Abstract
We consider a system of an arbitrary number of 1d linear Schrödinger equations on a bounded interval with bilinear control. We prove global exact controllability in large time of these N equations with a single control. This result is valid for an arbitrary potential with generic assumptions on the dipole moment of the considered particle. Thus, even in the case of a single particle, this result extends the available literature. The proof combines local exact controllability around finite sums of eigenstates, proved with Coron's return method, a global approximate controllability property, proved with Lyapunov strategy, and a compactness argument.
Highlights
We consider a system of an arbitrary number of 1d linear Schrödinger equations on a bounded interval with bilinear control
Adapting the ideas of [16], using favourable assumptions on the spectrum of AV and a compactness argument, we prove in Section 4 exact controllability locally around specific finite sums of eigenfunctions
Combining the properties of approximate controllability proved in Theorem 3.1 and local exact controllability proved in Theorem 4.1, we establish global exact controllability for (1.2), under the following hypotheses on the functions V, μ ∈ H4((0, 1), R)
Summary
Global approximate controllability with generic assumptions both on the potential and the dipole moment is obtained by the second author in [17] and extended to higher norms leading to the first global exact controllability result for a bilinear quantum system in [18]. The only exact simultaneous controllability results for infinite dimensional bilinear quantum systems were obtained in [16] by the first author locally around eigenstates in the case V = 0 for N = 2 or N = 3 This is proved either up to a global phase in arbitrary time or exactly up to a global delay in the case N = 2 and up to a global phase and a global delay in the case N = 3. We define the space l2r(N, C) := d ∈ l2(N, C) ; d0 ∈ R which is endowed with the natural metric
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