.  Google Scholar

[28]

I. P. de Jesus, J. 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.  Google Scholar

[29]

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

[30]

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

[31]

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

[32]

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

[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.  Google Scholar

[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.  Google Scholar

[35]

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

[36]

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

[37]

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

[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.  Google Scholar

[39]

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

[40]

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

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