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Pointwise Control of the Burgers Equation and Related Nash Equilibrium Problems: Computational Approach

Author

Listed:
  • A.M. Ramos

    (Universidad Complutense de Madrid)

  • R. Glowinski

    (Université Pierre et Marie Curie
    University of Houston)

  • J. Periaux

    (Dassault Aviation)

Abstract

This article is concerned with the numerical solution of multiobjective control problems associated with nonlinear partial differential equations and more precisely the Burgers equation. For this kind of problems, we look for the Nash equilibrium, which is the solution to a noncooperative game. To compute the solution of the problem, we use a combination of finite-difference methods for the time discretization, finite-element methods for the space discretization, and a quasi-Newton BFGS algorithm for the iterative solution of the discrete control problem. Finally, we apply the above methodology to the solution of several tests problems. To be able to compare our results with existing results in the literature, we discuss first a single-objective control problem, already investigated by other authors. Finally, we discuss the multiobjective case.

Suggested Citation

  • A.M. Ramos & R. Glowinski & J. Periaux, 2002. "Pointwise Control of the Burgers Equation and Related Nash Equilibrium Problems: Computational Approach," Journal of Optimization Theory and Applications, Springer, vol. 112(3), pages 499-516, March.
    • Handle: RePEc:spr:joptap:v:112:y:2002:i:3:d:10.1023_a:1017907930931
    DOI: 10.1023/A:1017907930931
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    References listed on IDEAS

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    1. A.M. Ramos & R. Glowinski & J. Periaux, 2002. "Nash Equilibria for the Multiobjective Control of Linear Partial Differential Equations," Journal of Optimization Theory and Applications, Springer, vol. 112(3), pages 457-498, March.
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    Cited by:

    1. B. Ivorra & A. M. Ramos & B. Mohammadi, 2007. "Semideterministic Global Optimization Method: Application to a Control Problem of the Burgers Equation," Journal of Optimization Theory and Applications, Springer, vol. 135(3), pages 549-561, December.
    2. Isaías Pereira Jesus, 2016. "Hierarchical Control for the Wave Equation with a Moving Boundary," Journal of Optimization Theory and Applications, Springer, vol. 171(1), pages 336-350, October.
    3. A.M. Ramos & R. Glowinski & J. Periaux, 2002. "Nash Equilibria for the Multiobjective Control of Linear Partial Differential Equations," Journal of Optimization Theory and Applications, Springer, vol. 112(3), pages 457-498, March.
    4. T. Roubíček, 2007. "On Nash Equilibria for Noncooperative Games Governed by the Burgers Equation," Journal of Optimization Theory and Applications, Springer, vol. 132(1), pages 41-50, January.
    5. A. M. Croicu & M. Y. Hussaini, 2008. "Multiobjective Stochastic Control in Fluid Dynamics via Game Theory Approach: Application to the Periodic Burgers Equation," Journal of Optimization Theory and Applications, Springer, vol. 139(3), pages 501-514, December.

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