Open accessJournal Article

# Controllability of bilinear quantum systems in explicit times via explicit control fields

04 Mar 2021-International Journal of Control (Informa UK Limited)-Vol. 94, Iss: 3, pp 724-734
Abstract: We consider the bilinear Schrodinger equation on a bounded one-dimensional domain and we provide explicit times such that the global exact controllability is verified. In addition, we show how to c...

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Open accessJournal Article
01 Jan 2021-Automatica
Abstract: A major application of the mathematical concept of graph in quantum mechanics is to model networks of electrical wires or electromagnetic wave-guides. In this paper, we address the dynamics of a particle trapped on such a network in presence of an external electromagnetic field. We study the controllability of the motion when the intensity of the field changes over time and plays the role of control. From a mathematical point of view, the dynamics of the particle is modeled by the so-called bilinear Schrodinger equation defined on a graph representing the network. The main purpose of this work is to extend the existing theory for bilinear quantum systems on bounded intervals to the framework of graphs. To this end, we introduce a suitable mathematical setting where to address the controllability of the equation from a theoretical point of view. More precisely, we determine assumptions on the network and on the potential field ensuring its global exact controllability in suitable spaces. Finally, we discuss two applications of our results and their practical implications to two specific problems involving a star-shaped network and a tadpole graph.

Topics: Controllability (62%), Quantum graph (57%), Schrödinger equation (52%) ... show more

7 Citations

Open accessPosted Content
Alessandro Duca1Institutions (1)
Abstract: A major application of the mathematical concept of graph in quantum mechanics is to model networks of electrical wires or electromagnetic wave-guides. In this paper, we address the dynamics of a particle trapped on such a network in presence of an external electromagnetic field. We study the controllability of the motion when the intensity of the field changes over time and plays the role of control. From a mathematical point of view, the dynamics of the particle is modeled by the so-called bilinear Schrodinger equation defined on a graph representing the network. The main purpose of this work is to extend the existing theory for bilinear quantum systems on bounded intervals to the framework of graphs. To this end, we introduce a suitable mathematical setting where to address the controllability of the equation from a theoretical point of view. More precisely, we determine assumptions on the network and on the potential field ensuring its global exact controllability in suitable spaces. Finally, we discuss two applications of our results and their practical implications to two specific problems involving a star-shaped network and a tadpole graph.

4 Citations

Open accessJournal Article
Abstract: In this work, we consider the bilinear Schrodinger equation (BSE) i∂tψ=−Δψ+u(t)Bψ in the Hilbert space L2(G,C) with G an infinite graph. The Laplacian −Δ is equipped with self-adjoint boundary cond...

Topics: Hilbert space (53%), Quantum state (51%), Laplace operator (51%)

4 Citations

Open accessJournal Article
Alessandro Duca1, Romain Joly1, Dmitry Turaev2Institutions (2)
Abstract: We study the Schr\"odinger equation $i\partial_t\psi=-\Delta\psi+V\psi$ on $L^2((0,1),\mathbb{C})$ where $V$ is a very high and localized potential wall. We aim to perform permutations of the eigenmodes and to control the solution of the equation. We consider the process where the position and the height of the potential wall change as follows. First, the potential increases from zero to a very large value, so a narrow potential wall is formed that almost splits the interval into two parts; then the wall moves to a different position, after which the height of the wall decays to zero again. We show that even though the rate of the variation of the potential's parameters can be arbitrarily slow, this process alternates adiabatic and non-adiabatic dynamics, leading to a non-trivial permutation of the eigenstates. Furthermore, we consider potentials with several narrow walls and we show how an arbitrarily slow motion of the walls can lead the system from any given state to an arbitrarily small neighborhood of any other state, thus proving the approximate controllability of the above Schr\"odinger equation by means of a soft, quasi-adiabatic variation of the potential.

1 Citations

Open accessPosted Content
Abstract: The aim of this work is to study the controllability of the bilinear Schrodinger equation on compact graphs. In particular, we consider the equation (BSE) $i\partial_t\psi=-\Delta\psi+u(t)B\psi$ in the Hilbert space $L^2(\mathscr{G},\mathbb{C})$, with $\mathscr{G}$ being a compact graph. The Laplacian $-\Delta$ is equipped with self-adjoint boundary conditions, $B$ is a bounded symmetric operator and $u\in L^2((0,T),\mathbb{R})$ with $T>0$. We provide a new technique leading to the global exact controllability of the (BSE) in $D(|\Delta|^{s/2})$ with $s\geq 3$. Afterwards, we introduce the "energetic controllability", a weaker notion of controllability useful when the global exact controllability fails. In conclusion, we develop some applications of the main results involving for instance star graphs.

Topics: Controllability (56%)

1 Citations

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Open accessBook
01 Jan 1968-
Abstract: From the Publisher: Engineers must make decisions regarding the distribution of expensive resources in a manner that will be economically beneficial. This problem can be realistically formulated and logically analyzed with optimization theory. This book shows engineers how to use optimization theory to solve complex problems. Unifies the large field of optimization with a few geometric principles. Covers functional analysis with a minimum of mathematics. Contains problems that relate to the applications in the book.

Topics: Vector optimization (61%), , Optimization problem (60%)

5,541 Citations

Open accessBook
01 Jan 1995-
Abstract: Linear operations in Banach spaces Entropy numbers, s-numbers, and eigenvalues Unbounded linear operators Sesquilinear forms in Hilbert spaces Sobolev spaces Generalized Dirichlet and Neumann boundary-value problems Second-order differential operators on arbitrary open sets Capacity and compactness criteria Essential spectra Essential spectra of general second-order differential operators Global and asymptotic estimates for the eigen-values of - + q when q is real. Estimates for the singular values of - + q when 1 is complete Bibliography Notation index Subject index

Topics: Operator theory (73%), Unbounded operator (68%), Spectral theorem (65%) ... show more

1,709 Citations

Open accessBook
01 Jan 1969-

557 Citations

Open accessJournal ArticleDOI: 10.1137/0320042
Abstract: This paper studies controllability of systems of the form ${{dw} / {dt}} = \mathcal {A}w + p(t)\mathcal {B}w$ where $\mathcal{A}$ is the infinitesimal generator of a $C^0$ semigroup of bounded linear operators $e^{\mathcal{A}t}$ on a Banach space X, $\mathcal{B}:X \to X$ is a $C^1$ map, and $p \in L^1 ([0,T];\mathbb{R})$ is a control. The paper (i) gives conditions for elements of X to be accessible from a given initial state $w_0$ and (ii) shows that controllability to a full neighborhood in X of $w_0$ is impossible for $\dim X = \infty$. Examples of hyperbolic partial differential equations are provided.

313 Citations

Open accessJournal Article
Tosio Kato1Institutions (1)