The Optimal Control of a Train

We consider the problem of determining an optimal driving strategy in a train control problem with a generalised equation of motion. We assume that the journey must be completed within a given time and seek a strategy that minimises fuel consumption. On the one hand we consider the case where continuous control can be used and on the other hand we consider the case where only discrete control is available. We pay particular attention to a unified development of the two cases. For the continuous control problem we use the Pontryagin principle to find necessary conditions on an optimal strategy and show that these conditions yield key equations that determine the optimal switching points. In the discrete control problem, which is the typical situation with diesel-electric locomotives, we show that for each fixed control sequence the cost of fuel can be minimised by finding the optimal switching times. The corresponding strategies are called strategies of optimal type and in this case we use the Kuhn–Tucker equations to find key equations that determine the optimal switching times. We note that the strategies of optimal type can be used to approximate as closely as we please the optimal strategy obtained using continuous control and we present two new derivations of the key equations. We illustrate our general remarks by reference to a typical train control problem.     

A comparison of phosphorus and fluorine containing IL lubricants for steel on aluminium

Ionic liquids have been shown to be highly effective lubricants for a steel on aluminium system. This work shows that the chemistry of the anion and cation are critical in achieving maximum wear protection. The performance of the ILs containing a diphenylphosphate (DPP) anion all showed low wear, as did some of the tris(pentafluoroethyl)trifluorophosphate (FAP) and bis(trifluoromethanesulfonyl)amide (NTf(2)) anion containing ILs. However, in the case of the FAP and NTf(2) based systems, a cation dependence was observed, with relatively poor wear resistance obtained in the case of an imidazolium FAP and two pyrrolidinium NTf(2) salts, probably due to tribocorrosion caused by the fluorine reaction with the aluminium substrate. The systems exhibiting poor performance generally had a lower viscosity, which also impacts on their tribological properties. Those ILs that exhibited low wear were shown to have formed protective tribofilms on the aluminium alloy surface.     

A philosophy for the modelling of realistic nonlinear systems

A nonlinear dynamical system is modelled as a nonlinear mapping from a set of input signals into a corresponding set of output signals. Each signal is specified by a set of real number parameters, but such sets may be uncountably infinite. For numerical simulation of the system each signal must be represented by a finite parameter set and the mapping must be defined by a finite arithmetical process. Nevertheless the numerical simulation should be a good approximation to the mathematical model. We discuss the representation of realistic dynamical systems and establish a stable approximation theorem for numerical simulation of such systems.

     

A Decomposition of the Descent Algebra of a Finite Coxeter Group

The purpose of this paper is twofold. First we aim to unify previous work by the first two authors, A. Garsia, and C. Reutenauer (see [2], [3], [4], [5] and [10]) on the structure of the descent algebras of the Coxeter groups of type A _n and B _n. But we shall also extend these results to the descent algebra of an arbitrary finite Coxeter group W. The descent algebra, introduced by Solomon in [14], is a subalgebra of the group algebra of W. It is closely related to the subring of the Burnside ring B(W) spanned by the permutation representations W/W _J, where the W _J are the parabolic subgroups of W. Specifically, our purpose is to lift a basis of primitive idempotents of the parabolic Burnside algebra to a basis of idempotents of the descent algebra.