We consider an optimal control problem with, possibly time-dependent, constraints on state and control variables, jointly. Using only elementary methods, we derive a sufficient condition for optimality. Although phrased in terms reminiscent of the necessary condition of Pontryagin, the sufficient condition is logically independent, as can be shown by a simple example.

When optical fibres are made by first constructing optical-fibre preforms, the fibre which is pulled from the heated preform is simply a scaled down version of the original preform structure. The expansion coefficient profile α(r) of the preform, which relates directly to the fabrication variables, can be determined from non-destructive optical retardation measurements δ(r) performed on the preform. In addition, the residual elastic stress distributions in a fabricated preform, which can be used to compare different fabrication procedures, have simple definitions as linear functionals of the expansion coefficients α(r). Thus, through the use of optical retardation data, an examination of different manufacturing procedures for preform fabrication is reduced to a problem in non-destructive manufacturing procedures for preform fabrication is reduced to a problem in non-destructive testing and analysis. The underlying numerical problem of evaluating the stres distributions reduces to solving and Abel-type integral equation for α(r), which involves an indeterminacy, followed by the evaluation of linear functionals defined on α(r). It is shown how the known inversion formulae for the Abel-type integral equation can be used formally to reduce the numerical problem of evaluating the radial stress to the evaluation of a linear functional defined on the data δ(r) which bypasses the indeterminacy. When only the radial stress is required, the problem of actually solving the Abel-type integral equation is avoided. Methods for evaluating the non-radial stresses, which avoid the indeterminacy, are also derived.

Solutions of the stiff system of linear differential equations

are obtained in a form yielding tight estimates of their properties, and conditions are obtained under which the operator norm of the map from r to the solution x does not become exponentially large for small values of ε. When these conditions are satisfied, the solutions are shown to be close to those of Ax + r = 0, save at any singular points of A, and in boundary layers. The behaviour of solutions near admissible singular points is also obtained.

The results are used to characterize those boundary-value problems for the above system in which the solution defines maps from the data that are of “moderate” operator norm. This leads to a constructive existence theory for a limited class of boundary-value problems for the nonlinear system

It is suggested that the treatment of more general classes of boundary-value problems may be simplified using these results. By the use of simple examples, the problems involving large operator norms are shown to be related to the stability properties of the possible branches of the outer solutions close to those of

The method of the Lie theory of extended groups has recently been formulated for Hamiltonian mechanics in a manner which is consistent with the results obtained using the Newtonian equation of motion. Here the method is applied to the three-dimensional time-independent harmonic oscillator and to the classical Kepler problem. The expected constants of motion are obtained. Previously unobserved relations between generators and invariants are also noticed.

For the class of continuous games where σi and fi {σi, φ(σ1, …, σN)} are the strategy of and payoff to player i for i = 1, …, N, it is proved that the set of weak type I optima defined in Paper I conicide with the set of solution of a matrix condition. The latter condition restricts the equilibrium solutions of an adjustment process. Numerical results for N = 2 and N = 3 indicate that the set of all equilibrium solutions coincides with the above sets. The optima of types I to IV from Paper I are described fairly completely for the given class of games.

The new optima and equilibria discussed in the preceding two papers are compared with the results of bargaining experiments between two and three players performed by Fouraker and Siegel. Experiments where players have complete or incomplete information are considered. There is clear evidence that the new optima are operating, and that traditional optima–Cournot, Pareto and competitive (threat)–are less satisfactory in explaining the course of the whole bargaining process.