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 controls, vector variational problems, gradient Young measures
@article{COCV_2003__9__437_0,
author = {Pedregal, Pablo},
title = {Some remarks on existence results for optimal boundary control problems},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {437--448},
year = {2003},
publisher = {EDP Sciences},
volume = {9},
doi = {10.1051/cocv:2003021},
mrnumber = {1998709},
zbl = {1066.49005},
language = {en},
url = {https://www.numdam.org/articles/10.1051/cocv:2003021/}
}
TY - JOUR AU - Pedregal, Pablo TI - Some remarks on existence results for optimal boundary control problems JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2003 SP - 437 EP - 448 VL - 9 PB - EDP Sciences UR - https://www.numdam.org/articles/10.1051/cocv:2003021/ DO - 10.1051/cocv:2003021 LA - en ID - COCV_2003__9__437_0 ER -
%0 Journal Article %A Pedregal, Pablo %T Some remarks on existence results for optimal boundary control problems %J ESAIM: Control, Optimisation and Calculus of Variations %D 2003 %P 437-448 %V 9 %I EDP Sciences %U https://www.numdam.org/articles/10.1051/cocv:2003021/ %R 10.1051/cocv:2003021 %G en %F COCV_2003__9__437_0
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
[1] , Direct methods in the Calculus of Variations. Springer (1989). | Zbl | MR
[2] , and, Optimal Control of Partial Differential Equations. Birkhäuser, Basel, ISNM 133 (2000). | MR
[3] , Rank-one convexity implies quasiconvexity on diagonal matrices. Intern. Math. Res. Notices 20 (1999) 1087-1095. | Zbl | MR
[4] , Compacité par compensation. Ann. Scuola Norm. Sup. Pisa Sci. Fis. Mat. (IV) 5 (1978) 489-507. | Zbl | MR | Numdam
[5] , 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. | Zbl | MR
[6] , A survey on compensated compactness, in Contributions to the modern calculus of variations, edited by L. Cesari. Pitman (1987) 145-183. | MR
[7] , Weak continuity and weak lower semicontinuity for some compensation operators. Proc. Roy. Soc. Edinburgh Sect. A 113 (1989) 267-279. | Zbl | MR
[8] , Parametrized Measures and Variational Principles. Birkhäuser, Basel (1997). | Zbl | MR
[9] , Optimal design and constrained quasiconvexity. SIAM J. Math. Anal. 32 (2000) 854-869. | Zbl | MR
[10] , Fully explicit quasiconvexification of the mean-square deviation of the gradient of the state in optimal design. ERA-AMS 7 (2001) 72-78. | Zbl | MR | EuDML
[11] , On Tartar's conjecture. Inst. H. Poincaré Anal. Non Linéaire 10 (1993) 405-412. | Numdam | Zbl | EuDML
[12] , On regularity for the Monge-Ampère equation. Preprint (1993).
[13] , Lower semicontinuity of variational integrals and compensated compactness, edited by S.D. Chatterji. Birkhäuser, Proc. ICM 2 (1994) 1153-1158. | Zbl | MR
[14] , 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] , The compensated compactness method applied to systems of conservation laws, in Systems of Nonlinear Partial Differential Eq., edited by J.M. Ball. Riedel (1983). | Zbl | MR
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