Abstract
This paper addresses the study of the hierarchical control for the one-dimensional wave equation in intervals with a moving boundary. This equation models the motion of a string where an endpoint is fixed and the other one is moving. When the speed of the moving endpoint is less than the characteristic speed, the controllability of this equation is established. We assume that we can act on the dynamic of the system by a hierarchy of controls. According to the formulation given by Stackelberg (Marktform und Gleichgewicht. Springer, Berlin, 1934), there are local controls called followers and global controls called leaders. In fact, one considers situations where there are two cost (objective) functions. One possible way is to cut the control into two parts, one being thought of as “the leader” and the other one as “the follower.” This situation is studied in the paper, with one of the cost functions being of the controllability type. 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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Nash, J.: Noncooperative games. Ann. Math. 54, 286–295 (1951)
Pareto, V.: Cours d’économie politique. Rouge, Laussane (1896)
von Stackelberg, H.: Marktform und Gleichgewicht. Springer, Berlin (1934)
Lions, J.-L.: Hierarchic control. Math. Sci., Proc. Indian Acad. Sci. 104, 295–304 (1994)
Lions, J.-L.: Contrôle de Pareto de Systèmes Distribués. Le cas d’ évolution C.R. Acad. Sc. Paris, série I. 302(11), 413–417 (1986)
Lions, J.-L.: Some remarks on Stackelberg’s optimization. Math. Mod. Methods Appl. Sci. 4, 477–487 (1994)
Díaz, J., Lions, J.-L.: On the approximate controllability of Stackelberg-Nash strategies. In: Díaz, J.I. (ed.) Ocean Circulation and Pollution Control Mathematical and Numerical Investigations, pp. 17–27. Springer, Berlin (2005)
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. 96(3), 343–356 (2002)
Glowinski, R., Ramos, A., Periaux, J.: Nash equilibria for the multi-objective control of linear differential equations. J. Optim. Theory Appl. 112(3), 457–498 (2002)
Glowinski, R., Ramos, A., Periaux, J.: Pointwise control of the burgers equation and related Nash equilibrium problems : computational approach. J. Optim. Theory Appl. 112(3), 499–516 (2002)
González, G., Marques-Lopes, F., Rojas-Medar, M.: On the approximate controllability of Stackelberg-Nash strategies for Stokes equations. Proc. Amer. Math. Soc. 141(5), 1759–1773 (2013)
Limaco, J., Clark, H., Medeiros, L.A.: Remarks on hierarchic control. J. Math. Anal. Appl. 359, 368–383 (2009)
Araruna, F.D., Fernández-Cara, E., Santos, M.C.: Stackelberg-Nash exact controllability for linear and semilinear parabolic equations. ESAIM: Control, Optim. Calc. Var. 21(3), 835–856 (2015)
Ramos, A.M., Roubicek, T.: Nash equilibria in noncooperative predator-prey games. Appl. Math. Optim. 56(2), 211–241 (2007)
Cui, L., Song, L.: Controllability for a wave equation with moving boundary. J. Appl. Math. (2014). doi:10.1155/2014/827698
Araruna, F.D., Antunes, G.O., Medeiros, L.A.: Exact controllability for the semilinear string equation in non cylindrical domains. Control Cybern. 33, 237–257 (2004)
Bardos, C., Chen, G.: Control and stabilization for the wave equation. Part III: domain with moving boundary. SIAM J. Control Optim 19, 123–138 (1981)
Cui, L., Liu, X., Gao, H.: Exact controllability for a one-dimensional wave equation in non-cylindrical domains. J. Math. Anal. Appl. 402, 612–625 (2013)
Cui, L., Song, L.: Exact controllability for a wave equation with fixed boundary control. Bound. Value Problems (2014). doi:10.1186/1687-2770-2014-47
Jesus, I.: Remarks on hierarchic control for the wave equation in moving domains. Arch. Math. 102, 171–179 (2014)
Miranda, M.: Exact controlllability for the wave equation in domains with variable boundary. Rev. Mat. Univ. Complut. Madr. 9, 435–457 (1996)
Miranda, M.: HUM and the wave equation with variable coefficients. Asympt. Anal. 11, 317–341 (1995)
Lions, J.-L.: L’analyse non Linéaire et ses Motivations Économiques. Masson, Paris (1984)
Lions, J.-L.: Contrôle optimal des systèmes gouvernés par des équations aux dérivées partielles. Dunod, Paris (1968)
Hörmander, L.: Linear partial differential operators. Die Grundlehren der mathematischen Wissenschaften, Bd. 116. Academic Press. Inc., Publishers, New York (1963)
Rockafellar, R.: Convex Analysis. Princeton University Press, Princeton (1969)
Brezis, H.: Functional analysis, Sobolev spaces and partial differential equations. Springer, Berlin (2010)
Ekeland, I., Teman, R.: Analyse convexe et problèmes variationnels Dunod. Gauthier-Villars, Paris (1974)
Acknowledgments
The author wants to express his gratitude to the anonymous reviewers for their questions and commentaries; they were very helpful in improving this article.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Roland Glowinski.
Rights and permissions
About this article
Cite this article
de Jesus, I.P. Hierarchical Control for the Wave Equation with a Moving Boundary. J Optim Theory Appl 171, 336–350 (2016). https://doi.org/10.1007/s10957-016-0984-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10957-016-0984-0