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Global exact controllability of bilinear quantum systems on compact graphs and energetic controllability.
TL;DR: The controllability of the bilinear Schrodinger equation on compact graphs was studied in this paper, where the authors introduced the notion of "energetic controLLability", which is useful when the global exact controllation fails.
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.
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TL;DR: In this article, the bilinear Schrodinger equation (BSE) is considered in the Hilbert space L2(G,C) with G an infinite graph. And the Laplacian −Δ is equipped with self-adjoint boundary cond...
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...
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Book•
01 Jan 1968
TL;DR: This book shows engineers how to use optimization theory to solve complex problems with a minimum of mathematics and unifies the large field of optimization with a few geometric principles.
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.
5,667 citations
TL;DR: For the observation or control of solutions of second-order hyperbolic equation in this paper, Ralston's construction of localized states [Comm. Pure Appl. Math, 22 (1969), pp.
Abstract: For the observation or control of solutions of second-order hyperbolic equation in $\mathbb{R}_t \times \Omega $, Ralston’s construction of localized states [Comm. Pure Appl. Math., 22 (1969), pp. ...
1,510 citations
Book•
01 Jan 1980
TL;DR: Bases in Banach Spaces - Schauder Bases Schauder's Basis for C[a,b] Orthonormal Bases in Hilbert Space The Reproducing Kernel Complete Sequences The Coefficient Functionals Duality Riesz Bases The Stability of Bases of Complex Exponentials as mentioned in this paper.
Abstract: Bases in Banach Spaces - Schauder Bases Schauder's Basis for C[a,b] Orthonormal Bases in Hilbert Space The Reproducing Kernel Complete Sequences The Coefficient Functionals Duality Riesz Bases The Stability of Bases in Banach Spaces The Stability of Orthonormal Bases in Hilbert Space Entire Functions of Exponential Type The Classical Factorization Theorems - Weierstrass's Factorization Theorem Jensen's Formula Functions of Finite Order Estimates for Canonical Products Hadamard's Factorization Theorem Restrictions Along a Line - The "Phragmen-Lindelof" Method Carleman's Formula Integrability on a line The Paley-Wiener Theorem The Paley-Wiener Space The Completeness of Sets of Complex Exponentials - The Trigonometric System Exponentials Close to the Trigonometric System A Counterexample Some Intrinsic Properties of Sets of Complex Exponentials Stability Density and the Completeness Radius Interpolation and Bases in Hilbert Space - Moment Sequences in Hilbert Space Bessel Sequences and Riesz-Fischer Sequences Applications to Systems of Complex Exponentials The Moment Space and Its Relation to Equivalent Sequences Interpolation in the Paley-Wiener Space: Functions of Sine Type Interpolation in the Paley-Wiener Space: Stability The Theory of Frames The Stability of Nonharmonic Fourier Series Pointwise Convergence Notes and Comments References List of Special Symbols Index
1,504 citations
TL;DR: A quantum graph as discussed by the authors is a graph equipped with a self-adjoint differential or pseudo-differential Hamiltonian, which is a special case of a combinatorial graph model.
Abstract: A quantum graph is a graph equipped with a self-adjoint differential or pseudo-differential Hamiltonian. Such graphs have been studied recently in relation to some problems of mathematics, physics and chemistry. The paper has a survey nature and is devoted to the description of some basic notions concerning quantum graphs, including the boundary conditions, self-adjointness, quadratic forms, and relations between quantum and combinatorial graph models.
681 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