Abstract
We consider a nonlinear partial differential control system describing phase transitions taking account of hysteresis effects. The control constraint is given by a multivalued mapping with nonconvex closed bounded values in a finite dimensional space depending on the phase variables. Existence of solutions and topological properties of the set of admissible “trajectory-control” pairs are discussed in detail.
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Krasnosel’skii, M.A., Pokrovskii, A.V.: Systems with Hysteresis. Nauka, Moscow (1983) (English edition Springer 1989)
Colli P., Kenmochi N., Kubo M.: A phase field model with temperature dependent constraint. J. Math. Anal. Appl. 256, 668–685 (2001)
Kenmochi N., Minchev E., Okazaki T.: On a system of nonlinear PDE’s with diffusion and hysteresis effects. Adv. Math. Sci. Appl. 14, 633–664 (2004)
Visintin, A.: Differential Models of Hysteresis. Springer, Berlin (1994)
Visintin, A.: Models of Phase Transitions. Birkhäuser, Boston (1996)
Krejčí P., Sprekels J.: A hysteresis approach to phase-field models. Nonlinear Anal. 39(5), 569–586 (2000)
Krejčí P., Sprekels J., Stefanelli U.: Phase-field models with hysteresis in one-dimensional thermoviscoplasticity. SIAM J. Math. Anal. 34, 409–434 (2002)
Brokate, M., Sprekels, J.: Hysteresis and Phase Transitions. Applied Mathematical Sciences, vol. 121. Springer, New York (1996)
Kenmochi N., Visintin A.: Asymptotic stability for nonlinear PDEs with hysteresis. Eur. J. Appl. Math. 5(1), 39–56 (1994)
Minchev E., Okazaki T., Kenmochi N.: Ordinary differential systems describing hysteresis effects and numerical simulations. Abstr. Appl. Anal. 7(11), 563–583 (2002)
Okazaki T.: Large time behaviour of solutions of nonlinear ordinary differential systems with hysteresis effect. Adv. Math. Sci. Appl. 14(1), 211–239 (2004)
Yamazaki N., Takahashi M., Kubo M.: Global attractors of phase transition models with hysteresis and diffusion effects. GAKUTO Int. Ser. Math. Sci. Appl. 14, 460–471 (2000)
Hoffmann K.H., Kenmochi N., Kubo M., Yamazaki N.: Optimal control problems for models of phase-field type with hysteresis of play operator. Adv. Math. Sci. Appl. 17(1), 305–336 (2007)
Timoshin S.A., Tolstonogov A.A.: Existence and properties of solutions of a control system with hysteresis effect. Nonlinear Anal. 74(13), 4433–4447 (2011)
Timoshin S.A., Tolstonogov A.A.: Bogolyubov-type theorem with constraints induced by a control system with hysteresis effect. Nonlinear Anal. 75(15), 5884–5893 (2012)
Kuratowski, C.: Topologie. I et II. Editions Jacques Gabay, Sceaux (1992)
Brezis, H.: Operateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert. Elsevier/North-Holland/Amsterdam, New York/London (1973)
Himmelberg C.J.: Measurable relations. Fundam. Math. 87, 53–72 (1975)
Tolstonogov A.A., Tolstonogov D.A.: L p -continuous extreme selectors of multifunctions with decomposable values: existence theorems. Set Valued Anal. 4, 173–203 (1996)
Hilpert, M.: On uniqueness for evolution problems with hysteresis. In: Rodrigues, J.F. (ed.) Mathematical Models for Phase Change Problems, pp. 377–388. Birkhäuser, Basel (1989)
Barbu, V.: Partial Differential Equations and Boundary Value Problems. Kluwer, Dordrecht (1988)
Tolstonogov A.A.: L p -continuous selections of fixed points of multifunctions with decomposable values. III: applications. Sib. Math. J. 40(6), 1380–1396 (1999)
Tolstonogov A.A.: On solutions of an evolution control system that depends on a parameter. Sb. Math. 194(9), 1383–1409 (2003)
Tolstonogov A.A.: L p -continuous selections of fixed points of multifunctions with decomposable values. I: existence theorems. Sib. Math. J. 40(3), 595–607 (1999)
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The research of PK was supported by GAČR Grant P201/10/2315 and RVO: 67985840, the research of AT and ST was supported by RFBR Grant no. 13-01-00287. The support of the Russian Academy of Sciences during the stay of PK in Irkutsk is gratefully acknowledged.
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Krejčí, P., Tolstonogov, A.A. & Timoshin, S.A. A control problem in phase transition modeling. Nonlinear Differ. Equ. Appl. 22, 513–542 (2015). https://doi.org/10.1007/s00030-014-0294-x
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DOI: https://doi.org/10.1007/s00030-014-0294-x