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June  2018, 7(2): 247-273. doi: 10.3934/eect.2018012

Robust Stackelberg controllability for linear and semilinear heat equations

1. 

Departamento de Control Automático, CINVESTAV-IPN, Apartado Postal 14-740, 0700, México, D.F., México

2. 

Instituto de Matemáticas, Universidad Nacional Autónoma de México, Circuito Exterior, C.U., 04510 D.F., México

The first author was supported by CONACyT (Mexico) and both authors were supported by project IN102116 of DGAPA, UNAM. (Mexico).

Received  May 2017 Revised  December 2017 Published  May 2018

In this paper, we present a Stackelberg strategy to control a semilinear parabolic equation. We use the concept of hierarchic control to combine the concepts of controllability with robustness. We have a control named the leader which is responsible for a controllability to trajectories objective. Additionally, we have a control named the follower, that solves a robust control problem. That means we solve for the optimal control in the presence of the worst disturbance case. In this way, the follower control is insensitive to a broad class of external disturbances.

Citation: Víctor Hernández-Santamaría, Luz de Teresa. Robust Stackelberg controllability for linear and semilinear heat equations. Evolution Equations and Control Theory, 2018, 7 (2) : 247-273. doi: 10.3934/eect.2018012
References:
[1]

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.

[2]

F. D. ArarunaS. D. B. de Menezes and M. A. Rojas-Medar, On the approximate controllability of Stackelberg-Nash strategies for linearized microplar fluids, Appl. Math. Optim., 70 (2014), 373-393.  doi: 10.1007/s00245-014-9240-x.

[3]

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

[4]

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

[5]

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

[6]

O. BodartM. González-Burgos and R. Pérez-García, Insensitizing controls for a heat equation with a nonlinear term involving the state and the gradient, Nonlinear Anal., 57 (2004), 687-711.  doi: 10.1016/j.na.2004.03.012.

[7]

J. I. Díaz, On the Von Neumann problem and the approximate controllability of Stackelberg-Nash strategies for some environmental problems, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM, 96 (2002), 343-356. 

[8]

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

[9]

L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI, 1998.

[10]

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.

[11]

L. A. Fernández and E. Zuazua, Approximate controllability for the semilinear heat equation involving gradient terms, J. Optim. Theor. Appl., 101 (1999), 307-328.  doi: 10.1023/A:1021737526541.

[12]

E. Fernández-Cara and S. Guerrero, Global Carleman inequalities for parabolic systems and applications to controllability, SIAM J. Control Optim., 45 (2006), 1395-1446.  doi: 10.1137/S0363012904439696.

[13]

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.

[14]

E. Fernández-Cara and E. Zuazua, Null and approximate controllability for weakly blowing up semilinear heat equations, Ann. I. H. Poincaré-AN, 17 (2000), 583-616.  doi: 10.1016/S0294-1449(00)00117-7.

[15]

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

[16]

R. GlowinskiA. Ramos 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.

[17]

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.

[18]

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.

[19]

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. Res. Inst. Math. Sci., 39 (2003), 227-274.  doi: 10.2977/prims/1145476103.

[20]

O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasi-Linear Equations of Parabolic Type, Translated from the Russian by S. Smith. Translations of Mathematical Monographs, Vol. 23 American Mathematical Society, Providence, R. I., 1968.

[21]

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.

[22]

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

[23]

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

[24]

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

[25]

C. McMillan and R. Triggiani, Min-max game theory and algebraic Riccati equations for boundary control problems with continuous input-solution map. II. The general case, Appl. Math. Optim., 29 (1994), 1-65.  doi: 10.1007/BF01191106.

[26]

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

[27]

V. Pareto, Cours d'économie politique, Travaux de Sciences Sociales, (1964), p424.  doi: 10.3917/droz.paret.1964.01.

[28]

T. Seidman and H. Z. Zhou, Existence and uniqueness of optimal controls for a quasilinear parabolic equation, SIAM J. Control Optim., 20 (1982), 747-762.  doi: 10.1137/0320054.

[29]

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

[30]

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

[31]

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

[32]

E. Zuazua, Exact boundary controllability for the semilinear wave equation, Nonlinear Partial Differential Equations and Their Applications, Vol. X (Paris 1987–1988), 357–391, Pitman Res. Notes Math. Ser., 220, Longman Sci. Tech., Harlow, 1991.

show all references

The first author was supported by CONACyT (Mexico) and both authors were supported by project IN102116 of DGAPA, UNAM. (Mexico).

References:
[1]

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.

[2]

F. D. ArarunaS. D. B. de Menezes and M. A. Rojas-Medar, On the approximate controllability of Stackelberg-Nash strategies for linearized microplar fluids, Appl. Math. Optim., 70 (2014), 373-393.  doi: 10.1007/s00245-014-9240-x.

[3]

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

[4]

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

[5]

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

[6]

O. BodartM. González-Burgos and R. Pérez-García, Insensitizing controls for a heat equation with a nonlinear term involving the state and the gradient, Nonlinear Anal., 57 (2004), 687-711.  doi: 10.1016/j.na.2004.03.012.

[7]

J. I. Díaz, On the Von Neumann problem and the approximate controllability of Stackelberg-Nash strategies for some environmental problems, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM, 96 (2002), 343-356. 

[8]

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

[9]

L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI, 1998.

[10]

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.

[11]

L. A. Fernández and E. Zuazua, Approximate controllability for the semilinear heat equation involving gradient terms, J. Optim. Theor. Appl., 101 (1999), 307-328.  doi: 10.1023/A:1021737526541.

[12]

E. Fernández-Cara and S. Guerrero, Global Carleman inequalities for parabolic systems and applications to controllability, SIAM J. Control Optim., 45 (2006), 1395-1446.  doi: 10.1137/S0363012904439696.

[13]

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.

[14]

E. Fernández-Cara and E. Zuazua, Null and approximate controllability for weakly blowing up semilinear heat equations, Ann. I. H. Poincaré-AN, 17 (2000), 583-616.  doi: 10.1016/S0294-1449(00)00117-7.

[15]

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

[16]

R. GlowinskiA. Ramos 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.

[17]

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.

[18]

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.

[19]

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. Res. Inst. Math. Sci., 39 (2003), 227-274.  doi: 10.2977/prims/1145476103.

[20]

O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasi-Linear Equations of Parabolic Type, Translated from the Russian by S. Smith. Translations of Mathematical Monographs, Vol. 23 American Mathematical Society, Providence, R. I., 1968.

[21]

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.

[22]

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

[23]

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

[24]

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

[25]

C. McMillan and R. Triggiani, Min-max game theory and algebraic Riccati equations for boundary control problems with continuous input-solution map. II. The general case, Appl. Math. Optim., 29 (1994), 1-65.  doi: 10.1007/BF01191106.

[26]

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

[27]

V. Pareto, Cours d'économie politique, Travaux de Sciences Sociales, (1964), p424.  doi: 10.3917/droz.paret.1964.01.

[28]

T. Seidman and H. Z. Zhou, Existence and uniqueness of optimal controls for a quasilinear parabolic equation, SIAM J. Control Optim., 20 (1982), 747-762.  doi: 10.1137/0320054.

[29]

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

[30]

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

[31]

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

[32]

E. Zuazua, Exact boundary controllability for the semilinear wave equation, Nonlinear Partial Differential Equations and Their Applications, Vol. X (Paris 1987–1988), 357–391, Pitman Res. Notes Math. Ser., 220, Longman Sci. Tech., Harlow, 1991.

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