Controle ótimo em equações de ondas via equilíbrios de Nash e Pareto
Resumo
The objective of this work is to making a theoretical and numerical study on bi-objective optimal control for the linear and semi-linear wave equations, via Nash and Pareto equilibrium strategies, in three-dimensional domains. Due to being related to an optimization process, we will use cost functional minimization, formulated in combination with some strategies related to Nash Equilibrium and Pareto Equilibrium. We will use the FreeFem++ software to transcribe the theoretical problems to the C++ programming language, describing the spatial discretization data through the Finite Element Method (FEM), and the discretization of its temporal evolution, with the Finite Difference Method.
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Referências
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partial differential equations, https://arxiv.org/abs/1908.11858
2. ARARUNA, E.FERNANDEZ CARA and M.C. SANTOS, ´ Stakelberg Nash Exact Controllability for linear and semilinear Parabolic Equations, ESAIM Control Optimisation and Calculus of Variations 21(3), July (2015).
3. ARARUNA, E. FERNANDEZ-CARA, S. GUERREIRO and M.C. SANTOS, New results on the Stackelberg–Nash
exact control of linear parabolic equations, Systems and Control Letters 104, 78-85 (2017).
4. ARARUNA, E. FERNANDEZ-CARA, S. GUERREIRO and SILVA, L.C, ´ Hierarchic control for the wave equation, J.
Optimiz. Theory App 178(4), 264-288 (2018).
5. CARVALHO, P.P., Some numerical results for control of 3D heat equations using Nash equilibrium, Comp. Appl. Math,
40 92 (2021).
6. CARVALHO, P.P, FERNANDEZ-CARA, E. and FERREL, J.B.L, ´ On the computation of Nash and Pareto equilibria
for some bi-objective control problems for the wave equation, Adv Comput Math, 46 73 (2020).
7. CARVALHO , P.P., FERNANDEZ-CARA, E., ´ On the Computation of Nash and Pareto Equilibria for Some Bi-objective
Control Problems, SeMA J., 61 49-78 (2013).
8. CARVALHO, P. P., & FERNANDEZ-CARA, E. Numerical Stackelberg–Nash Control for the Heat Equation, SIAM
Journal on Scientific Computing, 42(5), A2678-A2700, (2020).
9. CIARLET, P.H.G, Introduction to Numerical Linear Algebra and Optimisation, Cambridge University Press, Cambridge (1989).
10. D. BAUSO, Game theory with engineering applications, Advances in Design and Control, 30. Society for Industrial and
Applied Mathematics (SIAM), Philadelphia, PA, 2016.
11. D´IAZ, J.; LIONS, J.–L., On the approximate controllability of Stackelberg–Nash strategies, in: J.I. D´ıaz (Ed). Ocean
Circulation and Pollution Control Mathematical and Numerical Investigations, Springer, 2005.
12. FRED´ EREC HECHT, New development in Free ´ Fem++. Journal of numerical mathematics, vol.(20), p.1-14, De
Gruyter, 2012.
13. GARBOR KASSAY, VICENTIU RADULESCU, Equilibrium Problems and Applications (Mathematics in Science and ˇ
Engineering) (English Edition), 1 Edition (2019).
14. GLOWINSKI, R., Numerical Methods for Nonlinear Variational Problems, Springer, New York, NY, 1984.
15. J.F. NASH, Noncooperative games, Ann. Math, 54 (1951) 286-295.
16. LIONS, J.L., Contrˆole de Pareto de syst`emes distribu´es, Le cas d’´evolution. C. R. Acad. Sci. Paris, S´erie I 302(11),
413–417 (1986)
17. LIONS, J.L., Hierarchic Control, Proc. Indian Acad. Sci. Math. Sci. 104 (1994), no. 4, 295-304.
18. LIONS, J.– L., Quelques M´ethodes de R´esolutions des Probl`emes aux Limites non Lin´eaires, Dunod Gauthier-Villars,
Paris, 1969.
19. CAVALCANTI, MARCELO and DOMINGOS CAVALCANTI, VALERIA and ROSIER, CAROLE and ROSIER, LIONEL Numerical control of a semilinear wave equation on an interval, In: Auriol, J., Deutscher, J., Mazanti, G.,
Valmorbida, G. (eds.) Advances in Distributed Parameter Systems, pp. 69–89. Springer, Cham (2022)
20. C. MORAES, M. RINCON AND G. ANTUNES Numerical Method of the Exact Control for the Elastic String Problem
with Moving Boundary Bol. Soc. Parana Mat. 2023
21. PARETO, V. Cours d’´economie politique, Rouge, Laussane, Switzerland (1896).
22. RAMOS A. M, GLOWINSKI, R and J. PERIAUX, Nash equilibria for the multiobjective control of linear partial partial
differential equations, J. Optim.Theory Appl (2002) 457 - 498.
23. STACKELBERG, H. von., Marktform un Gleichgewicht, Springer, Berlin, Germany, 1934
Publicado
2025-02-22
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Funding data
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Conselho Nacional de Desenvolvimento Científico e Tecnológico
Grant numbers Edital CNPq Nº 09/2022
