Discussiones Mathematicae Differential Inclusions 16(1) (1996) 5-41

Yuncheng You

*Department of Mathematics, University of South
Florida
Tampa, FL 33620-5700, USA*

Slightly below the transition temperatures, the behavior of superconducting materials is governed by the Ginzburg-Landau (GL) equation which characterizes the dynamical interaction of the density of superconducting electron pairs and the exited electromagnetic potential. In this paper, an optimal control problem of the strength of external magnetic field for one-dimensional thin film superconductors with respect to a convex criterion functional is considered. It is formulated as a nonlinear coefficient control problem in Hilbert spaces. The global existence and regularity of solutions of the state equation, the existence of optimal control, and the maximum principle as a necessary condition satisfied by optimal control are proved. By proving the local Lipschitz continuity of the value functions and by using lower Dini derivatives, an optimal synthesis (i.e. optimal feedback control) is obtained via solving a differential inclusion.

**Keywords:** Ginzburg-Landau equation, optimal control, differential inclusion,
superconductivity.

**1991 Mathematics Subject Classification:** 35B27, 35K55, 35Q55, 49J20, 49N35,
82D55.

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