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Necessary conditions for optimal control of elliptic systems

Published online by Cambridge University Press:  17 February 2009

Hang Gao
Affiliation:
Department of Mathematics, Northeast Normal University, Changchun 130024, P. R. China.
Xunjing Li
Affiliation:
Department of Mathematics, Fudan University, Shanghai 200433, P. R. China.
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Abstract

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In this paper, we consider the system governed via the coefficients of a semilinear elliptic equation and give the necessary conditions for optimal control. Furthermore, we obtain the necessary conditions for an optimal domain in a domain optimization problem.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 2000

References

[1]Ahmed, N. U. and Teo, K. L., Optimal Control of Distributed Parameter System (North Holland, New York, 1981).Google Scholar
[2]Bonnans, J. F. and Casas, E., “A boundary Pontryagin's principle for the optimal control of state constrained elliptic systems”, Int. Ser. Numer. Math. 107 (1992) 241249.Google Scholar
[3]Bonnans, J. F. and Tiba, D., “Pontryagin principle in the control of semilinear elliptic variational inequalities”, Appl. Math. Optim. 23 (1991) 299312.CrossRefGoogle Scholar
[4]Casas, E., “Optimal control in coefficients of elliptic equations with state constraints”, Appl. Math. Optim. 26 (1992) 2137.CrossRefGoogle Scholar
[5]Casas, E. and Yong, J., “Maximum principle for state-constrained optimal control problems governed by quasilinear elliptic equation”, Diff. Int. Eqn. 8 (1995) 118.Google Scholar
[6]Clarke, F. H., Optimization and Nonsmooth Analysis (Wiley, New York, 1983).Google Scholar
[7]Fattorini, H. O. and Frankowska, H., “Necessary conditions for infinite dimensional control problems”, Math. Control Signal Systems 4 (1991) 4167.CrossRefGoogle Scholar
[8]Gilbarg, D. and Trudinger, N. S., Elliptic Partial Differential Equations of Second Order, 2nd ed. (Springer, Berlin, 1983).Google Scholar
[9]Ladyzhenskaya, O. A. and Ural'tseva, N. N., Linear and Quasilinear Elliptic Equations (Academic Press, New York, 1968).Google Scholar
[10]Li, X., “Vector-valued measure and the necessary conditions for the optimal control problems of linear systems”, in Proceedings of IFAC 3rd Symposium on Control of Distributed Parameter Systems, (Toulouse, 1982).Google Scholar
[11]Li, X. and Yong, J., “Necessary conditions of optimal control for distributed parameter systems”, SIAM J. Control Optim. 29 (1991) 895908.CrossRefGoogle Scholar
[12]Li, X. and Yong, J., Optimal Control Theory for Infinite Dimensional Systems (Birkhauser, Boston, 1995).CrossRefGoogle Scholar
[13]Lions, J. L., Optimal Control of Systems Governed by Partial Differential Equations (Springer, New York, 1971).CrossRefGoogle Scholar
[14]Yong, J., “Existence theory for optimal control of distributed parameter systems”, Kodai Math. J. 15 (1992) 193220.CrossRefGoogle Scholar
[15]Yong, J., “Pontryagin maximum principle for semilinear second order elliptic partial differential equations and variational inequalities with state constraints”, Diff. Int. Eqn. 5 (1992) 13071334.Google Scholar
[16]Yong, J., “Necessary conditions for minimax control problems of second order elliptic partial differential equations”, Kodai Math. J. 16 (1993) 469486.CrossRefGoogle Scholar