Simultaneous global exact controllability of an arbitrary number of 1d bilinear Schrödinger equations
01 Jan 2015-Journal de Mathématiques Pures et Appliquées (Elsevier Masson)-Vol. 103, Iss: 1, pp 228-254
TL;DR: In this article, the authors consider a system of an arbitrary number of linear Schrodinger equations on a bounded interval with bilinear control and prove global exact controllability in large time of these equations with a single control.
About: This article is published in Journal de Mathématiques Pures et Appliquées.The article was published on 2015-01-01 and is currently open access. It has received 40 citations till now. The article focuses on the topics: Controllability & Bounded function.
Citations
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TL;DR: In this paper, the one dimensional Schrodinger equation with a bilinear control is considered and a transformation mapping the solution to the linearized equation on a solution to an exponentially stable target linear equation is proposed.
31 citations
TL;DR: It is shown, for n ∈ N arbitrary, exact controllability in projections on the first n given eigenstates, based on Lie-algebraic control techniques applied to the finite-dimensional approximations coupled with classical topological arguments issuing from degree theory.
Abstract: We consider the bilinear Schrodinger equation with discrete-spectrum drift. We show, for n ∈ N arbitrary, exact controllability in projections on the first n given eigenstates. The controllability result relies on a generic controllability hypothesis on some associated finite-dimensional approximations. The method is based on Lie-algebraic control techniques applied to the finite-dimensional approximations coupled with classical topological arguments issuing from degree theory.
15 citations
Cites background from "Simultaneous global exact controlla..."
...In certain cases, when H is a function space on a a real interval, a description of reachable sets has been provided (see [4, 5, 21])....
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TL;DR: This work considers the bilinear Schrödinger equation on a bounded one-dimensional domain and provides explicit times such that the global exact controllability is verified and shows how to construct controls for the global approximate controllable.
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...
14 citations
TL;DR: In such spaces, the global exact controllability of the bilinear Schrodinger equation (BSE) is attained and examples of the main results involving star graphs and tadpole graphs are provided.
13 citations
Posted Content•
TL;DR: In this article, the authors consider an infinite number of one dimensional bilinear Schrodinger equations in a segment and prove the simultaneous local exact controllability in projection for any positive time and the simultaneous global exact control for sufficiently large time.
Abstract: We consider an infinite number of one dimensional bilinear Schrodinger equations in a segment. We prove the simultaneous local exact controllability in projection for any positive time and the simultaneous global exact controllability in projection for sufficiently large time.
11 citations
References
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Book•
17 Apr 2007
TL;DR: In this article, the controllability and the stabilization of nonlinear control systems in finite and infinite dimensions are studied, with a focus on specific phenomena due to nonlinearities.
Abstract: This book presents methods to study the controllability and the stabilization of nonlinear control systems in finite and infinite dimensions. The emphasis is put on specific phenomena due to nonlinearities. In particular, many examples are given where nonlinearities turn out to be essential to get controllability or stabilization. Various methods are presented to study the controllability or to construct stabilizing feedback laws. The power of these methods is illustrated by numerous examples coming from such areas as celestial mechanics, fluid mechanics, and quantum mechanics. The book is addressed to graduate students in mathematics or control theory, and to mathematicians or engineers with an interest in nonlinear control systems governed by ordinary or partial differential equations.
993 citations
TL;DR: In this paper, the controllability of systems of the form {dw} / {dt} = \mathcal {A}w + p(t) w + √ √ {B}w$ where W is the infinitesimal generator of a $C^0$ semigroup of bounded linear operators on a Banach space X and W is a control.
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.
335 citations
Book•
27 Jan 2005
TL;DR: This book is the first serious attempt to gather all of the available theory of these "nonharmonic Fourier series" in one place, combining published results with new results by the authors to create a unique source of such material for practicing applied mathematicians, engineers and other scientific professionals.
Abstract: This book uses techniques of Fourier series and functional analysis to deal with certain problems in differential equations The Fourier series and functional analysis are merely tools; the authors' real interest lies in the differential equations that they study It has been known since 1967 that a wide variety of sets {ewikt} of complex exponential functions play an important role in the control theory of systems governed by partial differential equations However, this book is the first serious attempt to gather all of the available theory of these "nonharmonic Fourier series" in one place, combining published results with new results by the authors, to create a unique source of such material for practicing applied mathematicians, engineers and other scientific professionals
247 citations
TL;DR: In this paper, the authors prove approximate controllability of the bilinear Schrodinger equation in the case of the uncontrolled Hamiltonian having a discrete non-resonant spectrum.
Abstract: We prove approximate controllability of the bilinear Schrodinger equation in the case in which the uncontrolled Hamiltonian has discrete non-resonant spectrum. The results that are obtained apply both to bounded or unbounded domains and to the case in which the control potential is bounded or unbounded. The method relies on finite-dimensional techniques applied to the Galerkin approximations and permits, in addition, to get some controllability properties for the density matrix. Two examples are presented: the harmonic oscillator and the 3D well of potential, both controlled by suitable potentials.
197 citations
"Simultaneous global exact controlla..." refers methods in this paper
...The hypotheses of this result were refined by Boscain, Caponigro, Chambrion, and Sigalotti in [6]....
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...A first strategy of the proof of approximate controllability is due to Chambrion, Mason, Sigalotti, and Boscain [10], which relies on the geometric techniques based on the controllability of the Galerkin approximations....
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