Control Norms for Large Control Times

Sergei Ivanov

Controllablity of a quantum particle in a 1D variable domain

Karine Beauchard (2010)

ESAIM: Control, Optimisation and Calculus of Variations

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We consider a quantum particle in a 1D infinite square potential well with variable length. It is a nonlinear control system in which the state is the wave function of the particle and the control is the length of the potential well. We prove the following controllability result : given $φ_{0}$ close enough to an eigenstate corresponding to the length and $φ_{f}$ close enough to another eigenstate corresponding to the length , there exists a continuous function $l : [0, T] \to ℝ_{+}^{*}$ with , such that and , and...

On the asymptotic properties of a simple estimate of the Mode

Christophe Abraham, Gérard Biau, Benoît Cadre (2010)

ESAIM: Probability and Statistics

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We consider an estimate of the mode of a multivariate probability density with support in $ℝ^{d}$ using a kernel estimate drawn from a sample . The estimate is defined as any in {} such that $f_{n} (x) = {max}_{i = 1, \dots, n} f_{n} (X_{i})$ . It is shown that behaves asymptotically as any maximizer ${\hat{θ}}_{n}$ of . More precisely, we prove that for any sequence ${(r_{n})}_{n \geq 1}$ of positive real numbers such that $r_{n} \to \infty$ and $r_{n}^{d} log n / n \to 0$ , one has $r_{n} ∥ θ_{n} - {\hat{θ}}_{n} ∥ \to 0$ in probability. The asymptotic normality of follows without further work.

Displaying similar documents to “Control Norms for Large Control Times”

Controllablity of a quantum particle in a 1D variable domain

On the asymptotic properties of a simple estimate of the Mode