Boundary value problems and periodic solutions for semilinear evolution inclusions

Nikolaos S. Papageorgiou

Commentationes Mathematicae Universitatis Carolinae (1994)

  • Volume: 35, Issue: 2, page 325-336
  • ISSN: 0010-2628

Abstract

top
We consider boundary value problems for semilinear evolution inclusions. We establish the existence of extremal solutions. Using that result, we show that the evolution inclusion has periodic extremal trajectories. These results are then applied to closed loop control systems. Finally, an example of a semilinear parabolic distributed parameter control system is worked out in detail.

How to cite

top

Papageorgiou, Nikolaos S.. "Boundary value problems and periodic solutions for semilinear evolution inclusions." Commentationes Mathematicae Universitatis Carolinae 35.2 (1994): 325-336. <http://eudml.org/doc/247638>.

@article{Papageorgiou1994,
abstract = {We consider boundary value problems for semilinear evolution inclusions. We establish the existence of extremal solutions. Using that result, we show that the evolution inclusion has periodic extremal trajectories. These results are then applied to closed loop control systems. Finally, an example of a semilinear parabolic distributed parameter control system is worked out in detail.},
author = {Papageorgiou, Nikolaos S.},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {evolution operator; multifunction; Hausdorff metric; extremal solution; periodic solution; Fredholm alternative; control system; parabolic system; evolution operators; boundary value problems; semilinear differential inclusions; Banach spaces; periodic trajectories; bang-bang controls; semilinear parabolic control systems},
language = {eng},
number = {2},
pages = {325-336},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Boundary value problems and periodic solutions for semilinear evolution inclusions},
url = {http://eudml.org/doc/247638},
volume = {35},
year = {1994},
}

TY - JOUR
AU - Papageorgiou, Nikolaos S.
TI - Boundary value problems and periodic solutions for semilinear evolution inclusions
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1994
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 35
IS - 2
SP - 325
EP - 336
AB - We consider boundary value problems for semilinear evolution inclusions. We establish the existence of extremal solutions. Using that result, we show that the evolution inclusion has periodic extremal trajectories. These results are then applied to closed loop control systems. Finally, an example of a semilinear parabolic distributed parameter control system is worked out in detail.
LA - eng
KW - evolution operator; multifunction; Hausdorff metric; extremal solution; periodic solution; Fredholm alternative; control system; parabolic system; evolution operators; boundary value problems; semilinear differential inclusions; Banach spaces; periodic trajectories; bang-bang controls; semilinear parabolic control systems
UR - http://eudml.org/doc/247638
ER -

References

top
  1. Anichini G., Nonlinear problems for systems of differential equations, Nonlinear Anal.-TMA 1 (1977), 691-699. (1977) Zbl0388.34011MR0592963
  2. Benamara M., Points Extremaux, Multi-applications et Fonctionelles Intégrales, Thèse du 3ème cycle, Université de Grenoble, 1975. 
  3. Diestel J., Uhl J., Vector Measures, Math. Surveys, Vol. 15, AMS, Providence, RI, 1977. Zbl0521.46035MR0453964
  4. Himmelberg C., Measurable relations, Fund. Math. 87 (1975), 57-91. (1975) Zbl0296.28003MR0367142
  5. Kartsatos A., Locally invertible operators and existence problems in differential systems, Tohoku Math. Jour. 28 (1976), 167-176. (1976) Zbl0356.34019MR0430385
  6. Klein E., Thompson A., Theory of Correspondences, Wiley, New York, 1984. Zbl0556.28012MR0752692
  7. Konishi Y., Compacité des résolvantes des opérateurs maximaux cycliquement monotones, Proc. Japan Acad. 49 (1973), 303-305. (1973) Zbl0272.47034MR0346600
  8. Papageorgiou N.S., On the theory of Banach space valued multifunctions. Part 1: Integration and conditional expectation, J. Multiv. Anal. 17 (1985), 185-206. (1985) MR0808276
  9. Papageorgiou N.S., Convergence theorems for Banach space valued integrable multifunctions, Intern. J. Math. and Math. Sci. 10 (1987), 433-442. (1987) Zbl0619.28009MR0896595
  10. Papageorgiou N.S., On multivalued evolution equations and differential inclusions in Banach spaces, Comm. Math. Univ. S.P. 36 (1987), 21-39. (1987) Zbl0641.47052MR0892378
  11. Papageorgiou N.S., Boundary value problems for evolution inclusions, Comment. Math. Univ. Carolinae 29 (1988), 355-363. (1988) Zbl0696.35074MR0957404
  12. Papageorgiou N.S., On evolution inclusion associated with time dependent convex subdifferentials, Comment. Math. Univ. Carolinae 31 (1990), 517-527. (1990) MR1078486
  13. Pavel N., Nonlinear Evolution Operators and Semigroups, Lecture Notes in Math. 1260, Springer, Berlin, 1987. Zbl0626.35003MR0900380
  14. Tanabe H., Equations in Evolution, Pitman, London, 1979. 
  15. Tolstonogov A., Extreme continuous selectors of multivalued maps and the ``bang-bang'' principle for evolution inclusions, Soviet Math. Doklady 317 (1991), 481-485. (1991) MR1121349
  16. Wagner D., Survey of measurable selection theorems, SIAM J. Control. Optim. 15 (1977), 859-903. (1977) Zbl0407.28006MR0486391
  17. Zecca P., Zezza P., Nonlinear boundary value problems in Banach spaces for multivalued differential equations on a non-compact interval, Nonlinear Anal.-TMA 3 (1979), 347-352. (1979) MR0532895

NotesEmbed ?

top

You must be logged in to post comments.