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

The Construction of Heavy Metal Identity through Heritage Narratives: A Case Study of Extreme Metal Bands in the North of England

Extreme and black metal is a music genre infused with ideologies of elitism, nationalism and exaggerated masculinity. In this paper, we explore the ways in which four bands from the north of England – Winterfylleth, Wodensthrone, Old Corpse Road and Oakenshield – construct mythologies, heritage narratives and identity through their own reflections on their music, metal and myths. These extreme metal bands in the North of England work inside the symbolic boundaries of their scene and exist within the imagined communities of their region. That is, the bands construct mythologies based around masculinity and around elitism, but also about “northernness.

Global well-posedness and asymptotic behavior in Besov-Morrey spaces for chemotaxis-Navier-Stokes fluids

In this work we consider the Keller-Segel system coupled with Navier-Stokes equations in $\mathbb{R}^{N}$ for $N\geq2$. We prove the global well-posedness with small initial data in Besov-Morrey spaces. Our initial data class extends previous ones found in the literature such as that obtained by Kozono-Miura-Sugiyama (J. Funct. Anal. 2016). It allows to consider initial cell density and fluid velocity concentrated on smooth curves or at points depending on the spatial dimension. Self-similar solutions are obtained depending on the homogeneity of the initial data and considering the case of chemical attractant without degradation rate. Moreover, we analyze the asymptotic stability of solutions at infinity and obtain a class of asymptotically self-similar ones.Comment: 22 pages. Some typos have been corrected. Some references have been updated/correcte