EDS EDP. Ferromagnetism. Some control considerations for ferromagnetic materials. Assymptotic. Groupe de Travail Contrôle, LJLL 13 février 2015

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1 Some control considerations for ferromagnetic materials EDP Itô Γ-convergence Hysteresis Assymptotic EDS Ferromagnetism Micromagnetism Control Magnetostatic Multi-scales Monte-Carlo Temperature Stéphane Labbé Joseph Fourier University, Jean Kuntzmann Laboratory. Groupe de Travail Contrôle, LJLL 13 février 2015

2 Plan Modeling of ferromagnetic materials Physical principles Modeling of ferromagnetic materials Physical principles The microscopic scale The micromagnetic model Link between scales One particle The stochastic model Results Illustrations Which noise for which model? Net of particles The deterministic model Stable configurations Control The sotchastic case

3 Ferromagnetic materials Modeling of ferromagnetic materials Physical principles Characterization Remanent magnetization. A critical temperatures separating linear and non-linear behavior. Microstructures and domain formation.

4 Ferromagnetic materials Modeling of ferromagnetic materials Physical principles Aimantation rémanente Comportement non linéaire Comportement linéaire Température de Curie Characterization Remanent magnetization. A critical temperatures separating linear and non-linear behavior. Microstructures and domain formation. Température

5 Ferromagnetic materials Modeling of ferromagnetic materials Physical principles Characterization Remanent magnetization. A critical temperatures separating linear and non-linear behavior. Microstructures and domain formation. Ste phane Labbe (Joseph Fourier University), Ferromagnetism,

6 Ferromagnetic materials Modeling of ferromagnetic materials Physical principles Characterization Remanent magnetization. A critical temperatures separating linear and non-linear behavior. Microstructures and domain formation.

7 Several scales Modeling of ferromagnetic materials Physical principles

8 Analogical and numerical experiments Modeling of ferromagnetic materials Physical principles

9 Not so easy to control... Modeling of ferromagnetic materials Physical principles

10 Several scales Modeling of ferromagnetic materials Physical principles atomic scale: every thing visible, modeling via DFT. microscopic scale: atom kernels are assimilated to punctual charges bearing a magnetic moment Mescopic scale: matter is modeled as continuous, ferromagnetic behavior are observable. Macroscopic scale: ferromagnetic behaviors are no more observable.

11 Several scales Modeling of ferromagnetic materials Physical principles atomic scale: characteristic time of the spin-orbit interactions Larmor scale: characteristic of the magnetic moment precession dynamic scale: characteristic time of the equilibrium relaxation adiabatic scale: slow dynamic of equilibrium states

12 Plan Modeling of ferromagnetic materials The microscopic scale Modeling of ferromagnetic materials Physical principles The microscopic scale The micromagnetic model Link between scales One particle The stochastic model Results Illustrations Which noise for which model? Net of particles The deterministic model Stable configurations Control The sotchastic case

13 Microscopic model Modeling of ferromagnetic materials The microscopic scale Given R the periodic net Z 3, we set R ε = εr and R ε,ω = R ε Ω. Static Atom kernels are seted in a cristalline configuration : R ε,ω, Magnetic moment beared by dirach measures : (µ x ) x Rε,Ω, avec µ = µ x δ x, x R ε,ω Electromagnetic interactions: Maxwell system in void. Heisenberg interaction energy: E Heis (µ) = A µ x µ y. x R ε,ω,y V(x)

14 Microscopic model Modeling of ferromagnetic materials The microscopic scale Dynamic Quasi instantaneous minimization of the local Heisenberg energy. Larmor precesion : x R ε,ω, dµ x dt = γµ x H(µ)(x), where H(µ)(x) is the electromagnetic files at point x generated by µ augmented by the external field. Heisenberg interactions can be modeled by introduction of an heuristic dissipation term.

15 Plan Modeling of ferromagnetic materials The micromagnetic model Modeling of ferromagnetic materials Physical principles The microscopic scale The micromagnetic model Link between scales One particle The stochastic model Results Illustrations Which noise for which model? Net of particles The deterministic model Stable configurations Control The sotchastic case

16 Basis of micromagnetism (1) Modeling of ferromagnetic materials The micromagnetic model Thermodynamical description of ferromagnetic materials: Micromagnetism, W.F. BROWN, in the 60 s. Notations Magnetic domain: Ω, open in R 3 Unit sphere: S 2 Magnetization: m, vector fields of Ω, whose values are in S 2 Energy functional: E defined on H 1 (Ω, R 3 ) with values in R Equilibrium State: element of H 1 (Ω, S 2 ) which minimizes E

17 Basis of micromagnetism (2) Modeling of ferromagnetic materials The micromagnetic model Energy functional E E : H 1 (Ω, S 2 ) R is defined by E(m)= A 2 Ω m H d (m) 2 m H ext 2 R 3 Ω H d (m): Demagnetizing field solution of the following equation (where m is an extension of m by 0 in R 3 ): { curl(hd ) = 0 in D (R 3, R 3 ) div(h d ) = div( m) in D (R 3, R 3 ). H ext : Zeeman, models the action of an external field (independant of m).

18 Basis of micro micromagnetism Modeling of ferromagnetic materials The micromagnetic model Dynamic: the Landau-Lifchitz system. Landau et Lifchitz m = m H(m) αm (m H(m)), t H(m) is the effective field. H(m) = de, with E(m) = e(m)dx. dm Ω Hypothesis: ϕ(m) = K 2 ( m 2 (m u) 2 ) where u is in L (R 3, S 2 ). Some remarks: Champ effectif H(m) = A m+h d (m)+k (m.u)u+h ext, Equilibrium states verify H(m) m 0,Ω = 0. For an autonomous external field, the energy of system s solutions is decreasing. The magnetization modulus is preserved overall the domain.

19 Plan Modeling of ferromagnetic materials Link between scales Modeling of ferromagnetic materials Physical principles The microscopic scale The micromagnetic model Link between scales One particle The stochastic model Results Illustrations Which noise for which model? Net of particles The deterministic model Stable configurations Control The sotchastic case

20 The static case Modeling of ferromagnetic materials Link between scales µ, minimizer of the energy under constraint E Heis (µ) + R 3 H d (µ) 2 dx Ω m, minimizer of the energy under constraint Ã Ω m 2 dx + R 3 H d (m) 2 dx Γ-converge, H 1 (Ω, R 3 ) ε 0 m H 1 (Ω; S 2 ). ε R ε,ω = εz d Ω B. Bidégaray, Q. Jouet and S. Labbé, Static ferromagnetic materials: from the microscopic to the mesoscopic scale, Communications in Contemporary Mathematics, 2013.

21 The dynamic case Modeling of ferromagnetic materials Link between scales Development of a particular notion of Gamma Convergence: convergence of trajectories. Static results induces dynamical results: notion of gradient flows (see for example works by E. Sandier and S. Serfaty in the Ginzburg Landau context). Work in progress with H. Pajot and B. Rufini.

22 Plan One particle The stochastic model Modeling of ferromagnetic materials Physical principles The microscopic scale The micromagnetic model Link between scales One particle The stochastic model Results Illustrations Which noise for which model? Net of particles The deterministic model Stable configurations Control The sotchastic case

23 Problem One particle The stochastic model Goal: to point out temperature like effects. Tool: introduction of a stochastic perturbation. What we want to emphazise : quantifiable thermal effects, hysteric phenomenas. The deterministic model: Find µ(t), de R + in S 2, such that dµ dt = µ H ext αµ (µ H ext ), The question: how to introduce the stochastic perturbation in the system preserving its structural properties? P. Étoré, S. Labbé et J. Lelong, Hysteretic behavior of a stochastic nano particle, Journal of Differential Equations, 2014.

24 The model: the studied EDS One particle The stochastic model Given (Ω, F, (F t ) t 0, P) a filtered probabilist space and W standard Brownian process adapted to the filtration (F t ) t 0 and valued in IR 3. We set dy t = µ t (b dt + ε dw t ) αµ t µ t (b dt + ε dw t ) µ t = Y t Y t Y 0 = y S 2, We remark that the process µ t preserves the structural properties of the deterministic model: preservation of the modulus in time, combination of to movements who, in the determinist case, are deriving from an hamiltonian dynamic of one and purely dissipative one for the other.

25 Plan One particle Results Modeling of ferromagnetic materials Physical principles The microscopic scale The micromagnetic model Link between scales One particle The stochastic model Results Illustrations Which noise for which model? Net of particles The deterministic model Stable configurations Control The sotchastic case

26 Some properties One particle Results Proposition When the pari of processes (Y, µ) is solution of the EDS we have d Y t 2 = 2ε 2 (α 2 + 1)dt so, Y t is deterministic. Comes from the Itô formula. We introduce h(t) = Y t h(t) = 2ε 2 (α 2 + 1)t + 1. Then, we are able to define an EDS for µ t b d(µ t b) = (µ t b) h (t) h(t) dt where L(x) is the edge by x operator. ε h(t) α ( (µ t b) 2 b 2) dt h(t) ( L(µt )b + α((µ t b)µ t b) ) dw t

27 Long time behavior One particle Results The EDS for µ t.b allows us to determine the long time behavior of the main EDS. Lemma t 1 ( sup µu b + α((µ u b)µ u b) ) dw u < a.s. t 0 h(u) This lemma is based on the Doob inequality for martingales and gives access to the proof of the theorem on long time behavior of the solutions Theorem µ t b b p.s. t

28 Convergence speed One particle Results We obtain a convergence rate for the L 1 norm Theorem lim E(h(t) b µ t b ) = ε2 (1 + α 2 ). t 2α We also prove the following corollary based upon the Markov inequality Corollary For all 0 < β < 1/2 and η > 0, P(t β ( b µ t b) η) 0.

29 The hysteresis model One particle Results Applied external field obeying to a slow dynamic b η (t) = (1 2t η) b t 1/η b(t) = (1 2t) b t 1 EDS for the processes Y t and µ t : dy η t = µ η t (b η (t) dt + ε dw t ) αµ η t µ η t (b η (t) dt + ε dw t ) µ η t = Y η t Y η 0 = b Y η t Then, we focus on the process Z η t = Y η t which is re-scaled in time. We set λ η t = Z η t Z η η t.

30 The hysteresis phenomena One particle Results We prove that Theorem t [0, 1 2 ], E(λη t This means that the trajectory of the process λ η t vanishing, which is an hysteric behavior. 1 b) 1 + ε2 (1+α 2 ) η b in expectation goes over 0 strictly when b is

31 Plan One particle Illustrations Modeling of ferromagnetic materials Physical principles The microscopic scale The micromagnetic model Link between scales One particle The stochastic model Results Illustrations Which noise for which model? Net of particles The deterministic model Stable configurations Control The sotchastic case

32 Almost sure convergence of µ t b for µ 0 = b, b = 1, ε = alpha=1.0 alpha=0.7 alpha=

33 2α 2 Convergence of t E( b µt.b) pour µ 0 = b, b = 1 et ε = 0.1. dashed line: level at 1. ε (1+α 2 ) The expectation is computed via a Monte-Carlo method using 100 samples. 50 alpha=1.0 alpha=0.7 alpha=

34 For α = 1, ε = 0.01 et η =

35 Zoom near t =

36 For α = 1, ε = et η =

37 Plan One particle Which noise for which model? Modeling of ferromagnetic materials Physical principles The microscopic scale The micromagnetic model Link between scales One particle The stochastic model Results Illustrations Which noise for which model? Net of particles The deterministic model Stable configurations Control The sotchastic case

38 Stratonovich versus Itô? Stratonovich versus Itô? One particle Which noise for which model? In the case of the Stratonovich integral, equal to the It one up to a finite variation process, the system is written as follows µ t = µ t (b t + ε W t ) α µ t µ t (b t + ε W t ), µ 0 S(R 3 ). Then, this Stratonovich integral is transformed into an It integral 1 : Then, µ t is solution of the Itô EDS: d µ t = A( µ t )dt + εa( µ t )dw t ε2 3 q=1 j=1 3 (A jq D j (A iq ))( µ t ) dt who implies d µ t = (A( µ t ) + ε 2 (α 2 + 1) µ t )dt + εa( µ t )dw t. d( µ t b) µt = b = ε 2 (α 2 + 1) b dt, b 1 voir Rogers and Williams, V.30

39 Stratonovich versus Itô? Stratonovich versus Itô? One particle Which noise for which model? Then, we have E[ µ t b] = α( b 2 E[( µ t b) 2 ]) ε 2 (α 2 + 1)E[ µ t b] E[ µ t b] E[ µ 0 b] e ε2 (α 2 +1)t = e ε2 (α 2 +1)t lim sup E[ µ t b] = lim inf t + t + e ε2 (α 2 +1)t α lim sup t + and similarly, we can show that E[ µ t b] ε 2 (α 2 + 1) t α( b 2 E[( µ s b) 2 ]) e ε2 (α 2 +1)s ds 0 t αe[( µ s b) 2 ]) e ε2 (α 2 +1)s ds 0 lim inf E[( µ t b) 2 ] t + lim inf E[ µ α t b] t + ε 2 (α 2 lim sup E[( µ t b) 2 ]. + 1) t + So far, the process µ t stays on the lower half sphere.

40 Differences: stability point of vue One particle Which noise for which model? Itô Asymptotically stable states remain asymptotically stables. Stratanovich Asymptotically stable states become stables. µ t b 1, p.s. when t goes to infinity. t+σ (µ s b)ds β < 1, p.s. when t, σ go to infinity. t

41 Plan Net of particles The deterministic model Modeling of ferromagnetic materials Physical principles The microscopic scale The micromagnetic model Link between scales One particle The stochastic model Results Illustrations Which noise for which model? Net of particles The deterministic model Stable configurations Control The sotchastic case

42 A net of par tiles and a lecture head y diple magntique z x δl l G. Carbou, S. Labbé, C. Prieur and S. Agarwall Control of a network of magnetic ellipsoidal samples. Mathematical Control and Related Fields, 2011.

43 Plan Net of particles Stable configurations Modeling of ferromagnetic materials Physical principles The microscopic scale The micromagnetic model Link between scales One particle The stochastic model Results Illustrations Which noise for which model? Net of particles The deterministic model Stable configurations Control The sotchastic case

44 Theorem There exists γ 0 > 0 (form parameter of the net) such that V l 3 γ 0, (3.1) There exists ν 0 > 0, et c > 0 such that, for every pertinent configuration m 0 associated to ε (m 0 i = ε i e 2 pour tout i), for all m init V ε(ν 0 ), the solution m of the system with M 0 associated to the initial condition m(0) = m init satisfies: t 0, m(t) V ε(ν 0 e ct ).

45 Unstable configuration y z x y z x

46 Stable configuration y z x y z x

47 Plan Net of particles Control Modeling of ferromagnetic materials Physical principles The microscopic scale The micromagnetic model Link between scales One particle The stochastic model Results Illustrations Which noise for which model? Net of particles The deterministic model Stable configurations Control The sotchastic case

48 Control Net of particles Control Given m and m two pertinent configurations. We choose the initial state in the neighborhood of m, then the configuration m is the goal. t [i l v, i l v + δ l +M, si m i = m i = e 2, ], M(t) = 0 si m v i = m i, M, si m i = m i = e 2, t / N [i l v, i l v + δ l ], M(t) = 0. v i=0

49 Control The magnetic dipole induces a magnetic field Ω i0 given by where H app(t, M)(i 0 ) = µ 0M 4π 1 r 3 (2 cos(θ)ur + sin(θ) u θ), r is the distance between the dipole and the considered particle: r = [(x 0 + vt i 0 l) 2 + δ 2 l 2 ] 1 2, u r et u θ are given by u r = 1 r θ is the angle ( e2, u r ). M is the control. x 0 + vt i 0 l δl 0, u θ = 1 r δl x 0 + vt i 0 l 0

50 Theorem Given γ 0 and c chosen in the stability zone. There exists γ 1 > 0 with γ 1 < γ 0, there exists ν 1 > 0, M > 0 and δ > 0 such that V l 3 γ 1, then we can claim the following stability result: given m and m two pertinent configurations t M(t) the control obtain auctioning the dipole. If m init V ε (ν 1 ), The solution m of the system associated to the initial condition m init satisfies: t T f, m(t) V ε (ν 1 e c(t T f ) ).

51 Plan Net of particles The sotchastic case Modeling of ferromagnetic materials Physical principles The microscopic scale The micromagnetic model Link between scales One particle The stochastic model Results Illustrations Which noise for which model? Net of particles The deterministic model Stable configurations Control The sotchastic case

52 The studied system Net of particles The sotchastic case Similarly to the study performed for one particle, where are now interested in a net of particles σ Σ l,n : dy σ,t = µ σ,t (H(µ t )(σ)dt + εdw σ,t ) α µ σ,t (µ σ,t (H(µ t )(σ)dt + εdw σ,t )), µ σ,t = Y σ,t Y σ,t, Y 0 = y (S 2 ) Σ l,n. Goal: to retrieve a phase transition behavior for configurations. Work in progress with J. Lelong and A. Kritoglou.

53 Examples of dynamics Net of particles The sotchastic case

54 Thank you for your attention.