# Minimax control of nonlinear evolution equations

Commentationes Mathematicae Universitatis Carolinae (1995)

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

## Access Full Article

top## Abstract

top## How to cite

topPapageorgiou, 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

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

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.