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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 & 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. Google Scholar

[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. Google Scholar

[3]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[8]

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

[9]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[15]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[22]

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

[23]

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

[24]

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

[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. Google Scholar

[26]

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

[27]

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

[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. Google Scholar

[29]

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

[30]

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

[31]

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

[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. Google Scholar

show all references

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. Google Scholar

[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. Google Scholar

[3]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[8]

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

[9]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[15]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[22]

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

[23]

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

[24]

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

[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. Google Scholar

[26]

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

[27]

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

[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. Google Scholar

[29]

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

[30]

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

[31]

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

[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. Google Scholar

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