Abstract
In this paper, we investigate the controllability for the one-dimensional plate equation in intervals with a moving boundary. This equation models the vertical displacement of a point x at time t in a bar with uniform cross section. We assume the ends of the bar with small and uniform variations. More precisely, we have introduced functions α(t) and β(t) modeling the motion of these ends. We present the following results: the existence and uniqueness of Nash equilibrium, the approximate controllability with respect to the leader control, and the optimality system for the leader control.
Similar content being viewed by others
Change history
22 August 2018
A Correction to this paper has been published: https://doi.org/10.1007/s10883-018-9417-8
References
Nash J. Noncooperative games. Ann Math 1951;54:286–295.
Pareto V. Cours d’économie politique. Switzerland: Rouge; 1896.
Stackelberg H. Marktform Und gleichgewicht. Berlin: Springer; 1934.
Lions J-L. Hierarchic control. Math Sci Proc Indian Acad Sci 1994;104:295–304.
Jesus I. Hierarchical control for the wave equation with a moving boundary. J Optim Theory Appl 2016;171:336–350.
Hörmander L, Vol. 116. Linear partial differential operators Die Grundlehren der mathematischen Wissenschaften Bd. New York: Academic Press Inc., Publishers; 1963.
Lions J-L. Contrôle de Pareto de systèmes distribués. Le cas d’ évolution. CR Acad Sc Paris, Sér I 1986;302(11):413–417.
Lions J-L. Some remarks on Stackelberg’s optimization. Math Models Methods Appl Sci 1994;4:477–487.
Díaz J, Lions J-L. On the approximate controllability of Stackelberg-Nash strategies. Ocean circulation and pollution control mathematical and numerical investigations. In: Díaz JI, editors. Berlin: Springer; 2005. p. 17–27.
Díaz J. On the von Neumann problem and the approximate controllability of Stackelberg-Nash strategies for some environmental problems. Rev R Acad Cien, Ser A Math 2002;96(3):343–356.
Glowinski R, Ramos A, Periaux J. Nash equilibria for the multi-objective control of linear differential equations. J Optim Theory Appl 2002;112(3):457–498.
Glowinski R, Ramos A, Periaux J. Pointwise control of the Burgers equation and related Nash equilibrium problems: computational approach. J Optim Theory Appl 2002;112(3):499–516.
González G, Lopes F, Rojas-Medar M. On the approximate controllability of Stackelberg-Nash strategies for Stokes equations. Proc Amer Math Soc 2013;141(5): 1759–1773.
Limaco J, Clark H, Medeiros LA. Remarks on hierarchic control. J Math Anal Appl 2009;359:368–383.
Araruna FD, Fernández-Cara E, Santos M. Stackelberg-Nash exact controllability for linear and semilinear parabolic equations, ESAIM : Control. Optim Calc Var 2015;21(3):835–856.
Araruna FD, Fernández-cara E, Guerrero S, Santos M. New results on the Stackelberg-Nash exact control of linear parabolic equations. Syst Control Lett 2017; 104:78–85.
Ramos A, Roubicek T. Nash equilibria in noncooperative Predator-Prey games. Appl Math Optim 2007;56(2):211–241.
Araruna FD, Antunes GO, Medeiros LA. Exact controllability for the semilinear string equation in non cylindrical domains. Control Cybern 2004;33:237–257.
Cui L, Song L. 2014. Exact controllability for a wave equation with fixed boundary control. Boundary Value Problems. https://doi.org/10.1186/1687-2770-2014-47.
Bardos C, Chen G. Control and stabilization for the wave equation. Part III: domain with moving boundary. SIAM J Control Optim 1981;19:123–138.
Cui L, Liu X, Gao H. Exact controllability for a one-dimensional wave equation in non-cylindrical domains. J Math Anal Appl 2013;402:612–625.
Milla Miranda M. Exact controlllability for the wave equation in domains with variable boundary. Rev Mat Univ Complut Madr 1996;9:435–457.
Milla Miranda M. HUM and the wave equation with variable coefficients. Asymptot Anal 1995;11:317–341.
Jesus I. Approximate controllability for a one-dimensional wave equation with the fixed endpoint control. J Differ Equ 2017;263:5175–5188.
Caldas C, Limaco J, Barreto R, Gamboa P. Exact controllability for the equation of the one dimensional plate in domains with moving boundary. Divulg Mat 2003;11:19–38.
Aubin J. L’analyse non linéaire et ses Motivations Économiques. Paris: Masson; 1984.
Lions J-L. Contrôle optimal des systèmes gouvernés par des équations aux dérivées partielles. Paris: Dunod; 1968.
Rockafellar R. Convex Analysis. Princeton: Princeton University Press; 1969.
Brezis H. Functional analysis, Sobolev spaces and partial differential equations. Berlin: Springer-Verlag; 2010.
Ekeland I, Temam R. analyse convexe et problèmes variationnels. Paris: Dunod, Gauthier-Villars; 1974.
Acknowledgments
We would like to thank the professor L. A. Medeiros by gentle suggestion of this problem and also to professor Marcos Travaglia for his comments on the manuscript. Moreover, the authors are grateful to the anonymous referees for their constructive comments and suggestions, which helped in improving the original manuscript significantly.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
de Jesus, I.P., Limaco, J. & Clark, M.R. Hierarchical Control for the One-dimensional Plate Equation with a Moving Boundary. J Dyn Control Syst 24, 635–655 (2018). https://doi.org/10.1007/s10883-018-9413-z
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10883-018-9413-z