We consider a linear Schr¨odinger equation, in a finite interval, with bilinear control, representing a quantum particle in an electric field (control). Similar results were proved for particular models in non-optimal spaces, in long time, and the proof relied on the Nash-Moser implicit function theorem to cope with an a priori loss of regularity. In this article, the model is more general, the spaces are optimal, there are no time constraints, and the proof relies on the classical inverse mapping theorem.
Ensuite, la même stratégie est appliquée aux équations de Schrödinger non linéaires et aux équations d'ondes non linéaires, montrant que la méthode fonctionne pour une large gamme de systèmes de contrôle bilinéaires. Nous considérons une équation de Schrödinger linéaire, sur un intervalle borné, à contrôle bilinéaire, représentant une particule quantique dans un champ électrique (le contrôle). Des résultats similaires ont déjà été établis, mais dans des espaces non optimaux, en temps long et leur preuve s'appuie sur le théorème de Nash-Moser, pour gérer une apparente perte de régularité.
La même stratégie est ensuite appliquée aux équations de Schrödinger non linéaires et aux équations d'ondes non linéaires, montrant qu'elle s'applique de manière assez générale aux systèmes. Mots clés : contrôle des équations aux dérivées partielles ; contrôle bilinéaire ; équation de Schrödinger ; systèmes quantiques; équation des vagues ; théorème de cartographie inverse.
Main result
A simpler proof
Additionnal results
- Generalization to higher regularities
- On the 3D ball with radial data
- Nonlinear Schr¨odinger equations
- Nonlinear wave equations
The first situation is the analogous result of Theorem 1, but with higher regularities: we prove the local exact controllability of (1) in smoother spaces and with smoother controls. Of course, the strategy can be used to go further and prove the local exact controllability of (1) around the ground state. The second situation is the analogous result of Theorem 1, but for the Schrödinger equation set on the three-dimensional unit sphere B3 for radial data.
The analysis is very close to the 1D case, since for these particular data the Laplacian behaves as in dimension 1. Note that this simpler situation has also been used by Anton to prove global existence for the nonlinear Schr¨odinger equation [ 8]. Other versions of this result with higher regularity can be proved: the system is exactly controllable, locally around the reference trajectory.
Other versions of this result with higher regularity can be proved: the system is exactly controllable, locally around the reference trajectory.
A brief bibliography
- A previous negative result
- Iterated Lie brackets
- Controllability results for Schr¨odinger and wave equations
- Other results about bilinear quantum systems
Therefore, Ball Marsden and Slemrod's negative result is sometimes only due to an 'unfortunate' choice of functional spaces, which does not allow the controllability. First, the controllability of finite dimensional quantum systems (i.e. modeled by an ordinary differential equation) is well understood. Finally, let us cite important papers on the controllability of PDEs, in which positive results are proved by applying geometrical control methods to the (finite dimensional) Galerkin approximations of the equation.
Let us also mention [51] by Mirrahimi, Rouchon, Turinici and [19] for explicit feedback controls, inspired by Lyapunov techniques. The controllability of Schr¨odinger equations with distributed and boundary controls, which act linearly on the state, has been studied since a long time. For linear equations, the controllability is equivalent to an observability inequality that can be proved with different techniques: multiplier methods (see [37] by Fabre, [48] by Machtyngier), microlocal analysis (see [47] by Lebeau, [28] by Burq ), Carleman estimate (see [43, 44] by Lasiecka, Triggiani, Zhang), or number theory (see [55] by Ramdani, Takahashi, Tenenbaum and Tucsnak).
Let us also mention that this negative result has been adapted to nonlinear Schr¨odinger equations in [40] by Ilner, Lange and Teismann. Let us emphasize that the local exact controllability result of [17] and the global approximate controllability of [53, 52] can be put together to obtain the global exact controllability of 1D models (see [52]).
Structure of this article
Notations
Optimal control techniques have also been explored for Schrödinger equations with a Hartee-type nonlinearity in [11, 12] by Baudouin, Kavian, Puel and in [35] by Cances, Le Bris, Pilot. Subsection 2.5 is devoted to the proof of the same result with higher regularities, i.e. Subsection 2.6 is devoted to the Schrödinger equation for radial data on the three-dimensional ball, i.e. 19). This subsection is devoted to the explanation of the existence, uniqueness, regularity results, and bounds for the weak solutions of the Cauchy problem.
Since our constant c1(t) is uniform on bounded sets, we easily get that N depends only on R, so the constant in the proposition depends only on T, µandRas argued. So, when u∈ C0([0, T],R), we can take the L2-scalar product of this equation with ψ; and the imaginary part of the resulting equality gives.
C 1 -regularity of the end-point map
Controllability of the linearized system
Proof of Theorem 1
Generalization to higher regularities
Thanks to several changes of variables, the first term of the right-hand side of (36) can be decomposed in the following way.
Case of the three dimensional ball with radial data
We conclude for this term as in Lemma 1, since the eigenvalues are the same and Corollary 4 still applies. However, this would require the analysis of the zeros of the Bessel functions and we have chosen to present the simplest result. The goal is to prove Theorem 4. First, let's introduce the following notations, which will be valid in all section 3. 39) The eigenvectors (ϕk)k∈N and eigenvalues (λk)k∈Nare.
Proposition 8 will be the consequence of the existence and uniqueness of a weak solutionζ for (43) (the preservation of the L2-norm can be proved as in the linear case). To clarify the definition of such a weak solution, let us introduce the operator Adefined by. Note that these formulas are only the result of the diagonalization of the matrix iA =. obtained by the decomposition into real and imaginary part.
Proof of Lemma 3: The proof of this Lemma is similar to that of Lemma 1. Let us prove that there exists a constant c = c(t) > 0 (uniformly bounded on bounded intervals often) such that.
C 1 -regularity of the end-point map
Controllability of the linearized system
The proof of Theorem 4 concludes with the same arguments as in Section 2.4 using the inverse mapping theorem and the preservation of the L2 norm. When the space domain is different, for example (0, a) with alarge, then λk = (kπ/a)2, thus a finite number of the quantities λk(λk−2) are negative: we get a new moment problem with a finite number moments of real-valued exponentials and an infinite number of trigonometric moments that can be easily solved by adapting the tools used in this article. In this paper, we have proposed a method for proving the local exact controllability of linear and nonlinear bilinear systems.
We have applied it to Schr¨odinger and wave equations, showing that it works for a wide range of problems. However, they all agreed that the linearized system fulfills a gap condition on the eigenvalues of the operator. This condition is not necessarily realized for the Schr¨odinger equation in higher space dimensions.
At the end of this appendix, the term "the" biorthogonal family of Θ refers to this unique biorthogonal family in Span{ξi;i∈Z}. Proposition 19 (1)If Θ is a Riesz basis of SpanΘ, then its biorthogonal familyΘ′ is also a Riesz basis of SpanΘ. 2) Θis a Riesz basis for SpanΘif and only if there exists C1, C2∈(0,+∞) such that for every scalar sequence(cj)j∈Z with finite support,. Let H be a Hilbert space, (ζj)j∈Z be an orthonormal family of H, V :H → SpanΘ an isomorphism such thatξj =V(ζj),∀j ∈ Z. Thus Θ′ is a Riesz basis for SpanΘ. 2) We assume that Θ is a Riesz basis for SpanΘ.
We proved that V is an isomorphism, so Θ is a Riesz basis of SpanΘ. 3) SpanΘ is a tight vector subspace of H, so we have the orthogonal decomposition H = SpanΘ + SpanΘ⊥ and the corresponding orthogonal projection P : H → SpanΘ. Proposition 20 The operator JΘ0 :SpanΘ→l2(Z,K) is an isomorphism if and only if Θ is a Riesz basis of SpanΘ. Estimation of control time for single-input systems on compact dummy groups.ESAIM Control Optim.
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