3,212 research outputs found
Solving 1D Conservation Laws Using Pontryagin's Minimum Principle
This paper discusses a connection between scalar convex conservation laws and
Pontryagin's minimum principle. For flux functions for which an associated
optimal control problem can be found, a minimum value solution of the
conservation law is proposed. For scalar space-independent convex conservation
laws such a control problem exists and the minimum value solution of the
conservation law is equivalent to the entropy solution. This can be seen as a
generalization of the Lax--Oleinik formula to convex (not necessarily uniformly
convex) flux functions. Using Pontryagin's minimum principle, an algorithm for
finding the minimum value solution pointwise of scalar convex conservation laws
is given. Numerical examples of approximating the solution of both
space-dependent and space-independent conservation laws are provided to
demonstrate the accuracy and applicability of the proposed algorithm.
Furthermore, a MATLAB routine using Chebfun is provided (along with
demonstration code on how to use it) to approximately solve scalar convex
conservation laws with space-independent flux functions
Mitigating the Curse of Dimensionality: Sparse Grid Characteristics Method for Optimal Feedback Control and HJB Equations
We address finding the semi-global solutions to optimal feedback control and
the Hamilton--Jacobi--Bellman (HJB) equation. Using the solution of an HJB
equation, a feedback optimal control law can be implemented in real-time with
minimum computational load. However, except for systems with two or three state
variables, using traditional techniques for numerically finding a semi-global
solution to an HJB equation for general nonlinear systems is infeasible due to
the curse of dimensionality. Here we present a new computational method for
finding feedback optimal control and solving HJB equations which is able to
mitigate the curse of dimensionality. We do not discretize the HJB equation
directly, instead we introduce a sparse grid in the state space and use the
Pontryagin's maximum principle to derive a set of necessary conditions in the
form of a boundary value problem, also known as the characteristic equations,
for each grid point. Using this approach, the method is spatially causality
free, which enjoys the advantage of perfect parallelism on a sparse grid.
Compared with dense grids, a sparse grid has a significantly reduced size which
is feasible for systems with relatively high dimensions, such as the -D
system shown in the examples. Once the solution obtained at each grid point,
high-order accurate polynomial interpolation is used to approximate the
feedback control at arbitrary points. We prove an upper bound for the
approximation error and approximate it numerically. This sparse grid
characteristics method is demonstrated with two examples of rigid body attitude
control using momentum wheels
Efimov Physics in Atom-Dimer Scattering of Lithium-6 Atoms
Lithium-6 atoms in the three lowest hyperfine states display universal
properties when the S-wave scattering length between each pair of states is
large. Recent experiments reported four pronounced features arising from Efimov
physics in the atom-dimer relaxation rate, namely two resonances and two local
minima. We use the universal effective field theory to calculate the atom-dimer
relaxation rate at zero temperature. Our results describe the four features
qualitatively and imply there is a hidden local minimum. In the vicinity of the
resonance at 685 G, we perform a finite temperature calculation which improves
the agreement of theory and experiment. We conclude that finite temperature
effects cannot be neglected in the analysis of the experimental data.Comment: 13 pages, 5 figures, final versio
Exact Relations for a Strongly-interacting Fermi Gas near a Feshbach Resonance
A set of universal relations between various properties of any few-body or
many-body system consisting of fermions with two spin states and a large but
finite scattering length have been derived by Shina Tan. We derive
generalizations of the Tan relations for a two-channel model for fermions near
a Feshbach resonance that includes a molecular state whose detuning energy
controls the scattering length. We use quantum field theory methods, including
renormalization and the operator product expansion, to derive these relations.
They reduce to the Tan relations as the scattering length is made increasingly
large.Comment: 25 pages, 8 figure
Efimov Physics in 6Li Atoms
A new narrow 3-atom loss resonance associated with an Efimov trimer crossing
the 3-atom threshold has recently been discovered in a many-body system of
ultracold 6Li atoms in the three lowest hyperfine spin states at a magnetic
field near 895 G. O'Hara and coworkers have used measurements of the 3-body
recombination rate in this region to determine the complex 3-body parameter
associated with Efimov physics. Using this parameter as the input, we calculate
the universal predictions for the spectrum of Efimov states and for the 3-body
recombination rate in the universal region above 600 G where all three
scattering lengths are large. We predict an atom-dimer loss resonance at (672
+/- 2) G associated with an Efimov trimer disappearing through an atom-dimer
threshold. We also predict an interference minimum in the 3-body recombination
rate at (759 +/- 1) G where the 3-spin mixture may be sufficiently stable to
allow experimental study of the many-body system.Comment: 27 pages, 9 figures, REVTeX4, published versio
Three-body Recombination of Lithium-6 Atoms with Large Negative Scattering Lengths
The 3-body recombination rate at threshold for distinguishable atoms with
large negative pair scattering lengths is calculated in the zero-range
approximation. The only parameters in this limit are the 3 scattering lengths
and the Efimov parameter, which can be complex valued. We provide semi-analytic
expressions for the cases of 2 or 3 equal scattering lengths and we obtain
numerical results for the general case of 3 different scattering lengths. Our
general result is applied to the three lowest hyperfine states of Lithium-6
atoms. Comparisons with recent experiments provide indications of loss features
associated with Efimov trimers near the 3-atom threshold.Comment: 4 pages, 4 figures, agrees with published versio
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