An algorithm is proposed to solve the problem of bang–bang constrained optimal control of non‐linear systems with free terminal time. The initial and terminal states are prescribed. The problem is reduced to minimizing a Lagrangian subject to equality constraints defined by the terminal state. A solution is obtained by solving a system of non‐linear equations. Since the terminal time is free, time‐optimal control is given a special emphasis. Second‐order sufficient conditions of optimality are also stated. The algorithm is demonstrated by a detailed study of the switching structure for stabilizing the F–8 aircraft in minimum time, and other examples. Copyright © 2005 John Wiley & Sons, Ltd.

In this paper an algorithm is proposed to solve the problem of time-optimal bang–bang control of nonlinear systems from a given initial state to a given terminal state. The problem is reduced to the problem of minimising a Lagrangian subject to an equality constraint defined by the terminal state. Then a solution is obtained by solving a system of nonlinear equations. Examples are given so as to illustrate the algorithm presented.

The flare reactions produced by epicutaneous tests with 68 undiluted allergenic pollen extracts were measured in 550 allergic patients. Skin test reactions greater than or equal to 2, greater than or equal to 5, greater than or equal to 10, greater than or equal to 20, and greater than or equal to 30 mm in diameter, respectively, were detected in approximately 67%, 22%, 10%, 3%, and 1% of the 34,700 skin tests. With the Kolmogorov-Smirnov difference test, the cumulative frequency of reaction diameters and loge-transformed diameters of all reactions and reactions to individual allergenic extracts differed significantly (p less than or equal to 0.01) from a normal distribution. The ability to identify specific differences between reactions to closely related pollen extracts was evaluated. Specific reactions could be reliably identified with greater than or equal to 10 mm diameter flares. This arbitrary conservative threshold was used to estimate the relative prevalence of positive reactions to each allergenic extract. Seven allergenic extracts elicited the first quartile of all positive reactions. Thirteen, 18, and 30 allergenic extracts, respectively, were needed to elicit the second, third, and fourth quartiles of all positive reactions. Reactions to amphiphilous, as well as anemophilous, pollens were detected. Skin test reactions to grasses were more prevalent than reactions to weeds and trees. The most informative allergenic extracts for the detection of patients who exhibited a positive reaction to any extract were from red fescue-grass pollens, mesquite, short ragweed, red clover, and timothy-grass pollens.

Observer variability in the pulmonary examination was assessed by having four blindfolded observers (two medical students and two pulmonary physicians) twice examine 31 patients with abnormal pulmonary findings. Examiners were consistent in the repetitive detection of pulmonary abnormalities in 74–89% of the examinations; conversely, 11–26% of the time they disagreed with themselves. Although pulmonary specialists recorded fewer (55% of observations) abnormal findings than did medical students (74%), they were significantly (p=0.008) less self-consistent than were the students. There was no clear trend in agreement between examiners (kappa=0.20−0.49). Each examiner’s findings were compared with those of physicians specially trained in pulmonary examination. Dichotomous variables (wheezes, crackles, rubs) were more reliably detected (kappa=0.30−0.70) than graded variables (tympany, dullness, breath sound intensity), where kappa=0.16−0.43. The authors suggest that dichotomous variables deserve greatest clinical reliance; that time in training, alone, does not improve clinical performance; and that there is a disconcertingly large amount of inter- and intraobserver disagreement in this fundamental clinical task.

SOLA International is a company based in Adelaide which produces a wide range of optical lenses, including those for spectacles and over 100 million of their product are being worn around the world today. Impact performance of lenses is a critical safety requirement, and yet the problem of fracture of lenses by impact of a ball bearing is not properly understood. However, it is known that both thinness and atness tend to reduce impact strength, while lens coat- ing facilitates the fracture process, and tends to focus the fracture load regime. Generally, crack initiation and propagation is a complex and not well understood issue. Ideally, SOLA International seeks a mathematical model which inputs lens geometry and material properties and is able to predict the deformation on impact. In particular, prediction of the stresses in the back surface coating is important. The mathematical model might then provide an alternative frame- work to laboratory tests, which are labourious and frequently not reproducible. The full solution of this problem involves many physical processes, including im- pact phenomenon, the relationship of static test results to impact tests, contact mechanics, random eects appearing through a ws and microcracks, fracture mechanics and nally crack propogation. In addition, the full problem in reality involves dynamic deformations of a visco-elastic body, coated with material of dieren t physical properties, and it is required to formulate conditions under which crack initiation will take place. A modest attempt towards some of these issues is outlined below.