92 research outputs found
Periodic excitations of bilinear quantum systems
A well-known method of transferring the population of a quantum system from
an eigenspace of the free Hamiltonian to another is to use a periodic control
law with an angular frequency equal to the difference of the eigenvalues. For
finite dimensional quantum systems, the classical theory of averaging provides
a rigorous explanation of this experimentally validated result. This paper
extends this finite dimensional result, known as the Rotating Wave
Approximation, to infinite dimensional systems and provides explicit
convergence estimates.Comment: Available online
http://www.sciencedirect.com/science/article/pii/S000510981200286
Generalized Scallop Theorem for Linear Swimmers
In this article, we are interested in studying locomotion strategies for a
class of shape-changing bodies swimming in a fluid. This class consists of
swimmers subject to a particular linear dynamics, which includes the two most
investigated limit models in the literature: swimmers at low and high Reynolds
numbers. Our first contribution is to prove that although for these two models
the locomotion is based on very different physical principles, their dynamics
are similar under symmetry assumptions. Our second contribution is to derive
for such swimmers a purely geometric criterion allowing to determine wether a
given sequence of shape-changes can result in locomotion. This criterion can be
seen as a generalization of Purcell's scallop theorem (stated in Purcell
(1977)) in the sense that it deals with a larger class of swimmers and address
the complete locomotion strategy, extending the usual formulation in which only
periodic strokes for low Reynolds swimmers are considered.Comment: 14 pages, 10 figure
Generic Controllability of 3D Swimmers in a Perfect Fluid
We address the problem of controlling a dynamical system governing the motion
of a 3D weighted shape changing body swimming in a perfect fluid. The rigid
displacement of the swimmer results from the exchange of momentum between
prescribed shape changes and the flow, the total impulse of the fluid-swimmer
system being constant for all times. We prove the following tracking results:
(i) Synchronized swimming: Maybe up to an arbitrarily small change of its
density, any swimmer can approximately follow any given trajectory while, in
addition, undergoing approximately any given shape changes. In this statement,
the control consists in arbitrarily small superimposed deformations; (ii)
Freestyle swimming: Maybe up to an arbitrarily small change of its density, any
swimmer can approximately tracks any given trajectory by combining suitably at
most five basic movements that can be generically chosen (no macro shape
changes are prescribed in this statement)
Hybrid control for low-regular nonlinear systems: application to an embedded control for an electric vehicle
This note presents an embedded automatic control strategy for a low
consumption vehicle equipped with an "on/off" engine. The main difficulties are
the hybrid nature of the dynamics, the non smoothness of the dynamics of each
mode, the uncertain environment, the fast changing dynamics, and low cost/ low
consumption constraints for the control device. Human drivers of such vehicles
frequently use an oscillating strategy, letting the velocity evolve between
fixed lower and upper bounds. We present a general justification of this very
simple and efficient strategy, that happens to be optimal for autonomous
dynamics, robust and easily adaptable for real-time control strategy. Effective
implementation in a competition prototype involved in low-consumption races
shows that automatic velocity control achieves performances comparable with the
results of trained human drivers. Major advantages of automatic control are
improved robustness and safety. The total average power consumption for the
control device is less than 10 mW
Small time reachable set of bilinear quantum systems
This note presents an example of bilinear conservative system in an infinite
dimensional Hilbert space for which approximate controllability in the Hilbert
unit sphere holds for arbitrary small times. This situation is in contrast with
the finite dimensional case and is due to the unboundedness of the drift
operator
Which notion of energy for bilinear quantum systems?
In this note we investigate what is the best L^p-norm in order to describe
the relation between the evolution of the state of a bilinear quantum system
with the L^p-norm of the external field. Although L^2 has a structure more easy
to handle, the L^1 norm is more suitable for this purpose. Indeed for every
p>1, it is possible to steer, with arbitrary precision, a generic bilinear
quantum system from any eigenstate of the free Hamiltonian to any other with a
control of arbitrary small L^p norm. Explicit optimal costs for the L^1 norm
are computed on an example
Optimal Strokes for Driftless Swimmers: A General Geometric Approach
Swimming consists by definition in propelling through a fluid by means of
bodily movements. Thus, from a mathematical point of view, swimming turns into
a control problem for which the controls are the deformations of the swimmer.
The aim of this paper is to present a unified geometric approach for the
optimization of the body deformations of so-called driftless swimmers. The
class of driftless swimmers includes, among other, swimmers in a 3D Stokes flow
(case of micro-swimmers in viscous fluids) or swimmers in a 2D or 3D potential
flow. A general framework is introduced, allowing the complete analysis of five
usual nonlinear optimization problems to be carried out. The results are
illustrated with examples coming from the literature and with an in-depth study
of a swimmer in a 2D potential flow. Numerical tests are also provided
Efficient finite dimensional approximations for the bilinear Schrodinger equation with bounded variation controls
This the text of a proceeding accepted for the 21st International Symposium
on Mathematical Theory of Networks and Systems (MTNS 2014). We present some
results of an ongoing research on the controllability problem of an abstract
bilinear Schrodinger equation. We are interested by approximation of this
equation by finite dimensional systems. Assuming that the uncontrolled term
has a pure discrete spectrum and the control potential is in some sense
regular with respect to we show that such an approximation is possible.
More precisely the solutions are approximated by their projections on finite
dimensional subspaces spanned by the eigenvectors of . This approximation is
uniform in time and in the control, if this control has bounded variation with
a priori bounded total variation. Hence if these finite dimensional systems are
controllable with a fixed bound on the total variation of the control then the
system is approximatively controllable. The main outcome of our analysis is
that we can build solutions for low regular controls such as bounded variation
ones and even Radon measures
Nonisotropic 3-level Quantum Systems: Complete Solutions for Minimum Time and Minimum Energy
We apply techniques of subriemannian geometry on Lie groups and of optimal
synthesis on 2-D manifolds to the population transfer problem in a three-level
quantum system driven by two laser pulses, of arbitrary shape and frequency. In
the rotating wave approximation, we consider a nonisotropic model i.e. a model
in which the two coupling constants of the lasers are different. The aim is to
induce transitions from the first to the third level, minimizing 1) the time of
the transition (with bounded laser amplitudes),
2) the energy of lasers (with fixed final time). After reducing the problem
to real variables, for the purpose 1) we develop a theory of time optimal
syntheses for distributional problem on 2-D-manifolds, while for the purpose 2)
we use techniques of subriemannian geometry on 3-D Lie groups. The complete
optimal syntheses are computed.Comment: 29 pages, 6 figure
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