9,302 research outputs found
Controllability on infinite-dimensional manifolds
Following the unified approach of A. Kriegl and P.W. Michor (1997) for a
treatment of global analysis on a class of locally convex spaces known as
convenient, we give a generalization of Rashevsky-Chow's theorem for control
systems in regular connected manifolds modelled on convenient
(infinite-dimensional) locally convex spaces which are not necessarily
normable.Comment: 19 pages, 1 figur
Local controllability of 1D linear and nonlinear Schr\"odinger equations with bilinear control
We consider a linear Schr\"odinger equation, on a bounded interval, with
bilinear control, that represents a quantum particle in an electric field (the
control). We prove the controllability of this system, in any positive time,
locally around the ground state. Similar results were proved for particular
models (by the first author and with J.M. Coron), in non optimal spaces, in
long time and the proof relied on the Nash-Moser implicit function theorem in
order to deal with an a priori loss of regularity. In this article, the model
is more general, the spaces are optimal, there is no restriction on the time
and the proof relies on the classical inverse mapping theorem. A hidden
regularizing effect is emphasized, showing there is actually no loss of
regularity. Then, the same strategy is applied to nonlinear Schr\"odinger
equations and nonlinear wave equations, showing that the method works for a
wide range of bilinear control systems
emgr - The Empirical Gramian Framework
System Gramian matrices are a well-known encoding for properties of
input-output systems such as controllability, observability or minimality.
These so-called system Gramians were developed in linear system theory for
applications such as model order reduction of control systems. Empirical
Gramian are an extension to the system Gramians for parametric and nonlinear
systems as well as a data-driven method of computation. The empirical Gramian
framework - emgr - implements the empirical Gramians in a uniform and
configurable manner, with applications such as Gramian-based (nonlinear) model
reduction, decentralized control, sensitivity analysis, parameter
identification and combined state and parameter reduction
Noise Response Data Reveal Novel Controllability Gramian for Nonlinear Network Dynamics
Control of nonlinear large-scale dynamical networks, e.g., collective
behavior of agents interacting via a scale-free connection topology, is a
central problem in many scientific and engineering fields. For the linear
version of this problem, the so-called controllability Gramian has played an
important role to quantify how effectively the dynamical states are reachable
by a suitable driving input. In this paper, we first extend the notion of the
controllability Gramian to nonlinear dynamics in terms of the Gibbs
distribution. Next, we show that, when the networks are open to environmental
noise, the newly defined Gramian is equal to the covariance matrix associated
with randomly excited, but uncontrolled, dynamical state trajectories. This
fact theoretically justifies a simple Monte Carlo simulation that can extract
effectively controllable subdynamics in nonlinear complex networks. In
addition, the result provides a novel insight into the relationship between
controllability and statistical mechanics.Comment: 9 pages, 3 figures; to appear in Scientific Report
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