Hostname: page-component-586b7cd67f-g8jcs Total loading time: 0 Render date: 2024-11-22T04:04:31.423Z Has data issue: false hasContentIssue false
  • English
  • Français

Minimax control of an elliptic variational bilateral problem

Published online by Cambridge University Press:  17 February 2009

Qihong Chen
Qihong Chen
Affiliation:
Department of Applied Mathematics, Shanghai University of Finance and Economics, 777 Guoding Road, Shanghai 200433, P. R. China: e-mail: chenqih@online.sh.cn.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

This paper deals with a minimax control problem for semilinear elliptic variational inequalities associated with bilateral constraints. The control domain is not necessarily convex. The cost functional, which is to be minimised, is the sup norm of some function of the state and the control. The major novelty of such a problem lies in the simultaneous presence of the nonsmooth state equation (variational inequality) and the nonsmooth cost functional (the sup norm). In this paper, the existence conditions and the Pontryagin-type necessary conditions for optimal controls are established.

Type
Research Article
Information
The ANZIAM Journal , Volume 44 , Issue 4 , April 2003 , pp. 539 - 559
Copyright
Copyright © Australian Mathematical Society 2003

References

[1]Barbu, V., Optimal control of variational inequalities (Pitman, London, 1984).Google Scholar
[2]Barron, E. N., “The Pontryagin maximum principle for minimax problems of optimal control”, Nonlinear Anal. 15 (1990) 1155–1165.CrossRefGoogle Scholar
[3]Cesari, L., Existence of solutions and existence of optimal solutions, Notes in Math. 979 (Springer, Berlin, 1983).Google Scholar
[4]Chen, Q., “Optimal control of semilinear elliptic variational bilateral problem”, Acta Math. Sinica 16 (2000) 123–140.CrossRefGoogle Scholar
[5]Ekeland, I., “On the variational principle”, J. Math. Anal. Appl. 47 (1974) 324–353.CrossRefGoogle Scholar
[6]Friedman, A., “Optimal control for variational inequalities”, SIAM J. Control Optim. 24 (1986) 439–451.CrossRefGoogle Scholar
[7]Gilbarg, D. and Trudinger, N. S., Elliptic partial differential equations of second order, 2nd ed. (Springer, Berlin, 1983).Google Scholar
[8]Himmelberg, C. J., Jacobs, M. Q. and Van Vleck, F. S., “Measurable multifunctions, selectors and Filippov's implicit functions lemma”, J. Math. Anal. Appl. 25 (1969) 276–284.CrossRefGoogle Scholar
[9]Li, X. and Yong, J., Optimal control theory for infinite dimensional systems (Birkhäuser, Boston, 1995).CrossRefGoogle Scholar
[10]Lions, J. L., Quelques méthodes de résolution des problemes aux limites non linéaires (Dunod, Paris, 1969).Google Scholar
[11]Neustadt, L., Optimization (Princeton Univ. Press, Princeton, NJ, 1976).Google Scholar
[12]Stampacchia, G., “Le probleme de Dirichlet pour les equations elliptiques du second ordre a coefficients discontinus”, Ann. Inst. Fourier Grenoble 15 (1965) 189–258.CrossRefGoogle Scholar
[13]Troianiello, G. M., Elliptic differential equations and obstacle problems (Plenum Press, New York, 1987).CrossRefGoogle Scholar
[14]. Yong, J., “A minimax control problem for second order elliptic partial differential equations”, Kodai Math. J. 16 (1993) 468–486.CrossRefGoogle Scholar