Abstract
A computing algorithm, based on the geometry of certain reachable sets, is presented for fixed terminal time optimal regular problems having differential equations\(\dot x = f(x ,u , t)\). Admissible controls must be measurable and have values in a setU, which must be compact, but need not be convex. Functionsf(x, u, t) andf x (x, u, t) must be continuous and Lipschitz inx andu, but existence off u (x, u, t) or second derivatives is not required.
The algorithm is based on taking a sequence of nonlinear steps, each of which linearizes\(\dot x = f(x ,u , t)\) in state only, about a current nominal control and trajectory. Small perturbations are assured by keeping the perturbed controlclose to the nominal control. In each nonlinear step, a regulator problem,linear in state, is solved by a convexity method of Barr and Gilbert (Refs. 1–2), which is undeterred by the possibility of singular arcs. The resulting control function is substituted into the original nonlinear differential equations, producing an improved trajectory.
Convergence of the algorithm is not proved, but demonstrated by a computing example, known to be singular. In addition, procedures are described for choosing parameters in the algorithm and for testing for theplausibility of convergence.
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Communicated by T. N. Edelbaum
The author wishes to thank his dissertation advisor, Dr. E. G. Gibert, for his numerous helpful comments and suggestions. Thanks also go to Dr. J. D. Ferguson, Dr. N. T. Hamilton, Dr. W. F. Powers, and Dr. A. C. Robinson for their assistance at various stages of this work. The University of Illinois at Chicago Circle Computer Center provided extensive free computer time and excellent service, without which this work would have been impossible. The prompt and accurate typing of Miss Wandell Poe and Miss Mary Rakauskas eliminated many difficulties in preparing the final copy.
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Katz, P.L. A geometric computing method for nonlinear optimal regulator problems with singular arcs. J Optim Theory Appl 15, 249–284 (1975). https://doi.org/10.1007/BF00933338
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DOI: https://doi.org/10.1007/BF00933338