Jack W. Macki

Mathematical models for hysteresis

The various existing classical models for hysteresis, Preisach, Ishlinskii, Duhem–Madelung, are surveyed, as well a more modern treatments by contemporary workers. The emphasis is on a clear mathematical description of the formulation and properties of each model. In addition the authors try to make the reader aware of the many open questions in the study of hysteresis.

AN INTRODUCTION TO OPTIMAL CONTROL THEORY.

Abstract : The report presents an introduction to some of the concepts and results currently popular in optimal control theory. The introduction is intended for someone acquainted with ordinary differential equations and real variables, but with no prior knowledge of control theory. The material covered includes the problems of controllability, controllability using special (e.g., bang-bang) controls, the geometry of linear time optimal processes, general existence of optimal controls, and the Pontryagin maximum principle. (Author)

Case study from industry

The shoulder of a packaging machine is a developable surface that guides the packing material without stretching or tearing. The shoulder is traditionally manufactured by bending a flexible plate along a given bending curve, also without stretching ortearing. In this paper, the shoulder geometry isdescribed mathematically b methods from classical differential geometry. For a given bending curve the generating lines of the (developable) shoulder surface are completely specified. It is shown that the bending curve can be chosen such that the resulting shoulder contains a planar triangle. The special case of a conical shoulder is also discussed, and the underlying bending curve is determined explicitly.The shoulder of a packaging machine is a developable surface that guides the packing material without stretching or tearing. The shoulder is traditionally manufactured by bending a flexible plate along a given bending curve, also without stretching or tearing. In this paper, the shoulder geometry is described mathematically by methods from classical differential geometry. For a given bending curve the generating lines of the (developable) shoulder surface are completely specified. It is shown that the bending curve can be chosen such that the resulting shoulder contains a planar triangle. The special case of a conical shoulder is also discussed, and the underlying bending curve is determined explicitly.

Classroom notes

An algorithm is developed that will select a hospitals charge structure to maximize reimbursements. Although the algorithm relies on the Kuhn–Tucker theory of nonlinear programming, it may be implemented using elementary sorting techniques.

APPLICATIONS OF CHANGE OF VARIABLES IN THE QUALITATIVE THEORY OF SECOND ORDER LINEAR ORDINARY DIFFERENTIAL EQUATIONS

This paper illustrates the systematic use of a simple change of variable formula to extend existing results on boundedness and stability of solutions to ordinary differential equations of second order. This approach can also be used on oscillation criteria and other qualitative properties.

Invariants for the time‐dependent harmonic oscillator

Lewis showed that in the case p (t) ≡1, h=1, I (t) = (1/2) {p2(t)[ρ (t) y′ (t) −y (t) ρ′ (t)]2+h2y2(t)/ρ2(t) } is constant in time if y (t) solves (p (t) y′) ′+q (t) y =0, and ρ (t) solves p (t) ρ3(t) L[ρ]=h2 (h constant). Recently, Eliezer and Gray showed that I (t) =const is just the conservation of angular momentum in an appropriate physical interpretation. We show, using a change of variable technique, that I (t) =const reduces to sin2ϑ+cos2ϑ=1. We discuss uniqueness and extendability of solutions to the above equation in ρ.Lewis showed that in the case p (t) ≡1, h=1, I (t) = (1/2) {p2(t)[ρ (t) y′ (t) −y (t) ρ′ (t)]2+h2y2(t)/ρ2(t) } is constant in time if y (t) solves (p (t) y′) ′+q (t) y =0, and ρ (t) solves p (t) ρ3(t) L[ρ]=h2 (h constant). Recently, Eliezer and Gray showed that I (t) =const is just the conservation of angular momentum in an appropriate physical interpretation. We show, using a change of variable technique, that I (t) =const reduces to sin2ϑ+cos2ϑ=1. We discuss uniqueness and extendability of solutions to the above equation in ρ.

Notice: Erratum and Comments onInitialization of the Simplex Algorithm: An Artificial-Free Approach

Two sets of authors have commented on the Classroom Note by H. Arsham that appeared in the December 1997 issue of SIAM Review (SIAM Rev., 39 (1997), pp. 736-744). The first group consists of Professors Richard W. Cottle of Stanford University, Jong-Shi Pang of Johns Hopkins University, Richard E. Stone of Burnsville, Minnesota, and Craig A. Tovey of Georgia Tech. The second group consists of Andreas Enge and Petra Huhn of the University of Augsburg, Germany. For the details, see http://epubs.siam.org/sirev/97014.htm.

Periodic solutions of a control problem via marginal maps

SummaryWe investigate the existence of periodic solutions to the control problem(1)

Three Notes on the Exponential of a Matrix andApplications

\dot x = f(t,x,u) + g(t), x \in R^n , u \in R^m ,