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Pointwise control of the Burgers equation and related Nash equilibrium problems: Computational approach

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2002
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Kluwer Academic/Plenum Publ
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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.
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RAMOS, A. M., GLOWINSKI, R., and PERIAUX, J., Nash Equilibria for the Multiobjectiûe Control of Partial Differential Equations, Journal of Optimization Theory and Applications, Vol. 112, pp. 457–498, 2002. SHIVAMOGGI, B. K., Theoretical Fluid Dynamics, Martinus Nijhoff Publishers,Dordrecht, Holland, 1985. DEAN, E. J., and GUBERNATIS, P., Pointwise Control of the Burgers Equation:A Numerical Approach, Computers and Mathematics with Applications,Vol. 22, pp. 93–100, 1991. BERGGREN, M., GLOWINSKI, R., and LIONS, J. L., A Computational Approach to Controllability Issues for Flow-Related Models, (I): Pointwise Control of the Viscous Burgers Equation, International Journal of Computational Fluid Dynamics, Vol. 7, pp. 237–252, 1996. GLOWINSKI, R., and LIONS, J. L., Exact and Approximate Controllability for Distributed Parameter Systems, II, Acta Numerica 1995, Edited by A. Iserles,Cambridge University Press, Cambridge, England, pp. 159–333, 1995. LIU, D. C., and NOCEDAL, J., On the Limited Memory BFGS Method for Large-Scale Optimization, Mathematical Programming, Vol. 45, pp. 503–528, 1989. DIAZ, J. I., Sobre la Controlabilidad Aproximada de Problemas No Lineales Disipatiûos,Jornadas Hispano–rancesas sobre Control de Sistemas Distribuidos,Edited by A. Valle, Universidad de Ma´laga, Ma´laga, Spain, pp. 41–48, 1990. DIAZ, J. I., Obstruction and Some Approximate Controllability Results for the Burgers Equation and Related Problems, Control of Partial Differential Equations and Applications, Edited by E. Casas, Marcel Dekker, New York, NY, pp. 63–76, 1996. DIAZ, J. I., and RAMOS, A.M., Numerical Experiments Regarding the Localized Control of Semilinear Parabolic Problems, Proceedings of the European Congress on Computational Methods in Applied Sciences and Engineering, Barcelona, Spain, 2000. RAMOS, A. M., GLOWINSKI, R., and PERIAUX, J., Pointwise Control of the Burgers Equation and Related Nash Equilibrium Problems: A Computational Approach, Matema´tica Aplicada Report MA-UCM 2001-6, Universidad Complutense de Madrid, 2001. NASH, J. F., Noncooperatiûe Games, Annals of Mathematics, Vol. 54, pp. 286–295, 1951. PARETO, V., Cours d’Economie Politique, Rouge, Lausanne,Switzerland, 1896. VON STACKELBERG, H., Marktform und Gleichgewicht, Springer, Berlin,Germany, 1934.
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