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.
DOI : 10.1051/cocv/2014052
Keywords: Controllability, Stackelberg–Nash strategies, Carleman inequalities
Araruna, F.D. 1 ; Fernández-Cara, E. 2 ; Santos, M.C. 1, 3
@article{COCV_2015__21_3_835_0,
author = {Araruna, F.D. and Fern\'andez-Cara, E. and Santos, M.C.},
title = {Stackelberg{\textendash}Nash exact controllability for linear and semilinear parabolic equations},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {835--856},
year = {2015},
publisher = {EDP Sciences},
volume = {21},
number = {3},
doi = {10.1051/cocv/2014052},
mrnumber = {3358632},
zbl = {1319.35280},
language = {en},
url = {https://www.numdam.org/articles/10.1051/cocv/2014052/}
}
TY - JOUR AU - Araruna, F.D. AU - Fernández-Cara, E. AU - Santos, M.C. TI - Stackelberg–Nash exact controllability for linear and semilinear parabolic equations JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2015 SP - 835 EP - 856 VL - 21 IS - 3 PB - EDP Sciences UR - https://www.numdam.org/articles/10.1051/cocv/2014052/ DO - 10.1051/cocv/2014052 LA - en ID - COCV_2015__21_3_835_0 ER -
%0 Journal Article %A Araruna, F.D. %A Fernández-Cara, E. %A Santos, M.C. %T Stackelberg–Nash exact controllability for linear and semilinear parabolic equations %J ESAIM: Control, Optimisation and Calculus of Variations %D 2015 %P 835-856 %V 21 %N 3 %I EDP Sciences %U https://www.numdam.org/articles/10.1051/cocv/2014052/ %R 10.1051/cocv/2014052 %G en %F COCV_2015__21_3_835_0
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, Tome 21 (2015) no. 3, pp. 835-856. doi: 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).
, 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
and , Global Carleman inequalities for parabolic systems and applications to controllability. SIAM J. Control Optim. 45 (2006) 1395–1446. | MR | Zbl | DOI
, , and , Local exact controllability of the Navier–Stokes system. J. Math. Pures Appl. 83 (2004) 1501–1542. | MR | Zbl | DOI
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
and , Exact controllability of the Navier–Stokes and Boussinesq equations. Russian Math. Surveys 54 (1999) 565–618. | MR | Zbl | DOI
, and , 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. | MR | Zbl | DOI
, and , A result concerning the global approximate controllability of the Navier–Stokes system in dimension 3. J. Math. Pures Appl. 98 (2012) 689–709. | MR | Zbl | DOI
, and , On the approximate controllability of Stackelberg–Nash strategies for Stokes equations. Proc. Amer. Math. Soc. 141 (2013) 1759–1773. | MR | Zbl | DOI
, Remarks on exact controllability for the Navier–Stokes equations. ESAIM Control Optim. Calc. Var. 6 (2001) 39–72. | MR | Zbl | Numdam | DOI
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
, Contrôle de Pareto de systèmes distribués. Le cas d’évolution. C.R. Acad. Sci. Paris, Sér. I 302 (1986) 413–417. | MR | Zbl
, Some remarks on Stackelberg’s optimization. Math. Models Methods Appl. Sci. 4 (1994) 477–487. | MR | Zbl | DOI
, Noncooperative games. Ann. Math. 54 (1951) 286–295. | MR | Zbl | DOI
V. Pareto, Cours d’économie politique. Rouge, Laussane, Switzerland (1896).
, and , Pointwise control of the Burgers equation and related Nash equilibria problems: A computational approach. J. Optim. Theory Appl. 112 (2001) 499–516. | MR | Zbl | DOI
, and , Nash equilibria for the multiobjective control of linear partial differential equations. J. Optim. Theory Appl. 112 (2002) 457–498. | MR | Zbl | DOI
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
, Exact controllability for the semilinear wave equation, J. Math. Pures Appl. 69 (1990) 1–31. | MR | Zbl
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