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

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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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Acknowledgements

This paper was written during a stay at the Institute of Mathematics of the University of Sevilla (IMUS). The authors are indebted to this Institute for its assistance.

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Authors and Affiliations

  1. Inst. Matemática, Universidade Federal Fluminense, Rio de Janeiro, RJ, Brazil

    Pitágoras P. Carvalho

  2. Dpto. Matemática, Universidade Estadual do Piauí, Teresina, PI, Brazil

    Pitágoras P. Carvalho

  3. Dpto. E.D.A.N., Universidad de Sevilla, Aptdo. 1160, 41080, Seville, Spain

    Enrique Fernández-Cara

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  1. Pitágoras P. Carvalho
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  2. Enrique Fernández-Cara
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Correspondence to Pitágoras P. Carvalho.

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Carvalho, P.P., Fernández-Cara, E. On the Computation of Nash and Pareto Equilibria for Some Bi-objective Control Problems. J Sci Comput 78, 246–273 (2019). https://doi.org/10.1007/s10915-018-0764-0

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  • DOI: https://doi.org/10.1007/s10915-018-0764-0

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