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 $6$-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