Stackelberg–Nash exact controllability for linear and semilinear parabolic equations
ESAIM: Control, Optimisation and Calculus of Variations, Volume 21 (2015) no. 3, pp. 835-856.

This paper deals with the application of Stackelberg–Nash strategies to the control of parabolic equations. We assume that we can act on the system through a hierarchy of controls. A first control (the leader) is assumed to choose the policy. Then, a Nash equilibrium pair (corresponding to a noncooperative multiple-objective optimization strategy) is found; this governs the action of the other controls (the followers). The main novelty in this paper is that, this way, we can obtain the exact controllability to a prescribed (but arbitrary) trajectory. We study linear and semilinear problems and, also, problems with pointwise constraints on the followers.

Received:
DOI: 10.1051/cocv/2014052
Classification: 34K35, 49J20, 35K10
Keywords: Controllability, Stackelberg–Nash strategies, Carleman inequalities
Araruna, F.D. 1; Fernández-Cara, E. 2; Santos, M.C. 1, 3

1 Dpto. de Matemática, Universidade Federal da Paraíba, 58051-900 João Pessoa – PB, Brasil
2 Dpto. EDAN and IMUS, University of Sevilla, Aptdo. 1160, 41080 Sevilla, Spain
3 Dpto. de Matemática, Universidade Federal de Pernambuco, 50740-540 Recife-PE, Brasil
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     title = {Stackelberg{\textendash}Nash exact controllability for linear and semilinear parabolic equations},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
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Araruna, F.D.; Fernández-Cara, E.; Santos, M.C. Stackelberg–Nash exact controllability for linear and semilinear parabolic equations. ESAIM: Control, Optimisation and Calculus of Variations, Volume 21 (2015) no. 3, pp. 835-856. doi : 10.1051/cocv/2014052. http://archive.numdam.org/articles/10.1051/cocv/2014052/

H. Brézis, Analyse Fonctionnelle, Théorie et Applications. Dunod, Paris (1999). | MR | Zbl

J.C. Cox and M. Rubinstein, Options Markets. Prentice-Hall. Englewood Cliffs, NJ (1985).

J.I. Díaz, On the von Neumann problem and the approximate controllability of Stackelberg–Nash strategies for some environmental problems. Rev. R. Acad. Cien., Ser. A. Math. 96 (2002) 343–356. | MR | Zbl

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, 1997. Springer, Berlin (2004) 17–27. | MR

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 | MR | Zbl

E. Fernández-Cara, S. Guerrero, O.Y. Imanuvilov and J.-P. Puel, Local exact controllability of the Navier–Stokes system. J. Math. Pures Appl. 83 (2004) 1501–1542. | DOI | MR | Zbl

A.V. Fursikov and O.Y. Imanuvilov, Controllability of evolution equations. Vol. 34 of Lecture Note Series. Research Institute of Mathematics, Seoul National University, Seoul (1996). | MR | Zbl

A.V. Fursikov and O.Y. Imanuvilov, Exact controllability of the Navier–Stokes and Boussinesq equations. Russian Math. Surveys 54 (1999) 565–618. | DOI | MR | Zbl

M. González-Burgos, S. Guerrero and J.-P. Puel, Local exact controllability to the trajectories of the Boussinesq system via a fictitious control on the divergence equation. Commun. Pure Appl. Anal. 8 (2009) 311–333. | DOI | MR | Zbl

S. Guerrero, O.Y. Imanuvilov and J.-P. Puel, A result concerning the global approximate controllability of the Navier–Stokes system in dimension 3. J. Math. Pures Appl. 98 (2012) 689–709. | DOI | MR | Zbl

F. Guillén-González, F.P. Marques-Lopes and M.A. Rojas-Medar, On the approximate controllability of Stackelberg–Nash strategies for Stokes equations. Proc. Amer. Math. Soc. 141 (2013) 1759–1773. | DOI | MR | Zbl

O.Y. Imanuvilov, Remarks on exact controllability for the Navier–Stokes equations. ESAIM Control Optim. Calc. Var. 6 (2001) 39–72. | DOI | Numdam | MR | Zbl

O.Y. Imanuvilov and M. Yamamoto, Carleman Estimate for a Parabolic Equation in a Sobolev Space of Negative Order and its Applications, Vol. 218 of Lect. Notes Pure Appl. Math. Dekker, New York (2001). | MR | Zbl

J.-L. Lions, Contrôle de Pareto de systèmes distribués. Le cas d’évolution. C.R. Acad. Sci. Paris, Sér. I 302 (1986) 413–417. | MR | Zbl

J.-L. Lions, Some remarks on Stackelberg’s optimization. Math. Models Methods Appl. Sci. 4 (1994) 477–487. | DOI | MR | Zbl

J.F. Nash, Noncooperative games. Ann. Math. 54 (1951) 286–295. | DOI | MR | Zbl

V. Pareto, Cours d’économie politique. Rouge, Laussane, Switzerland (1896).

A.M. Ramos, R. Glowinski and J. Periaux, Pointwise control of the Burgers equation and related Nash equilibria problems: A computational approach. J. Optim. Theory Appl. 112 (2001) 499–516. | DOI | MR | Zbl

A.M. Ramos, R. Glowinski and J. Periaux, Nash equilibria for the multiobjective control of linear partial differential equations. J. Optim. Theory Appl. 112 (2002) 457–498. | DOI | MR | Zbl

S.M. Ross, An introduction to mathematical finance. Options and other topics. Cambridge University Press, Cambridge (1999). | MR | Zbl

H. Von Stalckelberg, Marktform und gleichgewicht. Springer, Berlin, Germany (1934).

P. Wilmott, S. Howison and J. Dewynne, The mathematics of financial derivatives. Cambridge University Press, New York (1995). | MR | Zbl

E. Zuazua, Exact controllability for the semilinear wave equation, J. Math. Pures Appl. 69 (1990) 1–31. | MR | Zbl

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