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

[BIBTex]

OPTIMAL FEEDBACK CONTROL OF GINZBURG-LANDAU EQUATION FOR SUPERCONDUCTIVITY VIA DIFFERENTIAL INCLUSION

Yuncheng You

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

Abstract

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.

References

[1] V. Barbu, Optimal feedback controls for a class of nonlinear distributed parameter systems, SIAM J. Control and Optim. 21 (1983), 871-894.
[2] V. Barbu and Da Prato, Hamilton-Jacobi equations and synthesis of nonlinear control processes in Hilbert spaces, J. Differential Equations 48 (1983), 350-372.
[3] M.S. Berger and Y.Y. Chen, Symmetric vortices for the Ginzburg-Landau equations and the nonlinear desingularization phenomenon, J. Functional Analysis, (1989).
[4] L.D. Berkovitz, Optimal feedback controls, SIAM J. Control and Optim. 27 (1989), 991-1007.
[5] P. Blennerhassett, On the generation of waves by wind, Philos. Trans. Roy. Soc. London, Ser. A. 298 (1980), 451-494.
[6] H. Brezis, ``Analyse Fonctionnelle: Theorie et Applications'', Masson, Paris 1987.
[7] H.O. Fattorini, A unified theory of necessary conditions for nonlinear nonconvex control systems, Appl. Math. and Optim. 15 (1987), 141-185.
[8] V.L. Ginzburg and L.D. Landau, Concerning the theory of superconductivity, Soviet Physics JETP 20 (1950), 1064-1082.
[9] L.P. Gor'kov, Microscopic derivation of the Ginzburg-Landau equations in the theory of superconductivity, Soviet Physics JETP 36 (1959), 1918-1923.
[10] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer, New York 1981.
[11] S. Hou, Implicit function theorem in topological spaces, Applicable Analysis 13 (1982), 209-217.
[12] L. Jacobs and C. Rebbi, Interaction energy of superconducting vortices, Physics Rev. B 19 (1978), 4486-4494.
[13] Y. Kuramoto and T. Tsuzuki, On the formation of dissipative structures in reaction-diffusion systems, Progr. Theoret. Physics Suppl. 54 (1975), 687-699.
[14] Y. Lin and Y. Yang, Computation of superconductivity in thin films, IMA Preprint, Series #541, 1989.
[15] H.T. Moon, P. Huerre, and L.G. Redekopp, Transitions to chaos in the Ginzburg-Landau equation, Physica D, 7 (1983), 135-150.
[16] A. Pazy, ``Semigroups of Linear Operators and Applications to Partial Differential Equations'', Springer, New York 1983.
[17] J.D.L. Rowland and R.B. Vinter, Construction of optimal feedback controls, Systems and Control Letters 16 (1991), 357-367.
[18] Y. Yang, The Ginzburg-Landau equations for superconducting film and Meissner effect, J. Math. Phys. 31 (1990), 1284-1289.
[19] Y. You, A nonquadratic Bolza problem and a quasi-Riccati equation for distributed parameter systems, SIAM J. Control and Optim. 25 (1987), 904-920.
[20] Y. You, Nonlinear optimal control and synthesis of thermal nuclear reactors, in ``Distributed Parameter Control Systems: New Trends and Applications'', (G. Chen, et al, Ed.), 445-474, Marcel Dekker, New York 1991.