Some remarks on existence results for optimal boundary control problems

Pablo Pedregal

ESAIM: Control, Optimisation and Calculus of Variations (2003)

  • Volume: 9, page 437-448
  • ISSN: 1292-8119

Abstract

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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.

How to cite

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Pedregal, Pablo. "Some remarks on existence results for optimal boundary control problems." ESAIM: Control, Optimisation and Calculus of Variations 9 (2003): 437-448. <http://eudml.org/doc/245007>.

@article{Pedregal2003,
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.},
author = {Pedregal, Pablo},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {boundary controls; vector variational problems; gradient Young measures},
language = {eng},
pages = {437-448},
publisher = {EDP-Sciences},
title = {Some remarks on existence results for optimal boundary control problems},
url = {http://eudml.org/doc/245007},
volume = {9},
year = {2003},
}

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
PB - EDP-Sciences
VL - 9
SP - 437
EP - 448
AB - 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.
LA - eng
KW - boundary controls; vector variational problems; gradient Young measures
UR - http://eudml.org/doc/245007
ER -

References

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  1. [1] B. Dacorogna, Direct methods in the Calculus of Variations. Springer (1989). Zbl0703.49001MR990890
  2. [2] K.H. Hoffmann, G. Leugerin and F. Tröltzsch, Optimal Control of Partial Differential Equations. Birkhäuser, Basel, ISNM 133 (2000). MR1725022
  3. [3] S. Müller, Rank-one convexity implies quasiconvexity on diagonal matrices. Intern. Math. Res. Notices 20 (1999) 1087-1095. Zbl1055.49506MR1728018
  4. [4] F. Murat, Compacité par compensation. Ann. Scuola Norm. Sup. Pisa Sci. Fis. Mat. (IV) 5 (1978) 489-507. Zbl0399.46022MR506997
  5. [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. Zbl0427.35008MR533170
  6. [6] F. Murat, A survey on compensated compactness, in Contributions to the modern calculus of variations, edited by L. Cesari. Pitman (1987) 145-183. MR894077
  7. [7] P. Pedregal, Weak continuity and weak lower semicontinuity for some compensation operators. Proc. Roy. Soc. Edinburgh Sect. A 113 (1989) 267-279. Zbl0702.35054MR1037732
  8. [8] P. Pedregal, Parametrized Measures and Variational Principles. Birkhäuser, Basel (1997). Zbl0879.49017MR1452107
  9. [9] P. Pedregal, Optimal design and constrained quasiconvexity. SIAM J. Math. Anal. 32 (2000) 854-869. Zbl0995.49004MR1814741
  10. [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. Zbl0980.49021MR1856792
  11. [11] V. Sverak, On Tartar’s conjecture. Inst. H. Poincaré Anal. Non Linéaire 10 (1993) 405-412. Zbl0820.35022
  12. [12] V. Sverak, On regularity for the Monge-Ampère equation. Preprint (1993). 
  13. [13] V. Sverak, Lower semicontinuity of variational integrals and compensated compactness, edited by S.D. Chatterji. Birkhäuser, Proc. ICM 2 (1994) 1153-1158. Zbl0852.49010MR1404015
  14. [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. Zbl0437.35004
  15. [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). Zbl0536.35003MR725524

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