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Discussion of certain problems in system theory which have been or might be solved using some basic concepts from differential geometry. The problems considered involve differential equations, controllability, optimal control, qualitative behavior, stochastic processes, and bilinear systems. The main goal is to extend the essentials of linear theory to some nonlinear classes of problems.
Methods for transforming partial differential equations into forms more suitable for analysis and solution are investigated. The idea of Lie's infinitesimal contact transformation group is introduced to develop a systematic method which involves mostly algebraic manipulations. A thorough presentation of the application of this general method to the problem of similarity analysis in a broader sense - namely, the similarity between partial and ordinary differential equations, boundary value and initial value problems, and nonlinear and linear equations - is given with new and very general methods evolved for deriving the possible groups of transformations.
Basic problems for a special class of neutral functional differential equations (NFDE) are formulated, and some contributions to a general qualitative theory in the neighborhood of an equilibrium point are indicated. The properties of a NFDE (G,f) are examined to determine in what sense these properties are insensitive to small changes in (G,f) in the topology G x F. The special class of equations that is introduced includes retarded functional differential equations and difference equations.
Certain symmetry properties possessed by the solutions of linear differential equations are examined. For this purpose, some basic ideas from the theory of finite dimensional linear systems are used together with the work of Wei and Norman on the use of Lie algebraic methods in differential equation theory.
Two systems have been developed to resolve phase uncertainty without transmitting reference signals. In both methods signal is impressed on carrier as differential, rather than absolute, phase shift. At the receiver four-phase demodulation and logic process unambiguously resolves differential phase shift of input carrier.
Proton recoil distributions were obtained by using organic liquid scintillators of different size. The measured distributions are converted to neutron spectra by differentiation analysis for comparison to the unfolded spectra of the largest scintillator. The approximations involved in the differentiation analysis are indicated to have small effects on the precision of neutron spectra measured with the smaller scintillators but introduce significant error for the largest scintillator. In the case of the smallest cylindrical scintillator, nominally 1.2 by 1.3 cm, the efficiency is shown to be insensitive to multiple scattering and to the angular distribution to the incident flux. These characteristics of the smaller scintillator make possible its use to measure scalar flux spectra within media high efficiency is not required.
The differential maneuvering simulator for simulating two aircraft or spacecraft operating in a differential mode is described. Tests made to verify that the system could provide the required simulated aircraft motions are given. The mathematical model which converts computed aircraft motions into the required motions of the various projector gimbals is described.
Formulation and study of the initial value problem for neutral functional differential equations. The existence, uniqueness, and continuation of solutions to this problem are investigated, and an analysis is made of the dependence of the solutions on the initial conditions and parameters, resulting in the derivation of a continuous dependence theorem in which the fundamental mathematical principles underlying the continuous dependence problem for a very general system of nonlinear neutral functional differential equations are separated out.
Bounded topologies are considered for functional differential equations of the neutral type in which present dynamics of the system are influenced by its past behavior. A special bounded topology is generated on a collection of absolutely continuous functions with essentially bounded derivatives, and an application to a class of nonlinear neutral functional differential equations due to Driver (1965) is presented.
Numerical integration methods for the solution of initial value problems for ordinary vector differential equations may be modelled as discrete time feedback systems. The stability criteria discovered in modern control theory are applied to these systems and criteria involving the routine, the step size and the differential equation are derived. Linear multistep, Runge-Kutta, and predictor-corrector methods are all investigated.
General discussion of the theory of differential games with two players and zero sum. Games starting at a fixed initial state and ending at a fixed final time are analyzed. Strategies for the games are defined. The existence of saddle values and saddle points is considered. A stochastic version of a differential game is used to examine the synthesis problem.
Experimental results are developed for the relative elastic differential scattering of He(++) by He for collision energies in the range 4 equal to or less than E equal to or less than 75 eV. In the analysis of the data, semiclassical considerations are utilized, assuming that the dynamics of the scattering is governed solely by the B and E states of He2(++). It is shown that existing ab initio calculations for the intermolecular potentials predict differential cross sections which are not in particularly good agreement with the experimental data.
The use of a PLTZ block data composer as a matrix type phase modulator to record and process digital data by the differential phase mode in a holographic recording/processing system was investigated. The system has readout contrast ratios of between 10 and 15 to 1. The differential phase mode has the advantage that strain bias is not required and thickness and strain variations in the PLZT are cancelled out.
A condition for the Gateaux differentiability of the probability of misclassification as a function of a feature selection matrix B, assuming a maximum likelihood classifier and normally distributed populations, is given. It is also shown that if the probability of error has a local minimum at B then it is differentiable at B.
A differential phase shift keyed signal resolver resolves the differential phase shift in the incoming signal to determine the data content thereof overcoming phase uncertainty without requiring a transmitted reference signal.
Systems of ordinary differential equations in which the magnitudes of the eigenvalues (or time constants) vary greatly are commonly called stiff. Such systems of equations arise in nuclear reactor kinetics, the flow of chemically reacting gas, dynamics, control theory, circuit analysis and other fields. The research reported develops an A-stable numerical integration technique for solving stiff systems of ordinary differential equations. The method, which is called the generalized trapezoidal rule, is a modification of the trapezoidal rule. However, the method is computationally more efficient than the trapezoidal rule when the solution of the almost-discontinuous segments is being calculated.
The spherical means of the solutions of a linear partial differential equation Lu = f in a conical region are studied. The conical region is bounded by a surface generated by curvilinear ti surfaces. The spherical mean is the average of u over a constant ti surface. The conditions on the linear differential operator, L, and on the orthogonal coordinates (ti, eta, zeta) are established so that the spherical mean of the solution subjected to the appropriate boundary and initial conditions can be determined directly as a problem with only space variable. Conditions are then established so that the spherical mean of the solution in one concial region will be proportional to that of a known solution in another conical region. Applications to various problems of mathematical physics and their physical interpretations are presented.
The results of an intensive analysis of a differential orbit improvement method utilizing observational data for a 550-kilometer altitude, near-circular, near-equatorial satellite orbit are presented. Observations of the Small Astronomy Satellite (SAS-1) are in the form of direction cosines as measured at two ground interferometer tracking stations near the Equator during the first 22 orbital revolutions (approximately 37 hours) after launch of the spacecraft. Numerical results, in both tabular and graphical form, are displayed for numerous iterated fittings of various observational arcs by differential correction of the orbital elements. Parameters varied in these comparative cases include the time duration of the observational data block, the number of pairs of direction cosine data and the number of tracking station passes included in the solution, the distribution of such passes between the two available tracking stations, and the acceptance criterion for the observational residuals in the least squares fitting procedure. It was found that three observational pairs of direction cosine data, the minimum number possible for a uniquely determined solution in theory, are sufficient to promote covergence to an accurate solution, if properly selected.