Small-time approximate controllability of bilinear Schrödinger equations and diffeomorphisms
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| Format: | Preprint |
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2024
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| author | Beauchard, Karine Pozzoli, Eugenio |
| author_facet | Beauchard, Karine Pozzoli, Eugenio |
| contents | We consider Schrödinger PDEs, posed on a boundaryless Riemannian manifold $M$, with bilinear control. We propose a new method to prove the global $L^2$-approximate controllability. Contrarily to previous ones, it works in arbitrarily small time and does not require a discrete spectrum.
This approach consists in controlling separately the radial part and the angular part of the wavefunction thanks to the control of the group ${\rm Diff}_c^0(M)$ of diffeomorphisms of $M$ and the control of phases, which refer to the possibility, for any initial state $ψ_0\in L^2(M,\mathbb{C})$, diffeomorphism $P\in {\rm Diff}_c^0(M)$ and phase $φ\in L^2(M,\mathbb{R})$ to reach approximately the states $(\det DP)^{1/2}(ψ_0\circ P)$ and $e^{i φ}ψ_0 $. The control of the radial part uses the transitivity of the group action of ${\rm Diff}_c^0(M)$ on positive densities proved by Moser.
We develop this approach on two examples of Schrödinger equations, posed on $\mathbb{T}^d$ or $\mathbb{R}^d$, for which the small-time control of phases was recently proved. We prove that it implies the small-time control of flows of vector fields thanks to Lie bracket techniques. Combining this property with the simplicity of the group ${\rm Diff}_c^0(M)$ proved by Thurston, we obtain the control of the group ${\rm Diff}_c^0(M)$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_02383 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Small-time approximate controllability of bilinear Schrödinger equations and diffeomorphisms Beauchard, Karine Pozzoli, Eugenio Optimization and Control Analysis of PDEs We consider Schrödinger PDEs, posed on a boundaryless Riemannian manifold $M$, with bilinear control. We propose a new method to prove the global $L^2$-approximate controllability. Contrarily to previous ones, it works in arbitrarily small time and does not require a discrete spectrum. This approach consists in controlling separately the radial part and the angular part of the wavefunction thanks to the control of the group ${\rm Diff}_c^0(M)$ of diffeomorphisms of $M$ and the control of phases, which refer to the possibility, for any initial state $ψ_0\in L^2(M,\mathbb{C})$, diffeomorphism $P\in {\rm Diff}_c^0(M)$ and phase $φ\in L^2(M,\mathbb{R})$ to reach approximately the states $(\det DP)^{1/2}(ψ_0\circ P)$ and $e^{i φ}ψ_0 $. The control of the radial part uses the transitivity of the group action of ${\rm Diff}_c^0(M)$ on positive densities proved by Moser. We develop this approach on two examples of Schrödinger equations, posed on $\mathbb{T}^d$ or $\mathbb{R}^d$, for which the small-time control of phases was recently proved. We prove that it implies the small-time control of flows of vector fields thanks to Lie bracket techniques. Combining this property with the simplicity of the group ${\rm Diff}_c^0(M)$ proved by Thurston, we obtain the control of the group ${\rm Diff}_c^0(M)$. |
| title | Small-time approximate controllability of bilinear Schrödinger equations and diffeomorphisms |
| topic | Optimization and Control Analysis of PDEs |
| url | https://arxiv.org/abs/2410.02383 |