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Some remarks on existence results for optimal boundarycontrol problems

Published online by Cambridge University Press:  15 September 2003

Pablo Pedregal
Pablo Pedregal*
Affiliation:
ETSI Industriales, Universidad de Castilla-La Mancha, 13071 Ciudad Real, Spain; Pablo.Pedregal@uclm.es.
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Abstract

An optimal control problem when controls act on the boundary can also be understood as a variational principle under differential constraints and no restrictions on boundary and/or initial values. From this perspective, some existence theorems can be proved when cost functionals depend on the gradient of the state. We treat the case of elliptic and non-elliptic second order state laws only in the two-dimensional situation. Our results are based on deep facts about gradient Young measures.

Keywords

Boundary controlsvector variational problemsgradient Young measures.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 9 , January 2003 , pp. 437 - 448
Copyright
© EDP Sciences, SMAI, 2003

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References

B. Dacorogna, Direct methods in the Calculus of Variations. Springer (1989).
K.H. Hoffmann, G. Leugerin and F. Tröltzsch, Optimal Control of Partial Differential Equations. Birkhäuser, Basel, ISNM 133 (2000).
Müller, S., Rank-one convexity implies quasiconvexity on diagonal matrices. Intern. Math. Res. Notices 20 (1999) 1087-1095. CrossRef
Murat, F., Compacité par compensation. Ann. Scuola Norm. Sup. Pisa Sci. Fis. Mat. (IV) 5 (1978) 489-507.
F. Murat, Compacité par compensation II, in Recent Methods in Nonlinear Analysis Proceedings, edited by E. De Giorgi,E. Magenes and U. Mosco. Pitagora, Bologna (1979) 245-256.
F. Murat, A survey on compensated compactness, in Contributions to the modern calculus of variations, edited by L. Cesari. Pitman (1987) 145-183.
Pedregal, P., Weak continuity and weak lower semicontinuity for some compensation operators. Proc. Roy. Soc. Edinburgh Sect. A 113 (1989) 267-279. CrossRef
P. Pedregal, Parametrized Measures and Variational Principles. Birkhäuser, Basel (1997).
Pedregal, P., Optimal design and constrained quasiconvexity. SIAM J. Math. Anal. 32 (2000) 854-869. CrossRef
Pedregal, P., Fully explicit quasiconvexification of the mean-square deviation of the gradient of the state in optimal design. ERA-AMS 7 (2001) 72-78.
Sverak, V., Tartar's, On conjecture. Inst. H. Poincaré Anal. Non Linéaire 10 (1993) 405-412. CrossRef
V. Sverak, On regularity for the Monge-Ampère equation. Preprint (1993).
V. Sverak, Lower semicontinuity of variational integrals and compensated compactness, edited by S.D. Chatterji. Birkhäuser, Proc. ICM 2 (1994) 1153-1158.
Tartar, L., Compensated compactness and applications to partal differential equations, in Nonlinear analysis and mechanics: Heriot-Watt Symposium, Vol. IV, edited by R. Knops. Pitman Res. Notes Math. 39 (1979) 136-212.
L. Tartar, The compensated compactness method applied to systems of conservation laws, in Systems of Nonlinear Partial Differential Eq., edited by J.M. Ball. Riedel (1983).