Local controllability of 1D linear and nonlinear Schrödinger equations with bilinear control
01 Nov 2010-Journal de Mathématiques Pures et Appliquées (Elsevier Masson)-Vol. 94, Iss: 5, pp 520-554
TL;DR: In this paper, Masson et al. considered a linear Schrodinger equation with bilinear control and proved the exact controllability of the system in any positive time, locally around the ground state.
About: This article is published in Journal de Mathématiques Pures et Appliquées.The article was published on 2010-11-01 and is currently open access. It has received 146 citations till now. The article focuses on the topics: Implicit function theorem & Nonlinear system.
Citations
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TL;DR: In this communication, state-of-the-art quantum control techniques are reviewed and put into perspective by a consortium of experts in optimal control theory and applications to spectroscopy, imaging, as well as quantum dynamics of closed and open systems.
Abstract: It is control that turns scientific knowledge into useful technology: in physics and engineering it provides a systematic way for driving a dynamical system from a given initial state into a desired target state with minimized expenditure of energy and resources As one of the cornerstones for enabling quantum technologies, optimal quantum control keeps evolving and expanding into areas as diverse as quantum-enhanced sensing, manipulation of single spins, photons, or atoms, optical spectroscopy, photochemistry, magnetic resonance (spectroscopy as well as medical imaging), quantum information processing and quantum simulation In this communication, state-of-the-art quantum control techniques are reviewed and put into perspective by a consortium of experts in optimal control theory and applications to spectroscopy, imaging, as well as quantum dynamics of closed and open systems We address key challenges and sketch a roadmap for future developments
572 citations
TL;DR: In this article, the authors prove an approximate controllability result for the bilinear Schrodinger equation for the problem of controlling the rotation of a bipolar rigid molecule confined on a plane by means of two orthogonal external fields.
Abstract: In this paper we prove an approximate controllability result for the bilinear Schrodinger equation. This result requires less restrictive non-resonance hypotheses on the spectrum of the uncontrolled Schrodinger operator than those present in the literature. The control operator is not required to be bounded and we are able to extend the controllability result to the density matrices. The proof is based on fine controllability properties of the finite dimensional Galerkin approximations and allows to get estimates for the $L^{1}$ norm of the control. The general controllability result is applied to the problem of controlling the rotation of a bipolar rigid molecule confined on a plane by means of two orthogonal external fields.
109 citations
TL;DR: In this article, the Schrodinger equation is shown to be controllable in Sobolev spaces H s, s> 0, generically with respect to the potential, where the potential has a random time-dependent amplitude.
95 citations
TL;DR: This paper extends this finite dimensional result, known as the rotating wave approximation, to infinite dimensional systems and provides explicit convergence estimates.
71 citations
TL;DR: Weakly coupled systems are a class of infinite dimensional conservative bilinear control systems with discrete spectrum as discussed by the authors, which can be precisely approached by finite dimensional Galerkin approximations.
Abstract: Weakly coupled systems are a class of infinite dimensional conservative bilinear control systems with discrete spectrum. An important feature of these systems is that they can be precisely approached by finite dimensional Galerkin approximations. This property is of particular interest for the approximation of quantum system dynamics and the control of the bilinear Schrodinger equation. The present study provides rigorous definitions and analysis of the dynamics of weakly coupled systems and gives sufficient conditions for an infinite dimensional quantum control system to be weakly coupled. As an illustration we provide examples chosen among common physical systems.
67 citations
References
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Book•
15 Apr 2004
TL;DR: Control Theory from the Geometric Viewpoint as mentioned in this paper is a recent addition to the geometric control theory monograph/textbook literature having Jurdjevic (1997) as its closest neighbor and Nijmeijer and van der Schaft (1995), Isidori (1996) and Bloch (2003) as more distant ones.
Abstract: Geometrical methods have had a profound impact in the development of modern nonlinear control theory. Fundamental results such as the orbit theorem, feedback linearization, disturbance decoupling or the various controllability tests for nonlinear systems are all deeply rooted on a geometric view of control theory. It is perhaps surprising, and possibly debatable, that in order to understand and appreciate the “essence” of linear control systems one has to delve into the intricacies of Lie brackets and Lie algebras. This is because only the geometric perspective offers the tools to study the properties of control systems that are invariant under (nonlinear) changes of coordinates and can therefore be considered intrinsic. Consider, for example, an inverted pendulum or the ball and beam system. It should be apparent that reachability or optimality properties for these systems do not depend on the particular reference frame chosen to write their equations of motion. These are intrinsic properties of these physical systems and thus require geometric techniques for its study. “Control Theory from the Geometric Viewpoint” is a recent addition to the geometric control theory monograph/textbook literature having Jurdjevic (1997) as its closest neighbor and Nijmeijer and van der Schaft (1995), Isidori (1996) and Bloch (2003) as more distant ones. The book evolved from lecture notes for graduate courses taught by the first author at the International School for Advanced Studies in Trieste, Italy. The lecture notes style can be felt throughout the 24 chapters of the book treating a large number of topics ranging from controllability and reachability analysis to higher order conditions for optimality. This lecture notes style, patent on the relatively large number of treated topics in 400 pages, is the book’s main handicap and merit. If, on the one hand, most chapters can be independently read thus allowing the reader to immediately dive into the topic of choice and quickly reach the zenith result, on the other hand, there is a certain lack of fluidity when one tries to read the book chapters consecutively. In the remaining lines I will try to articulate my own opinion, naturally conditioned by my taste and background, on the choice of topics and presentation as I go through some of the individual chapters.
1,325 citations
"Local controllability of 1D linear ..." refers background in this paper
...The controllability of (17) is linked to the rank of the Lie algebra spanned by H0 and H1 (see for instance [5] by Albertini and D’Alessandro, [7] by Altafini, [26] by Brockett, see also [3] by Agrachev and Sachkov, [32] by Coron for a more general discussion)....
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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
"Local controllability of 1D linear ..." refers background or methods in this paper
...The controllability of (17) is linked to the rank of the Lie algebra spanned by H0 and H1 (see for instance [5] by Albertini and D’Alessandro, [7] by Altafini, [26] by Brockett, see also [3] by Agrachev and Sachkov, [32] by Coron for a more general discussion)....
[...]
...This strategy is coupled with the return method and quasi-static deformations in [14,17] and with power series expansions in [15,17] (see [30,32] by Coron for a presentation of these technics)....
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...The controllability of (17) is linked to the rank of the Lie algebra spanned by H0 and H1 (see for instance [5] by Albertini and D’Alessandro, [7] by Altafini, [27] by Brockett, see also [3] by Agrachev and Sachkov, [33] by Coron for a more general discussion)....
[...]
...Therefore, the way the Lie algebra rank condition could be used directly in infinite dimension is not clear (see also [32] for the same discussion on other examples)....
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TL;DR: In this article, the controllability of nonlinear systems described by the equation dx/dt - F(x,u) was discussed and it was shown that strong accessibility implies strong accessibility for a large class of manifolds including Euclidean spaces.
718 citations
01 Jan 2000
TL;DR: The Lecture Notes in Chemistry (LNC) series as discussed by the authors provides an accessible introduction to the field to postgraduate students and nonspecialist researchers from related areas, providing a source of advanced teaching material for specialized seminars, courses and schools, and being readily accessible in print and online.
Abstract: The series Lecture Notes in Chemistry (LNC), reports new developments in chemistry and molecular science quickly and informally, but with a high quality and the explicit aim to summarize and communicate current knowledge for teaching and training purposes. Books published in this series are conceived as bridging material between advanced graduate textbooks and the forefront of research. They will serve the following purposes: • provide an accessible introduction to the field to postgraduate students and nonspecialist researchers from related areas, • provide a source of advanced teaching material for specialized seminars, courses and schools, and • be readily accessible in print and online. The series covers all established fields of chemistry such as analytical chemistry, organic chemistry, inorganic chemistry, physical chemistry including electrochemistry, theoretical and computational chemistry, industrial chemistry, and catalysis. It is also a particularly suitable forum for volumes addressing the interfaces of chemistry with other disciplines, such as biology, medicine, physics, engineering, materials science including polymer and nanoscience, or earth and environmental science. Both authored and edited volumes will be considered for publication. Edited volumes should however consist of a very limited number of contributions only. Proceedings will not be considered for LNC. The year 2010 marks the relaunch of LNC.
374 citations
TL;DR: In this paper, the exact boundary controllability of linear and nonlinear Korteweg-de Vries equations on bounded domains with various boundary conditions is studied, for sufficiently small initial and final states.
Abstract: The exact boundary controllability of linear and nonlinear Korteweg-de Vries equation on bounded domains with various boundary conditions is studied. When boundary conditions bear on spatial derivatives up to order 2, the exact controllability result by Russell-Zhang is directly proved by means of Hilbert Uniqueness Method. When only the first spatial derivative at the right endpoint is assumed to be controlled, a quite different analysis shows that exact controllability holds too. From this last result we derive the exact boundary controllability for nonlinear KdV equation on bounded domains, for sufficiently small initial and final states.
339 citations
"Local controllability of 1D linear ..." refers background in this paper
...Therefore, the controllability result of Theorem 1 enters the classical framework of local controllability results for nonlinear systems, proved with fixed point arguments (see, for instance, [55] by Rosier, [28] by Cerpa and Crépeau, [58] by Russell and Zhang, [63] by Zhang, [64] by Zuazua; this list is not exhaustive)....
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