Some remarks on existence results for optimal boundary control problems
ESAIM: Control, Optimisation and Calculus of Variations, Tome 9 (2003), pp. 437-448.

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.

DOI : 10.1051/cocv:2003021
Classification : 49J20, 49J45
Mots-clés : boundary controls, vector variational problems, gradient Young measures 
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     title = {Some remarks on existence results for optimal boundary control problems},
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     pages = {437--448},
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Pedregal, Pablo. Some remarks on existence results for optimal boundary control problems. ESAIM: Control, Optimisation and Calculus of Variations, Tome 9 (2003), pp. 437-448. doi : 10.1051/cocv:2003021. http://archive.numdam.org/articles/10.1051/cocv:2003021/

[1] B. Dacorogna, Direct methods in the Calculus of Variations. Springer (1989). | MR | Zbl

[2] K.H. Hoffmann, G. Leugerin and F. Tröltzsch, Optimal Control of Partial Differential Equations. Birkhäuser, Basel, ISNM 133 (2000). | MR

[3] S. Müller, Rank-one convexity implies quasiconvexity on diagonal matrices. Intern. Math. Res. Notices 20 (1999) 1087-1095. | MR | Zbl

[4] F. Murat, Compacité par compensation. Ann. Scuola Norm. Sup. Pisa Sci. Fis. Mat. (IV) 5 (1978) 489-507. | Numdam | MR | Zbl

[5] 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. | MR | Zbl

[6] F. Murat, A survey on compensated compactness, in Contributions to the modern calculus of variations, edited by L. Cesari. Pitman (1987) 145-183. | MR

[7] P. Pedregal, Weak continuity and weak lower semicontinuity for some compensation operators. Proc. Roy. Soc. Edinburgh Sect. A 113 (1989) 267-279. | MR | Zbl

[8] P. Pedregal, Parametrized Measures and Variational Principles. Birkhäuser, Basel (1997). | MR | Zbl

[9] P. Pedregal, Optimal design and constrained quasiconvexity. SIAM J. Math. Anal. 32 (2000) 854-869. | MR | Zbl

[10] P. Pedregal, Fully explicit quasiconvexification of the mean-square deviation of the gradient of the state in optimal design. ERA-AMS 7 (2001) 72-78. | EuDML | MR | Zbl

[11] V. Sverak, On Tartar's conjecture. Inst. H. Poincaré Anal. Non Linéaire 10 (1993) 405-412. | EuDML | Numdam | Zbl

[12] V. Sverak, On regularity for the Monge-Ampère equation. Preprint (1993).

[13] V. Sverak, Lower semicontinuity of variational integrals and compensated compactness, edited by S.D. Chatterji. Birkhäuser, Proc. ICM 2 (1994) 1153-1158. | MR | Zbl

[14] L. Tartar, 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. | Zbl

[15] 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). | MR | Zbl

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