Asymptotics for optimal design problems for the Schr\"odinger equation with a potential

We study the problem of optimal observability and prove time asymptotic observability estimates for the Schr\"odinger equation with a potential in $L^{\infty}(\Omega)$, with $\Omega\subset \mathbb{R}^d$, using spectral theory. An elegant way to model the problem using a time asymptotic observability constant is presented. For certain small potentials, we demonstrate the existence of a nonzero asymptotic observability constant under given conditions and describe its explicit properties and optimal values. Moreover, we give a precise description of numerical models to analyze the properties of important examples of potentials wells, including that of the modified harmonic oscillator

The uniform controllability property of semidiscrete approximations for the parabolic distributed parameter systems in Banach spaces

The problem we consider in this work is to minimize the L^q-norm (q > 2) of the semidiscrete controls. As shown in [LT06], under the main approximation assumptions that the discretized semigroup is uniformly analytic and that the degree of unboundedness of control operator is lower than 1/2, the uniform controllability property of semidiscrete approximations for the parabolic systems is achieved in L^2. In the present paper, we show that the uniform controllability property still continue to be asserted in L^q. (q > 2) even with the con- dition that the degree of unboundedness of control operator is greater than 1/2. Moreover, the minimization procedure to compute the ap- proximation controls is provided. An example of application is imple- mented for the one dimensional heat equation with Dirichlet boundary control

Controllability of the Schr\"odinger equation on unbounded domains without geometric control condition

We prove controllability of the Schr\"odinger equation in $\mathbb{R}^d$ in any time $T > 0$ with internal control supported on nonempty, periodic, open sets. This demonstrates in particular that controllability of the Schr\"odinger equation in full space holds for a strictly larger class of control supports than for the wave equation and suggests that the control theory of Schr\"odinger equation in full space might be closer to the diffusive nature of the heat equation than to the ballistic nature of the wave equation. Our results are based on a combination of Floquet-Bloch theory with Ingham-type estimates on lacunary Fourier series.Comment: 12 pages, 3 figures. We removed partly erroneous statements on fractional Laplacian

An investigation of the mathematical formulation of quantum theory and its physical interpretation, 1900-1927

Imperial Users onl

Exact controllability for quasi-linear perturbations of KdV

We prove that the KdV equation on the circle remains exactly controllable in arbitrary time with localized control, for sufficiently small data, also in presence of quasi-linear perturbations, namely nonlinearities containing up to three space derivatives, having a Hamiltonian structure at the highest orders. We use a procedure of reduction to constant coefficients up to order zero, classical Ingham inequality and HUM method to prove the controllability of the linearized operator. Then we prove and apply a modified version of the Nash-Moser implicit function theorems by H\"ormander.Comment: 39 page

Final report on estimation and statistical analysis of spatially distributed random processes

Includes bibliographical references.Final report;Supported by the NSF. ECS-8312921prepared by Alan S. Willsky, Bernard C. Levy, George C. Verghese