Catastrophe Theory and the Problem of Stellar Collapse

Recently, a new mathematical tool called "catastrophe theory" has been developed by the topologists Thom, Zeeman, Mather, and others in an attempt to mathematically explain the discontinuities of observed behavior due to smooth changes in the basic parameters of physical, social, and biological processes. It has been shown that the number of mathematically distinct ways in which such discontinuities may arise is small when compared with the dimension of the process, and a complete classification of all distinct types has been made for processes depending upon five or less parameters. The purpose of this note is two-fold: first, to serve as a very brief introduction to the subject of catastrophe theory and secondly, to illustrate the theory by applying it to the determination of equilibrium configurations for stellar matter which has reached the endpoint of thermonuclear evolution, the problem of "stellar collapse". It will be seen that catastrophe theory enables us to give a very satisfactory explanation for the observed phenomenon of unstable equilibrium configurations and the appearance of the so-called Chandrasekhar and Oppenheimer-Landau-Volkoff crushing points

Singularity Theory for Nonlinear Optimization Problems

Techniques from the theory of singularities of smooth mappings are employed to study the reduction of nonlinear optimization problems to simpler forms. It is shown how singularity theory ideas can be used to : 1) reduce decision space dimensionality; (2) transform the constraint space to simpler form for primal algorithms; (3) provide sensitivity analysis

Algorithms for Stochastic Inflow Nonlinear Objective Water Reservoir Control Problem

In earlier IIASA Working Papers, algorithms to determine the optimal control of a water reservoir network with stochastic inflows and nonlinear utilities have been proposed. Both studies utilize a dynamic programming-type approach, coupled with approximations of one type or another, in order to yield a computational algorithm in which the bulk of the calculation is carried out by efficient (and rapid) network flow algorithms. The purpose of this note is to present a synthesis of the work and to spell out the precise steps of an algorithm in sufficient detail to enable a computer program to be constructed

Polyhedral Dynamics - II: Geometrical Structure as a Basis for Decision Making in Complex Systems

The tools of polyhedral dynamics and dynamic programming are combined through the medium of cross-impact analysis to attack problems of organizational structure. It is argued that the standard cross-impact approaches to such problems are deficient in that they ignore the true multi-dimensional nature of such systems, as well as providing no systematic mechanism for rational decision making. The results of the analysis are illustrated by applications to the structuring of a large scientific organization and by the analysis of a simplified version of a problem arising in the energy field

Some Recent Developments in the Theory and Computation of Linear Control Problems

Recent analysis and computational results for the solution of linear dynamics-quadratic cost control processes are presented. It is shown that, if the number of system inputs and outputs is less than the number of state variables, a substantial reduction in computing effort may be achieved by utilizing the new equations, termed "generalized X-Y" functions over the standard matrix Riccati equation solution. In addition to the basic X-Y equations, the paper also discusses the reduced algebraic equation for infin-infinite-interval problems, infinite-dimensional problems, the discrete-time case, and Kalman filtering problems. Numerical experiments are also reported

Connectivity and Stability in Ecological and Energy Systems

One of the more enduring topics of methodological interest at IIASA has been the problem of ascertaining and describing stability characteristics for large-scale systems. These points have been of particular applied interest in the ecology and energy areas where the terms "resilience" and "hypotheticality" have been used to intuitively characterize the type of stability of greatest practical interest. Our primary purpose in this note is to present some new results in stability theory which have great relevance to the aforementioned studies. These results deal with the problem of "connective" stability, in which the basic question is how large a perturbation in structure the system can withstand and still remain asymptotically stable. In many ways, these results are similar in spirit to structural stability questions in which the invariance of the topological features of the system trajectory is the central issue. However, the two theories are not the same as connective stability deals with the stability of a point under structural perturbation, while structural stability is concerned with trajectories. In addition, connective stability is a quantitative theory as precise numerical estimates can be given for the magnitude of the allowed perturbation, while structural stability is primarily qualitative. Therefore, we feel justified in presenting these results in order to provide systems analysts with another tool to probe the stability characteristics of applied systems. A secondary objective of this note is to point out the connections between the notion of connective stability and the idea of a system's connectivity pattern. All of these results will be illustrated with examples from energy and ecology

Invariant Theory, the Riccati Group, and Linear Control Problems

The classical algebraic theory of invariants is applied to the linear-quadratic-gaussian (LQG) control problem to derive a canonical form under a certain matrix transformation group. The particular group of transformations, termed here the "Riccati group," is induced from the matrix Riccati equation characterizing the LQG problem solution. Examples of the invariant-theoretic approach are given along with a discussion of topics meriting further study, including geometric interpretation of the group orbits, extension of the Riccati group, and connections with the generalized X-Y functions

Resiliency, Stability and the Domain of Attraction

The paper demonstrates that, by suitable application of linear control theory, it is possible to modify the domain of attraction of a critical point for certain nonlinear systems