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Journal of Scientific Computing

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15-06-2018

On the Computation of Nash and Pareto Equilibria for Some Bi-objective Control Problems

Authors: Pitágoras P. Carvalho, Enrique Fernández-Cara

Published in: Journal of Scientific Computing | Issue 1/2019

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Abstract

This article is concerned with the numerical solution of some multi-objective optimal control problems for systems governed by linear and semilinear parabolic equations. More precisely, for such problems, we look for Nash and Pareto equilibria, which respectively correspond to appropriate noncooperative and cooperative strategies. First, we study the linear case and then some semilinear problems. In order to compute the solutions, we combine finite difference methods for the time discretization, finite element methods for the space discretization and fixed-point algorithms for the iterative solution of the discrete control problems. We also illustrate these techniques with several numerical experiments.

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Araruna, F.D. , Fernández-Cara, E. ,Santos, M.C.: Stackelberg–Nash exact controllability for linear and semilinear parabolic equations. In: ESAIM: COCV, vol. 21, pp. 835–856 (2015)

Araruna, F.D., Fernández-Cara, E., Guerrero, S., Santos, M.C.: New results on the Stackelberg–Nash exact control of linear parabolic equations. Syst. Control Lett. 104, 78–85 (2017)MathSciNetCrossRefMATH

Brézis, H.: Functional Analysis. Sobolev Spaces and Partial Differential Equations. Springer, New York (2011)MATH

Díaz, J.I.: 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(3), 343–356 (2002)MathSciNetMATH

Díaz, J.I., Lions, J.-L.: On the approximate controllability of Stackelberg–Nash strategies. In: Díaz, J.I. (ed.) Ocean Circulation and Pollution Control - A Mathematical and Numerical Investigation, pp. 17–27. Springer-Verlag, Berlin (2004)CrossRef

Guillén, F.: On the approximate controllability of Stackelberg–Nash strategies for Stokes equations. Proc. Am. Math. Soc. 141(5), 1759–1773 (2013)MathSciNetCrossRefMATH

Lions, J.-L.: Contrôle de Pareto de systèmes distribués. Le cas d’évolution. C. R. Acad. Sci. Paris, Série I 302(11), 413–417 (1986)

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

Nash, J.F.: Noncooperative games. Ann. Math. 54, 286–295 (1951)MathSciNetCrossRefMATH

10.

Pareto, V.: Cours d’conomie politique. Rouge, Laussane (1896)

11.

Ramos, A.M., Glowinski, R., Periaux, J.: Nash equilibria for the multi-objective control of linear partial partial differential equations. J. Optim. Theory Appl. 112, 457–498 (2002)MathSciNetCrossRefMATH

12.

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

Title: On the Computation of Nash and Pareto Equilibria for Some Bi-objective Control Problems
Authors: Pitágoras P. Carvalho
Enrique Fernández-Cara
Publication date: 15-06-2018
Publisher: Springer US
Published in: Journal of Scientific Computing / Issue 1/2019
Print ISSN: 0885-7474
Electronic ISSN: 1573-7691
DOI: https://doi.org/10.1007/s10915-018-0764-0

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