Minimax control of nonlinear evolution equations

Nikolaos S. Papageorgiou

Commentationes Mathematicae Universitatis Carolinae (1995)

  • Volume: 36, Issue: 1, page 39-56
  • ISSN: 0010-2628

Abstract

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In this paper we study the minimax control of systems governed by a nonlinear evolution inclusion of the subdifferential type. Using some continuity and lower semicontinuity results for the solution map and the cost functional respectively, we are able to establish the existence of an optimal control. The abstract results are then applied to obstacle problems, semilinear systems with weakly varying coefficients (e.gȯscillating coefficients) and differential variational inequalities.

How to cite

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Papageorgiou, Nikolaos S.. "Minimax control of nonlinear evolution equations." Commentationes Mathematicae Universitatis Carolinae 36.1 (1995): 39-56. <http://eudml.org/doc/22078>.

@article{Papageorgiou1995,
abstract = {In this paper we study the minimax control of systems governed by a nonlinear evolution inclusion of the subdifferential type. Using some continuity and lower semicontinuity results for the solution map and the cost functional respectively, we are able to establish the existence of an optimal control. The abstract results are then applied to obstacle problems, semilinear systems with weakly varying coefficients (e.gȯscillating coefficients) and differential variational inequalities.},
author = {Papageorgiou, Nikolaos S.},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {minimax problem; optimal control; subdifferential; strong solution; Mosco convergence; obstacle problems; differential variational inequalities; evolution triple; compact embedding; monotone operator; -convergence; minimax optimization problem; saddle point; adjoint equation; duality theory; necessary conditions; Pontryagin maximum principle; nonlinear parametric optimal control},
language = {eng},
number = {1},
pages = {39-56},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Minimax control of nonlinear evolution equations},
url = {http://eudml.org/doc/22078},
volume = {36},
year = {1995},
}

TY - JOUR
AU - Papageorgiou, Nikolaos S.
TI - Minimax control of nonlinear evolution equations
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1995
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 36
IS - 1
SP - 39
EP - 56
AB - In this paper we study the minimax control of systems governed by a nonlinear evolution inclusion of the subdifferential type. Using some continuity and lower semicontinuity results for the solution map and the cost functional respectively, we are able to establish the existence of an optimal control. The abstract results are then applied to obstacle problems, semilinear systems with weakly varying coefficients (e.gȯscillating coefficients) and differential variational inequalities.
LA - eng
KW - minimax problem; optimal control; subdifferential; strong solution; Mosco convergence; obstacle problems; differential variational inequalities; evolution triple; compact embedding; monotone operator; -convergence; minimax optimization problem; saddle point; adjoint equation; duality theory; necessary conditions; Pontryagin maximum principle; nonlinear parametric optimal control
UR - http://eudml.org/doc/22078
ER -

References

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  1. Attouch H., Variational Convergence for Functionals and Operators, Pitman, London, 1984. MR0773850
  2. Balder E., Necessary and sufficient conditions for L 1 -strong-weak lower semicontinuity of integral functionals, Nonlin. Anal. 11 (1987), 1399-1404. (1987) MR0917861
  3. Berge C., Espaces Topologiques et Fonctions Multivoques, Dunod, Paris, 1966. 
  4. Brézis H., Operateurs Maximaux Monotones et Semigroupes de contractions dans les Espaces de Hilbert, North Holland, Amsterdam, 1973. 
  5. Brézis H., Monotonicity methods in Hilbert spaces and some applications to nonlinear partial differential equations, in Contributions to Nonlinear Functional Analysis, ed. by E. Zarantonello, Academic Press, New York, 1971, pp. 101-156. MR0394323
  6. Cornet B., Existence of slow solutions for a class of differential inclusions, J. Math. Anal. Appl. 96 (1983), 130-147. (1983) Zbl0558.34011MR0717499
  7. Dal Maso G., An Introduction to Γ -convergence, Birkhäuser, Boston, 1993. Zbl0816.49001MR1201152
  8. Krasnosel'skii M.A., Pokrovskii A.V., Systems with Hysteresis, Springer-Verlag, New York, 1988. Zbl0665.47038
  9. Kenmochi N., Some nonlinear parabolic variational inequalities, Israel J. Math. 22 (1975), 304-331. (1975) MR0399662
  10. Moreau J.-J., Evolution problem associated with a moving convex set in a Hilbert space, J. Differential Equations 26 (1977), 347-374. (1977) MR0508661
  11. Mosco U., Convergence of convex sets and of solutions of variational inequalities, Advances in Math. 3 (1969), 510-585. (1969) Zbl0192.49101MR0298508
  12. Papageorgiou N.S., On evolution inclusions associated with time dependent convex subdifferentials, Comment. Math. Univ. Carolinae 31 (1990), 517-527. (1990) Zbl0711.34076MR1078486
  13. Wagner D., Survey of measurable selection theorems, SIAM J. Control Optim. 15 (1977), 859-903. (1977) Zbl0407.28006MR0486391
  14. Yamada Y., On evolution equations generated by subdifferential operators, J. Fac. Sci. Univ. Tokyo 23 (1976), 491-515. (1976) Zbl0343.34053MR0425701
  15. Yotsutani S., Evolution equations associated with subdifferentials, J. Math. Soc. Japan 31 (1978), 623-646. (1978) MR0544681
  16. Zhikov V., Kozlov S., Oleinik O., Ngoan K., Averaging and G -convergence of differential operators, Russian Math. Surveys 34 (1979), 69-147. (1979) MR0562800
  17. Ahmed N.U., Optimization and Identification of Systems Governed by Evolution Equations on Banach Spaces, Longman Publ. Co., Essex, United Kingdom, 1988. MR0982263
  18. Ahmed N.U., Optimal control of infinite dimensional uncertain systems, J. Optim. Th. Appl. 80 (1994), 261-272. (1994) Zbl0798.49032MR1259659
  19. Tanimoto S., Duality in the optimal control of non-well posed distributed systems, J. Math. Anal. Appl. 171 (1992), 277-282. (1992) Zbl0765.49002MR1192506

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