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This article is designed to give through the study of difference equation (discrete dynamical systems) a view of and an introduction to the general theory of the stability of dynamical systems in its most modern aspect. Much of what is presented here is known, although not perhaps as well known as it should be, and there are some things that are new. One of these has to do with a connectedness property of the positive limit sets of the solutions of difference equations which provides a means through the use of Liapunov functions of establishing the existence of equilibrium points (fixed points) and oscillations (periodic points). Another is the generalization of the usual concept of a vector Liapunov function, and this leads to a possible method of designing control systems where the measure of the error or the performance criterion is a vector rather than a scalar. Applications of the theory are illustrated by simple examples., Sponsored in part by Air Force Office of Scientific Research, Arlington, Va., Army Research Office, Washington, D.C., and National Science Foundation, Washington, D.C.

Stochastic control problems of either the optimal stopping or impulsive control type on diffusion models are considered, but there is also a continuously acting control. Various computationally convenient Markov chain approximations to the problems are developed, and it is shown that the costs for the sequence of approximations converge to the optimal cost for original problems., Prepared in cooperation with Office of Naval Research, Arlington, Va.

Linear algebraic equations are derived for identifying the parameters of a linear multivariable differential system based on input-output data observed noise-free over a finite time interval. The technique circumvents the need for estimating the initial state of the system. The formulation allows for unknown input disturbances to the extent that such disturbances can be modeled by the free response of a linear differential equation. It is shown that a unique solution is obtained only if the component functions of the input-output data, together with a certain number of their derivatives, are linearly independent on the observation interval.

This paper is concerned with the development of a complete abstract invariant as well as a set of canonical forms under dynamic compensation for linear systems characterized by proper, rational transfer matrices. More specifically, it is shown that one can always associate with any proper rational transfer matrix, T(s), a special lower left triangular matrix, (Xi sub T)(s), called the interactor. This matrix is then shown to represent an abstract invariant under dynamic compensation which, together with the rank of T(s), represents a complete abstract invariant. A set of canonical forms under dynamic compensation is also developed along with appropriate dynamic compensation.

The report describes an investigation concerning parametric sonar applications to under-ice operations. Two basic tasks are undertaken. These are first, a data gathering program in an environment which partially simualtes the under-ice environment; and second, a theoretical investigation which extrapolates the results obtained to the under-ice environment. To gather data in a representative environment, a parametric sonar operating in a sub-bottom profiling mode was used in a downward looking mode analogous to upward scanning of the water to sea-ice interface. Results were analyzed and compared to possible under-ice operations in the theoretical investigations.

Advances in chemical assay methods for cholesterol are briefly summarized.

Finite Amplitude Sonar, also referred to as Parametric or Non-Linear Sonar, requires generation of high acoustic source levels. These high levels require high electrical powers and may stress transmitter and transducer components beyond normal linear ranges. Unless guarded against, such stresses introduce distortions into the signal paths and degrade the unique properties of Finite Amplitude operation. This report describes various transducer and transmitter configurations which can be applied to Finite Amplitude Sonar and discusses the potential advantages and disadvantages of each. Two opposing requirements are considered--high source levels and minimum signal distortion.

In the study of probabilities on geometrical objects, there have been some recent attempts to formulate general theories of random sets, notably by Kendall and Matheron. It is the purpose here to make a contribution in this direction by demonstrating the existence of a strong law of large numbers for random sets taking values in the class of compact subsets of (R sup n). The result is proved first under the assumption of convexity and then extended to the general case., Revision of report dated Jul 74.

EUBER STRESS FUNCTIONS FOR INFINITE DOMAINS, BY M. E. Gurtin. Aug 61, 15p. (Technical rept. no. 13) (Contract Nonr-56225, Proj. NR 064-431) Unclassified report DESCRIPTORS: (*Stresses, Materials, Solids, Elasticity, Deformation, Theory.) (Mechanical properties, *Applied mathematics, Vector analysis, Tensor analysis, Operators (Mathematics).) Open-ended Terms: Hel holtz theorem. Displacement equations of equilibrium in linear theory of homogeneous and isotropic elastic solids are discussed. Use is made of Helmholtz's theorem which is of interest in mathematical physics as applied to elastostatics. The Papkovich-Neuber solution of stress functions for infinite domains is applied.

Hydromagnetic flow in the vicinity of a two-dimensional stagnation point is considered, when the electrical conductivity is small and the applied magnetic field is at an arbitrary angle to the surface. First, inviscid flow is treated. Perturbation in the magnetic Reynolds number is singular; the previously found logarithmic singularity in vorticity at the surface of an arbitrary body is symptomatic of this fact. Then viscous effects are included, and it becomes evident that similarity solutions by no means tell the whole story. The principal functions are tabulated. (Author)

Given a fixed multivariable linear system in normal form, a design procedure by Anakwa and Pearson is applied to the dynamic compensation of a helicopter in the longitudinal mode to achieve dynamic decoupling between horizontal and vertical velocity, and collective pitch control and longitudinal cyclic pitch control, as well as a desired degree of stability, using only output feedback. Complete computer programs of the design procedure are included. Results of a computer simulation are discussed as well as the sensitivity due to variations in the helicopter parameters. These demonstrate the applicability and feasibility of the design procedures.

General projection methods for nonhomogeneous linear functional differential equations (FDE) dx/dt = L(x sub t) + f(t) are considered. It is shown that by an appropriate choice of spaces these can be treated as an equivalent abstract equation in a Banach space where the solution of the homogeneous equation generates a strongly continuous semigroup. Well-known approximation results for such semigroups (e.g., Trotter's theorem) are used to develop a general framework for obtaining approximate solutions of the original FDE. Convergence results are easily established. Applications to optimal control of linear FDE are discussed. In particular, the original infinite-dimensional problem is replaced by a 'projected' finite-dimensional problem which is more easily solved. Convergence of solutions of these finite-dimensional problems to solutions of the original problem is obtained., Sponsored in part by National Science Foundation, Washington, D.C. Grant NSF-GP-28931.

This is an expository paper describing the current state of representing space curves and surfaces by means of splines. It brings together in more readable form the content of a number of technical reports. As its title indicates, intensive use is made of pictures to both motivate the analysis and depict the spline in question.

The various levels of microprogramming languages are categorized, and the problems specific to the efficient compilation of microcode for horizontal architectures are considered. A general model of the compilation and optimization processes is developed, and techniques are suggested for treating the optimization process as a number of distinct phases. Where analytic solutions are not feasible, heuristic techniques are suggested.

The nonlinear Volterra integral equation is considered. The author discusses topologies on the collection of functions g such that the solution of the equation varies continuously with the data g and f, where the topology on f is the uniform convergence on compact intervals. A necessary and sufficient condition (on such a topology) for the continuous dependence to hold is given. In a particular case where a Lipschitz condition is added, it is shown that there exists a smallest topology which satisfies the condition. (Modified author abstract)

The problems of firmware development are briefly considered with specific references to the AN/UYK-17 SPE and, in particular, the Microprogrammed Control Unit (MCU). Alternatives treated are definition of a hospitable target machine, and direct generation of microcode from a higher level language. Various forms and levels of higher level languages are discussed. A motivation and source level description of a proposed language, PUMPKIN, are included. Three levels of this language are briefly described (base, base augmented with run time support, extensible base). Areas for proposed future research are outlined. (Author)

The manual contains descriptions and schematics of the proportional bandwidth spectrum analyzer (PBSA). It contains a brief description of the overall system, more detailed functional descriptions of the parts of the system, and schematics with timing diagrams and magnetic core layouts. The manual is divided along these lines into the three parts: General information; theory of operation; and, schematics. The PBSA is a signal processing portion of a developmental device to be used for research in underwater acoustic reception.

The initial value problem for the dynamical equations of elasticity and thermoelasticity is not well posed in the class of smooth motions so that an existence theory in the large can be established only for thermodynamic processes with shock waves. On the other hand, in this class of processes there is no uniqueness of solutions. It is shown in the report that, by visualizing a thermoelastic material as the limit of a family of thermoviscoelastic materials of the Maxwell type, one obtains a shock admissibility criterion, the viscosity criterion., Sponsored in part by Grant NSF-GP-28931.

The nonlinear wave equation, (phi sub tt) - delta phi + (phi sup 3) = 0, has many solutions that are periodic in time and localized in space, all with infinite energies. The search for spherically symmetric solutions that are well represented by the simple approximation, phi(r,t) about = A(r) sin omega t, leads to a discrete spectrum of solutions (phi sub N(r,t;omega)). These solutions are discussed in the report. (Modified author abstract)

This work aims at observing and measuring phase velocity, attenuation, and dispersion of stress waves in fiber-reinforced rods by the generation of longitudinal waves. The propagation of pulses in square bars of glass-epoxy fiber reinforced material is observed experimentally. Fiber orientation is varied from parallel to perpendicular to the bar's axis. Phase velocities are compared with those predicted from taking an effective Young's modulus for the bar. Geometric dispersion is considered by examining sinusoidal vibrations of composite rods of circular cross-section. Dispersion curves are predicted based on a lateral inertia correction and measured using a resonance method. The experimental results are not very extensive but nevertheless sufficient for qualitative conclusions to be advanced. Damping measurements are taken from oscilloscope records of free oscillations of the round composite rods. Again the results are not as complete as desired, but some conclusions can be drawn. (Author, modified-PL)

A theory of existence of optimal controls for control systems governed by nonlinear evolution equations in Banach space is presented. The main technique is to imbed the problem into the theory of nonlinear evolution equations as presented in another paper and make use of certain approximation theorems presented there. Applications to a nonlinear wave equation with control in the damping coefficient and a nonlinear conservation law model for highway traffic control are presented. (Author), Prepared in cooperation with Rensselaer Polytechnic Inst., Troy, N.Y. Dept. of Mathematics.

The concept of varying window size in a working set dispatcher to control working set size and number of page faults is examined. A space-time cost equation is developed and used to compare different dispatching algorithms and different types of secondary storage based on the simulated execution of real programs. A general approach is indicated for studying the relative merit of different dispatching algorithms and their interaction with different hardware configurations. (Author)

The paper is the last of a series of technical reports on the application of bicubic splines to surface representation. In particular, the report considers the representation of sphere-like surfaces by bicubic splines. Torus-like and cylinder-like surfaces are also considered. In all cases, the representations obtained are in terms of cardinal splines on rectangular meshes. This is made possible by the choice of coordinates employed. Both the parametric and non-parametric representations of surfaces are considered. In addition, an approximate representation for multiply connected regions is examined. (Author), See also report dated Oct 72, AD-751 675.

Theorems that give sufficient conditions for continuous dependence on parameters are the objects of the investigation. The concept of one theorem is better than the other is defined and a best result is a theorem which is better than all the other theorems in a given class. The author proves the existence of a best result and characterizes it. An application to ordinary differential equations is presented. (Author)

The paper deals with the addition of an interactive graphics terminal to even a small to medium size computer, enabling one to construct an 'n-dimensional blackboard'. On his device, with the use of control dials, joysticks, and other analog input devices, a mathematician can display and manipulate two-dimensional projections of n-dimensional geometrical constructs with much greater speed and accuracy than he can bring to this usual blackboard sketches. This has been implemented in a prototype version on the microprogrammed mini-computer BUGS system at Brown University. This paper describes the facilities of the BUGS blackboard, and reports on the observed interaction between the physical facilities available on the BUGS system and the sorts of mathematical results that the use of these facilities in the BUGS blackboard seem to encourage. (Author)

The execution of a program in a multiprogrammed system has to be interrupted frequently for reference to information stored in different levels of memory. To discuss the strategy of the basic decision algorithm appropriate for the system, it is necessary to know something about the manner in which these references to stored information are made. In dealing with a large program - such as an assembler or a compiler - it is impossible, in practice, to predict the references deterministically and it has been recognized for a long time that one has to resort to probabilistic models in this context. The choice of the correct probabilistic model is far from being an obvious one. It is the purpose of the paper to elucidate the problem by considering an analytic model and to relate the results of the analysis to actual measurements.

The term compumetrics was coined by Richard Hamming to describe the emerging science of the performance of computing systems. The term implies the emphasis on quantitative methods - analytic, simulation and empirical. Although - or perhaps because - activity in the field has been increasing recently, there exists as yet no systematic introduction to the subject. It is to fill this need that the present report has been written. The problem of evaluating a computing system has some of the flavor of operations research, looking for optimal, or at least reasonably good combinations, of given components. Much of the work being done proceeds in this spirit, employing the traditional techniques of operations research: probabilistic modelling, queueing theory, mathematical programming of various types, and data analysis. At the same time it differs from the typical problem of operations research in that the authors deal with a system whose components are known even on the most detailed level.

The paper is devoted to the problem of existence of periodic solutions for some nonautonomous neutral functional differential equations. It is essentially an application of a basic theorem on the Fredholm alternative for periodic solutions of some linear neutral equations recently obtained by one of the authors and of a generalized Leray-Schauder theory developed by the second one. Although their proofs are surprisingly simple, the obtained results are nontrivial extensions to the neutral case of a number of recent existence theorems for periodic solutions of functional differential equations., Supported in part by Grant NGL-40-002-015.

Many problems in analysis are concerned with the dependence upon parameters of fixed points of maps. For contraction mappings, criteria are relatively easy to obtain and have been known for some time. In the study of solutions of functional differential equations, more general results were needed. It is the purpose of this paper to give a rather general fixed-point theorem for condensing maps depending on a parameter, to prove continuous dependence and to indicate how many of the previous results are special cases. (Author), Sponsored in part by Grant NGL-40-002-015.

The paper is concerned with sequential Monte Carlo methods for optimizing a system under constraints. The authors wish to minimize f(x) where (q sub i) (x) 2 or = O, i = 1,...,m, must hold. The (q sub i) (x) can be calculated, but f(x) can only be observed in the presence of noise. A general approach, based on an adaptation of a version of stochastic approximation to the penalty function method, is discussed, and a convergence theorem proved. (Author Modified Abstract), Sponsored in part by Grant NSF-GK-31073X.

The paper discusses the SIMALE which is a very high speed, dynamically microprogrammed, parallel processing computer. It will soon be integrated into the Brown University Graphics System where it will be used primarily for real-time picture transformations. (Author), Pub. in SIGMICRO, v3 n4 22p 1973.

In the paper the author proves that if a periodic neutral differential equation with uniformly stable D operator (which includes retarded equations) has an uniformly asymptotically stable equilibrium point, and one perturbs the equation slightly then a periodic solution of the same period will exist. Since a generalization of a Browder's asymptotic fixed point theorem is available, the author will not assume that the period is greater than the lag, and this makes possible to extend the results for existence of an equilibrium point if the system and the perturbations are autonomous. (Author), Prepared in cooperation with Universidade de Campinas, Brazil.

The report is concerned with a Voigt material, which corresponds to a model having a spring and dashpot in parallel. The problem of longitudinal wave propagation in a semi-infinite rod which is initially unstrained and at rest and whose end is given an impulsive velocity, which is then maintained, is considered. Formulae for the stress and velocity distributions in the rod are obtained and their values are plotted against time for different positions in the rod and against position in the rod for different times, dimensionless coordinates being used. (Author)

Optimal control problems for systems described by differential equations with a delayed argument depending on state, control and time are considered. Unlike previous work in this area, there is no restrictive assumption on the monotonicity of the delayed argument with respect to time. The class of admissible controls consists of bounded measurable functions. The terminal point of the trajectory is assumed to be free or constrained by an inequality. Necessary conditions for optimality in the form of a maximum principle are derived via Gabasov's method. The problem of discontinuity of the adjoint variables, as well as other essential features of the maximum principle, are discussed. An example given in the Appendix shows that the optimal behavior of the delay need not necessarily consist in zeroing the delay on the whole time interval. (Author)

The general problem of representing disk-like surfaces is analyzed using bicubic splines. This is made possible via the introduction of a number of coordinate transformations. The representations obtained use cardinal splines which are considered in detail. Consideration is given to the use of intrinsic coordinates and also to the imposition of a uniform mesh. (Author)

Consider the linear control system in Hilbert space given by dx/dt = Ax + Bu. Here A is the infinitesimal generator of a C sub o semigroup of bounded linear operators T(t), t > or = O, on a real Hilbert space E. The author assumes that T(t) is such that norm//T(t)//(sub L(E,E)) < or = M(e sup(- omega t)) for some constants M > or = 1 and omega > O. B is a bounded linear operator from a real Hilbert space H to E and N(B) is properly contained in H. The author attempts to synthesize a feedback control u(t) = f(x(t)) whose active part is bounded, preserves the property of exponential asymptotic stability possessed by the uncontrolled system u = O, and is suboptimal in some sense. The synthesis is formally obtained but leads to a nonlinear singular evolution equation for the state variable X(t). The theory of maximal dissipative sets is then applied to show that the state evolution equation possesses a unique solution when the synthesis is modified in an appropriate multivalued way at the singularities. (Author)

The report covers the work which has been done on applications of modern control theory to the design of actual systems. Procedures have been developed that permit control system designers familiar with frequency domain techniques to understand and apply modern forms of control compensation and to indicate how the more modern theory can be used to improve system performance. Specific applications that have been looked at are the design of large scale power plants and aircraft control. Closely related to applications is also the work done on the development of numerical methods for synthesis and numerical solutions of the partial differentials that arise in stochastic control, monte carlo optimization, a computer program for parameter adaptive control when there is a time lag, digital compensation, iterative methods improving performance and for small noise stochastic control. New results have been obtained that should provide new methods for achieving guaranteed cost control and/or asymptotic stability where there are unknown time-varying parameters. (Author)

The extremum properties of the generalized Rayleigh quotient related to flutter instability are investigated. It is shown that, in addition to the well-known stationary property, under centain circumstances the quotient exhibits maximum-minimum properties which are in contrast to those of the classical Rayleigh quotient. One consequence is that a Rayleigh-Ritz type of stability analysis using these results leads to a lower bound as opposed to an upper bound in the classical case. The results are applied to multiple-parameter systems and a physical interpretation is given for the generalized Rayleigh quotient, leading to the proof of a convexity theorem. (Author), Prepared in cooperation with Waterloo Univ. (Ontario).

An energy balance was applied to the expansion process that occurs in a Wilson cloud chamber, and the nonadiabatic effects were thereby quantified. Chamber pressure, volume, and optical transmission were recorded continuously on a high-speed oscillograph, and this yielded highly accurate data both for chamber pressure against volume and for onset of condensation as a function of time. Both moist and dry argon were tested, and comparisons with theoretical values of heat transfer were made. Effective polytropic coefficients were defined, and the polytropic coefficients for the wet argon tests were used to calculate critical supersaturation. These actual critical supersaturations were compared with those computed for equivalent adiabatic expansions. The critical supersaturations were compared with the data of Wilson and Powell and also with the predictions from classical nucleation theory. (Author)

Some time ago, in 'Stability and oscillations in functional differential equations', the author announced a result on the Fredholm alternative for the existence of periodic solutions of a non-homogeneous linear neutral functional differential equation (NFDE). The aim of the paper is to indicate a proof of this result, and at the same time, use the method of proof to give a brief survey of some recent developments in the theory of NFDE which have applications far beyond the problem of periodic solutions. (Author)

Kestin proposed a method for adjusting the coefficients of a scalar stochastic approximation process, and proved w.p.1. convergence. A family of multidimensional processes for function minimization are treated here. Each method consists of a sequence of trucated one-dimensional procedures of the Kestin type. The methods seem to offer a number of advantages over the usual Kiefer-Wolfowitz procedures, and are more natural analogs of the schemes in common use in deterministic optimization theory. (Author), Supported in part by Grants AF-AFOSR-2078-71, and NSF-GK-31073.

The influence of a (a) coupling between flexural and extensional deformation and (b) coupling between structure and acoustic volume on the dynamic response of piezoelectric ceramic transducer elements mounted on metal diaphragms is analyzed using three analytical methods: (1) classical boundary value techniques; (2) simple direct variational procedures; and (3) finite element methods. The analyses are able to predict the voltage output of the transducer, including resonant amplitudes and shapes, with reasonable accuracy and, also, indicate critical front and back acoustic volume design parameters needed to control resonance. The finite element model includes a general formulation for axisymmetric, layered shells of revolution (which degenerates to a circular plate), whose average normal displacement is coupled to the long-wavelength motion of air in adjacent cavities (acoustic stiffness), ports (acoustic mass), and porous plugs (acoustic damping). The methods outlined here are also applicable to window-enclosure response to sonic boom excitation and to skull-brain impact studies. (Author)

In order to determine the instability behavior of continuous nonconservative systems, the transcendental characteristic equation usually must be solved numerically. This procedure is often quite complicated, and it is therefore of interest to obtain some qualitative information on the nature of instability. In the paper expressions for the slopes of the loading-frequency curves are derived, necessary conditions for flutter and divergence instability are found, and the application of these results is discussed. (Author)

Let T(t) be a C(sub 0) semigroup on a Banach space E with infinitesimal generator A. It is shown that A is the infinitesimal generator of a C(sub 0) semigroup on F where F is the Banach space D(A) endowed with the graph norm. (Author)

The paper considers the linear control system dx/dt = Ax + Bu. Here A is infinitesimal generator of a strongly continuous group of bounded linear operators T(t) on a Hilbert space E, B is a bounded linear operator from a Hilbert space H to E. The author gives sufficient conditions for the existence of a bounded linear operator K from E to H so that the control system with feedback control law u(t) = Kx(t) has the zero solution asymptotically stable. The results reduce to a well-known theorem of Kalman in the case E,H are finite dimensional. (Author)