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

Comment on ``Quantum Phase of Induced Dipoles Moving in a Magnetic Field''

It has recently been suggested that an Aharonov-Bohm phase should be capable of detection using beams of neutral polarizable particles. A more careful analysis of the proposed experiment suffices to show, however, that it cannot be performed regardless of the strength of the external electric and magnetic fields.Comment: 2 pages, latex file, no figure

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

Attenuation of Persistent L∞-Bounded Disturbances for Nonlinear Systems

A version of nonlinear generalization of the L1-control problem, which deals with the attenuation of persistent bounded disturbances in L∞-sense, is investigated in this paper. The methods used in this paper are motivated by [23]. The main idea in the L1-performance analysis and synthesis is to construct a certain invariant subset of the state-space such that achieving disturbance rejection is equivalent to restricting the state-dynamics to this set. The concepts from viability theory, nonsmooth analysis, and set-valued analysis play important roles. In addition, the relation between the L1-control of a continuous-time system and the l1-control of its Euler approximated discrete-time systems is established

Do local manufacturing firms benefit from transactional linkages with multinational enterprises in China?

This paper examines the linkage effects of foreign direct investment (FDI) on firm-level productivity in Chinese manufacturing. It is found that FDI generates positive vertical linkage effects in Chinese manufacturing at both the national and regional levels, and limited positive horizontal spillovers at the regional level. While OECD firms gain from both vertical and (probably) horizontal linkages, Hong Kong, Macao and Taiwanese firms benefit only from backward linkage effects. In the domestic sector, in which we are most interested, both state-owned enterprises (SOEs) and non-SOEs are hurt by competition from foreign firms in the same industries. While SOEs gain from vertical linkages with foreign firms, non-SOEs are unable to do so. The patterns of productivity spillovers from FDI in Chinese manufacturing seem to be determined by one key factor – the technological capabilities of the firms involved. Important data limitations and policy implications of this research are discussed