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...continuous feedback exists; for these situations we proposed two solutions: either a discontinuous feedback or an averaging procedure that weakens the monotonic property of Lyapounov approaches [44], =-=[24]-=-. 5.4.3. Propagator space methods Traditionally, the numerical simulations consider some description of the interaction of the laser and the system, of which the most used is the dipole approximation,...

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...e admissible laser field. This can be studied via the general accessibility criteria [3,23] based on Lie brackets; more specific results can be found in [26]. A detailed presentation has been made in =-=[6]-=-. Even if positive results of controllability for systems, with Hamiltonian defined by (3), have been obtained, finding efficient numerical algorithms to determine the control field remains a very dif...

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...ive controllability properties for the controlled Schrödinger equation, by means of the dipolar term uW1 and the polarizability term u2W2. This question has already been tackled by various authors in =-=[3]-=-, [4] (for finite dimensional approximations) and in [5] (for the infinite dimensional version of the problem, when Ω is a bounded set of Rn and W1, W2 are smooth functions). All these contributions r...

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...e systems is called controllable. Positive results of controllability are obtained by applying the Lie algebra criteria [4], [33] and especially the specific result in [36]. This study is detailed in =-=[8]-=-. The theoretical results of controllability do not offer automatically a method to determine the laser field. Very often this task is formulated as a cost functional to be minimized. Several techniqu...

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...e contrôles de fortes amplitudes [7]), du point de vue mathématique, ce terme a permis de montrer la contrôlabilité dans des cas où le moment dipolaire est insuffisant pour conclure (voir par exemple =-=[6]-=-, [10], [3]). Pour V ∈ L2 ((0, 1), R), on note λk,V et ϕk,V les valeurs propres (en ordre croissant) et vecteurs propres de l’opérateur AV défini sur le domaine D(AV ) := H2 ∩ H1 0 ((0, 1), C) par ∗ C...

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...e contrôles de fortes amplitudes [7]), du point de vue mathématique, ce terme a permis de montrer la contrôlabilité dans des cas où le moment dipolaire est insuffisant pour conclure (voir par exemple =-=[6]-=-, [10], [3]). Pour V ∈ L2((0, 1),R), on note λk,V et ϕk,V les valeurs propres (en ordre croissant) et vecteurs propres de l’opérateur AV défini sur le domaine D(AV ) := H 2 ∩H10 ((0, 1),C) par ∗CMLS U...

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... present controllability properties for the controlled Schrödinger equation, using the dipolar term uW1 and the polarizability term u2W2. This question has already been tackled by various authors in =-=[3]-=-, [4] (for finite dimensional approximations) and in [5] (for the infinite dimensional version of the problem, when Ω is a bounded set of Rn and W1,W2 are smooth functions). All the results in these c...

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...that if D is the rectangle [0, 1]n, Hypothesis 1.1 iii) hold generically with respect to V in the set G := {V ∈ C∞(D,R) ; V (x1, . . . , xn) = V1(x1) + · · ·+ Vn(xn), with Vk ∈ C∞([0, 1],R) } . As in =-=[CGLT]-=-, we use a time-periodic oscillating control of the form u(t, ψ) := α(ψ) + β(ψ) sin ( t ε ) . (1.4) Following classical techniques (see e.g. [SVM]) of dynamical systems in finite dimension let us intr...