Control of a network of magnetic ellipsoidal samples

Abstract

International audienceIn this work,we present a mathematical study of stability and controllability of one-dimensional network of ferromagnetic particles. The control is the magnetic field generated by a dipole whose position and whose amplitude can be selected. The evolution of the magnetic field in the network of particules is described by the Landau-Lifschitz equation. First, we model a network of ellipsoidal shape ferromagnetic particles. Then, we prove the stability of relevant configurations and discuss the controllability by the means of the external magnetic field induced by the magnetic dipole. Finally some numerical results illustrate the stability and the controllability results

Optimal Shape of Isolated Ferromagnetic Domains

“…In many papers related to our setting, a perfect ellipsoid shape is assumed [1,2,4,7,10,11,22]. This is a common simplification due to the fact that the self-demagnetization of ellipsoids can be treated analytically.…”

“…The assumption of bounded support in Theorem 2.2 is used in the proof of Lemma 3.1 in order to assure the boundedness of the second moment of u ǫ (in x ′ −direction) and in the proof of Theorem 2.2 for the conclusion that no mass is lost in the limit process (in direction of large |x 1 |). For Theorem 2.2, the assumption seems to be necessary: Indeed, consider two disjoint needles of mass 1 2 , with optimal cross-section and with increasing distance in lateral direction as ǫ → 0. We then have the uniform bound E[u ǫ ] 1, however, we do not have compactness of the sequence u ǫ in any L p -space.…”

“…for some constant µ > 0, thus minimizing (3) can be reduced to (1). In [41] it has been shown that minimizers of our energy functional (1) with volume constraint (4) exist in all dimensions n and for all prescribed masses µ ≥ 0. For 2 ≤ n ≤ 7, (regular and local) minimizer are open and bounded with smooth boundary.…”

“…1 L * A(t) is the minimizer among the class of functions with support I L * . We haveA L * (τ ) = √ 2 √ πL * |τ | J 1 (L * |τ |)[24, p.11 (8)], where J 1 is the Bessel function of the…”

Control and stabilization of steady-states in a finite-length ferromagnetic nanowire

“…and thus this energy functional is naturally nonincreasing along a solution of (1). Since the two terms at the right-hand side of (1) are orthogonal, every steady-state of (1) must satisfy m ∧ h(m) = 0, and accordingly the set of steady-states coincides with the set of extremal points of the energy E.…”