Smooth control of nanowires by means of a magnetic field

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1 Smooth control of nanowires by means of a magnetic field Gilles Carbou, Stéphane Labbé, Emmanuel Trélat Abstract We address the problem of control of the magnetic moment in a ferromagnetic nanowire by means of a magnetic field. Based on theoretical results for the 1D Landau-Lifschitz equation, we show a robust controllability result. AMS classification: Primary: 58F15, 58F17; Secondary: 53C35. Keywords: Landau-Lifschitz equation, control 1 Model and control result The magnetic moment u of a ferromagnetic material is usually modeled as a unitary vector field, solution of the Landau-Lifschitz equation u t = u H e u u H e ), 1) where the effective field H e is given by H e = u + h d u) + H a. The demagnetizing field h d u) is solution of the magnetostatic equations div B = div H + u) = and curl H =, where B is the magnetic induction. The applied field is denoted by H a see [3, 1, 17, ] for more details on the modelization). Existence results have been established for the Landau-Lifschitz equation in [4, 5, 13, 1], numerical aspects have been investigated in [11, 15, 16], and asymptotic properties have been proved in [1, 6, 1, 18, ]; control issues were addressed in [9]. MAB, UMR 5466, CNRS, Université Bordeaux 1, 351 cours de la Libération, 3345 Talence cedex, France. carbou@math.u-bordeaux1.fr Laboratoire Jean Kuntzmann, Tour IRMA, 51 rue des Mathématiques, B.P. 53, 3841 Grenoble Cedex 9, France. stephane.labbe@imag.fr Université d Orléans, Laboratoire MAPMO, CNRS, UMR 668, Fédération Denis Poisson, FR 964, Bat. Math., BP 6759, 4567 Orléans cedex, France. emmanuel.trelat@univ-orleans.fr 1

2 Here we restrict ourselves to a one dimensional model, i.e., we consider a ferromagnetic nanowire, submitted to an external magnetic field applied along the axis of the wire and which is our control. The model then writes see []) u t = u h δu) u u h δ u)), ) where h δ u) = u x u e u 3 e 3 +δe 1. Here, e 1, e, e 3 ) is the canonical basis of IR 3 and the nanowire is the real axis IRe 1. The magnetic field is written δt)e 1, where the function δ ) is our control. Setting hu) = u xx u e u 3 e 3, this yields u t = u hu) u u hu)) δu e 1 + u u e 1 )). 3) When δ, stationary solutions do exist, of the form M x) = th x 4) 1 ch x and are called Bloch walls. Their stability properties were studied in [7]. When δ ) δ is constant, the solution writes where u δ t, x) = R δt M x + δt), 5) 1 R θ = cosθ sinθ sin θ cosθ is the rotation of angle θ around the axis IRe 1. It corresponds to a rotation plus translation of the above wall along the nanowire. Notice the invariance of 3) through translations x x σ and rotations R θ around the axis e 1. This generates a three-parameters family of particular solutions defined by u δ,θ,σ t, x) = M Λ u δ t, x) = R δt+θ M x + δt σ) 6) called travelling wall profiles. Controlling these walls position plus speed) might be relevant for coding and transporting some information. This is our aim here to derive a controllability result, with an eye on possible applications such as rapid recording. In [9], control properties were proven with piecewise constant controls. However, practical applications require the control to be smooth. Recall that the control here is an external magnetic field applied along the nanowire. The main result of [9] strongly uses the fact that the control is a piecewise constant function and our aim is here to extend this result to the case of smooth controls, hence closer to practical issues.

3 Theorem 1. There exist ε > and δ > such that, for all δ 1, δ IR satisfying δ i δ, i = 1,, for all σ 1, σ IR, for every ε, ε ), there exist T > and a control function δ ) C IR, IR) such that, for every solution u of 3) associated with the control δ ) and satisfying θ 1 IR u, ) u δ1,θ1,σ1, ) H ε, 7) there exists a real number θ such that ut, ) u δ,θ,σ T, ) H ε. 8) Moreover, there exists real numbers θ and σ, with θ θ + σ σ ε, such that ut, ) u δ,θ,σ t, ) H. 9) t + In the proof of the main result, we shall choose control laws δ ) so that { δ1 if t, δt) = δ + σ1 σ t if t T, 1) where T > is large, δ [,T] is a smooth function such that t δ remains small, and the function δ is smooth overall IR. Notice that this control shares robustness properties in H norm. The time T is required to be large enough. It follows from this result that the family of travelling wall profiles 6) is approximately controllable in H norm, locally in δ and globally in σ, in time sufficiently large. Proof of Theorem 1 Similarly as in [7, 8, 9], it is relevant to first reexpress the Landau-Lifschitz equation in adapted coordinates..1 Preliminaries The following formulas, easy to establish, will be useful next: d dθ R θ = sinθ cosθ = R θ+ π e 1e T 1 = R π R θ e 1 e T 1 ; cosθ sinθ v e 1 = R π v + v 1e 1 ; R θ u R θ v = R θ u v); a b c) = ba.c) ca.b); R θ IRe 1 ) = IRe 1. 3

4 It is clear from Equation ) that the solution u has a constant norm. Up to normalizing, assume this norm is equal to 1. Set vt, x) = R δt)t ut, x δt)t)); then, v has a constant norm too, equal to 1. Using the above formulas, computations lead to v t = v hv) v v hv)) δv x + v 1 v e 1 ) t δv x v 3 e + v e 3 ), 11) where we recall that hv) = v xx v e v 3 e 3. Define M 1 x) = 1 chx th x and M = 1. In the frame M x), M 1 x), M ), the solution v : IR + IR S IR 3 writes in the form vt, x) = 1 r 1 t, x) r t, x) M x) + r 1 t, x)m 1 x) + r t, x)m. Note that: M x) = 1 ch x M 1x), M 1x) = 1 chx M, M x) = sh x ch x M 1x) 1 ch x M ; e 1 = thx M + 1 ch x M 1x), e = M, e 3 = 1 chx M th x M 1 x); hm ) = ch x M ; M M 1 = M, M M = M 1, M 1 M = M ; Then, easy but lengthy computations, ) not reported here, show that v is solution r1 of 11) if and only if r = satisfies r where Rt, δ, δ, x, r, r x, r xx ) = δ and A = r t = Ar + Rt, δ, δ, x, r, r x, r xx ), 1) ) l r + Gr)r l xx + H 1 x, r)r x + H r)r x, r x ) + Px, r) δbx, r) t δcx, r), ) L L with L = L L xx + 1 th x)id; 13) 4

5 l = x + thxid; Gr) is the matrix defined by r 1 r r + 1 r 1 Gr) = 1 r 1 r r1 1 r + 1 r 1r ; 1 r 1 r H 1 x, r) is the matrix defined by H 1 x, r) = r 1 r r 1 r r + r r1 1 r chx ; r r 3 1 r r + r 1 r H r) is the quadratic form on IR defined by H r)x, X) = 1 r )X T X + r T X) 1 r r 1 + r 1 r ) 3/ ; 1 r r r 1 Px, r) = P 1 x, r), with P x, r) and Px, r) =r 1 r 1 1) ch x r sh x 1r ch x r 1 r 1 ch x r1 1 r sh x ch x + r3 1 + r 1 1 r ) + r 1 r, P x, r) = r 1 1 r 1 1) ch x + shx r 1 ch x r r 1 ch x r 1 r 1 r shx ch x + r r, Bx, r) = x + thx)r r 1 + r1 ch x r 1 r 1)r, Cx, r) = x + th x )) 1 1 r 1 r +. 1 chx 1) ) + thx 1 r 5

6 It is clear that there holds Gr) = O r ), H 1 x, r) = O r ), H r) = O r ), Px, r) = O r ), Bx, r) = O r + r x ), Cx, r) = O r + r x ), uniformly with respect to the variable x IR. Then, we infer that there exists a constant C > such that, if r IR = r 1 and δ 1, then, for all p, q IR, for all x, t, ε IR, Rt, δ, ε, x, r,p, q) IR C δ r IR + δ p IR + t ε + t ε p IR 14) + r IR q IR + r IR p IR + r IR p IR + r IR ). From this a priori estimate, one might consider Rt, δ, δ, x, r, r x, r xx ) as a remainder term in Equation 1). The proof uses stability properties established for the linear operator A, so as to establish. We next follow the same lines as in [9].. Change of coordinates The operator L is a self-adjoint operator on L IR), of domain H IR), and L = l l with l = x + thxid one has l = x + thxid). It follows that L is nonpositive, and that kerl = kerl is the one dimensional subspace of L IR) 1 generated by ch x. In particular, the operator L, restricted to the subspace E = kerl), is negative. Remark 1. On the subspace E: the norms L) 1/ f L IR) and f H 1 IR) are equivalent; the norms Lf L IR) and f H IR) are equivalent; the norms L) 3/ f L IR) and f H 3 IR) are equivalent. Writing A = JL, with J = ) 1 1, 1 1 it is clear that the kernel of A is kera = kerl kerl; it is the two dimensional space of L IR ) generated by ) 1 ) a 1 x) = 1 and a x) = ch x. ch x 6

7 Moreover, combining the facts that L ker L) is negative and that Spec J = {1+ i, 1 i}, it follows that the operator A, restricted to the subspace E = kera), is negative. In what follows, solutions r of 1) are written as the sum of an element of ker A and of an element of E. Since Equation 11) is invariant with respect to translations in x and rotations around the axis e 1, for every Λ = θ, σ) IR, M Λ x) = R θ M x σ) is solution of 11). Define R Λ x) = MΛ x), M 1 x) M Λ x), M ), the coordinates of M Λ x) in the mobile frame M 1 x), M x)). The mapping Ψ : IR E H IR) Λ, W) rx) = R Λ x) + Wx) is a diffeomorphism from a neighborhood U of zero in IR E into a neighborhood V of zero in H IR). Indeed, if r = R Λ + W with W E, then, by definition, r, a 1 L = R Λ, a 1 L and r, a L = R Λ, a L. 15) Conversely, if Λ IR satisfies 15)), then W = r R Λ E. The mapping h : IR IR, defined by hλ) = R Λ, a 1 L, R Λ, a L ) is smooth and satisfies dh) = Id, thus is a local diffeomorphism at, ). It follows easily that Ψ is a local diffeomorphism at zero. Therefore, every solution r of 1), as long as it stays 1 in the neighborhood V, can be written as rt, ) = R Λt) ) + Wt, ), 16) where Wt, ) E and Λt) IR, for every t, and Λt), Wt, )) U. In these new coordinates, Equation 1) leads to see [7] for the details of computations) W t t, x) = AWt, x) + Rt, δ, ε, Λt), x, Wt, x), W x t, x), W xx t, x)), Λ t) = MΛt), Wt, ), W x t, )), 17) where R : IR IR IR IR IR H IR) ) H 1 IR) ) L IR) ) E and M : IR H 1 IR) ) L IR) ) IR are nonlinear mappings, for which there exist constants K > and η > such that Rt, δ, ε, Λ,, W, W x, W xx ) H 1 IR)) ) K Λ IR + δ + t ε + W H IR)) W H3 IR)) + Kt ε, 18) 1 This a priori estimate will be a consequence of the stability property derived next. This decomposition is actually quite standard and has been used e.g. in [14] to establish stability properties of static solutions of semilinear parabolic equations, and in [, 19] to prove stability of travelling waves. 7

8 ) MΛ, W, W x ) K Λ IR + W H 1 IR)) W H 1 IR)), 19) for every W E, every δ IR, every t, and every Λ IR satisfying Λ IR η. Note that, since L is selfadjoint, it follows that AW E, for every W E, and thus 17) makes sense..3 Asymptotic estimates ) W1 Denoting W =, define on H W IR) ) IR the function VW) = 1 ) L W L = 1 L IR)) LW 1 L IR) + 1 LW L IR). ) Remark. It follows from Remark 1 that, on the subspace E = kera), VW) is a norm, which is equivalent to the norm W H IR )). Consider a solution W, Λ) of 17), such that W, ) = W ) and Λ) = Λ. Since L is selfadjoint, one has d L VWt, )) = AW, W 1 dt L W ) L IR)) ) ) L) 1/ L) + L) 1/ Rt, δ, ε, Λ,, W, W x, W xx ), 3/ W 1 L) 3/. W L IR)) 1) Concerning the first term of the right-hand side of 1), one computes ) L AW, W 1 L = L) 3/ W W 1 L IR)) L)3/ W L IR)), L IR )) and, using Remark 1, there exists a constant C 1 > such that ) L AW, W 1 L C W 1 W H 3 IR)). ) L IR)) Concerning the second term of the right-hand side of 1), one deduces from the Cauchy-Schwarz inequality, from Remark 1, and from the estimate 18), that ) ) L) 1/ L) L) 1/ Rt, δ, ε, Λ,, W, W x, W xx ), 3/ W 1 L) 3/ W L IR)) Rt, δ, ε, Λ,, W, W x, W xx ) H 1 IR)) W H 3 IR)) ) K Λ IR + δ + t ε + W H IR)) W H 3 IR)) + t ε W H 3 IR)) K Λ IR + δ + t ε + W H IR)) + 1 ) ξ ξ W + H3IR)) t ε, 3) 8

9 where, to get the last line, we used the inequality d dt VW) t ε W H3 IR)) ξ t ε + 1 ξ W H 3 IR)) ; here, ξ denotes some real number to be chosen later. One infers from 1), ) and 3) that C 1 +K + ξ t δ. Λ IR + δ + t δ + W H IR)) + 1 ξ Fix ǫ > ; then, under the a priori estimates Λt) IR + δ + t δ + Wt, ) H IR)) + 1 ξ C 1 K )) W H 3 IR)) and there holds ξ t δ ǫ, d dt VWt, )) C 1 Wt, ) H 3 IR)) + ǫ C 1 Wt, ) H IR)) + ǫ C VWt, )) + ǫ. The existence of a constant C > follows from Remark. Therefore, choosing ξ > large enough, there exist constants C 3 > and C 4 > such that, if W, ) H IR)) C1 6K, if the a priori estimate max Λs) IR s t C 1 6K 4) holds, and if the control function δ ) is chosen so that and for every t, then δt) + t δt) C 1 6K 5) t δt) ǫ/ξ 6) Ws, ) H IR)) C 3e C4s W, ) H IR)) + C 3ǫ, 7) 9

10 for every s [, T], and moreover, one deduces from 17), 19), and 7) that, if the a priori estimate 4) holds, then Λt) IR Λ) IR + C 1C 3 4 t W, ) H IR)) e C4s ds t + KC3 W, ) H IR)) e C4s ds Λ) IR + C 1C 3 4C 4 W, ) H IR)) + K C 3 C 4 W, ) H IR)). 8) From the above a priori estimates, we infer that, if the quantity Λ) IR + W, ) H IR)) is small enough, and if the control function δ fits the conditions 5) and 6), then Λt) IR remains small, for every t, and Wt, ) H IR)) is exponentially decreasing to. Finally we must choose a smooth control function such that ut, x) is close to u δ1,θ1,σ1 t, x) at initial time, and close to u δ,θ,σ t, x) for large times. Hence, we can choose the function δ such that δt) = δ 1 for t. Then, with the reasoning above, we enforce vt, x) to remain close to M x), that is, the solution ut, x) follows the profile u δt),θ1,σ1 t, x). At times t T, we require ut, x) to be close to u δ,θ,σ t, x) for some θ ; one must have, for t T, and hence, σ 1 + δt)t = σ + δ t, δt) = δ + σ 1 σ. t To conclude, observe that it is possible to choose a function δ and a time T > large enough, such that δ is smooth on IR and satisfies the above requirements and the estimates 5) and 6). The first part of the theorem, on the interval [, T], then follows from the above considerations. For the second part, we use a stronger version of the estimate 7), namely, Ws, ) H IR)) C 3e C4s W, ) H IR)) + ξ t δt). Since t δt) is integrable, it follows from the above estimate, and from 17) and 19), that Λ t) IR is integrable on [, + ). Hence, Λt) has a limit in IR, denoted Λ = θ, σ ), as t tends to +. The theorem follows with θ = θ + θ and σ = σ + σ. References [1] François Alouges, Tristan Rivière, and Sylvia Serfaty. Néel and cross-tie wall energies for planar micromagnetic configurations. ESAIM Cont. Optim. Calc. Var. 8 ),

11 [] Andrea L. Bertozzi, Andreas Münch, Michael Shearer, and Kevin Zumbrun. Stability of compressive and undercompressive thin film travelling waves. European J. Appl. Math., 13) 1), [3] F. Brown. Micromagnetics. Wiley, New York 1963). [4] Gilles Carbou and Pierre Fabrie. Time average in micromagnetism. J. Differential Equations, 147 ), ). [5] Gilles Carbou and Pierre Fabrie. Regular solutions for Landau-Lifschitz equation in a bounded domain. Differential Integral Equations, 14 ), ). [6] Gilles Carbou, Pierre Fabrie and Olivier Guès. On the ferromagnetism equations in the non static case. Comm. Pure Appli. Anal., 3, ). [7] Gilles Carbou and Stéphane Labbé. Stability for static walls in ferromagnetic nanowires. Discrete Contin. Dyn. Syst. Ser. B 6, 6), [8] Gilles Carbou and Stéphane Labbé. Stability for walls in Ferromagnetic Nanowires. Numerical Mathematics and Advanced Applications: Proceedings of ENUMATH 5, the 6th European Conference on Numerical Mathematics and Advanced Applications, Santiago de Compostela, Spain, July 5 Hardcover), Springer 6). [9] Gilles Carbou, Stéphane Labbé, and Emmanuel Trélat. Control of travelling walls in a ferromagnetic nanowire. Discrete Contin. Dyn. Syst. 7), suppl. [1] Antonio DeSimone, Robert V. Kohn, Stefan Müller, and Felix Otto. Magnetic microstructures a paradigm of multiscale problems. In ICIAM 99 Edinburgh), Oxford Univ. Press, Oxford, ). [11] Houssem Haddar and Patrick Joly. Stability of thin layer approximation of electromagnetic waves scattering by linear and nonlinear coatings. J. Comput. Appl. Math., 143, 1 36 ) [1] Laurence Halpern and Stéphane Labbé. Modélisation et simulation du comportement des matériaux ferromagétiques. Matapli, 66, ). [13] J.-L. Joly, G. Métivier, J. Rauch. Global solutions to Maxwell equations in a ferromagnetic medium. Ann. Henri Poincaré, 1, 37-34, ). [14] Todd Kapitula. Multidimensional stability of planar travelling waves. Trans. Amer. Math. Soc., 349 1), ). [15] S. Labbé. Simulation numérique du comportement hyperfréquence des matériaux ferromagnétiques. Thèse de l Université Paris ). [16] Stéphane Labbé and Pierre-Yves Bertin. Microwave polarisability of ferrite particles with non-uniform magnetization. Journal of Magnetism and Magnetic Materials, 6, ). 11

12 [17] L. Landau et E. Lifschitz. Electrodynamique des milieux continues. cours de physique théorique, tome VIII ed. Mir) Moscou 1969). [18] Tristan Rivière and Sylvia Serfaty. Compactness, kinetic formulation, and entropies for a problem related to micromagnetics. Comm. Partial Differential Equations, 8 1-), ). [19] V. Roussier. Stability of radially symmetric travelling waves in reactiondiffusion equations. Ann. Inst. H. Poincaré Anal. Non Linéaire, 13) 4), [] D. Sanchez. Behaviour of the Landau-Lifschitz equation in a ferromagnetic wire. preprint MAB 5). [1] A. Visintin. On Landau Lifschitz equation for ferromagnetism. Japan Journal of Applied Mathematics, 1, ). [] H. Wynled. Ferromagnetism. Encyclopedia of Physics, Vol. XVIII /, Springer Verlag, Berlin 1966). 1