In order to understand the numerical behavior of a certain class of periodic optimal control problems, a relatively simple problem is posed. The complexity of the extremal paths is uncovered by determining an analytic approximation to the solution by using the Lindstedt-Poincaré asymptotic series expansion. The key to obtaining this series is in the proper choice of the expansion parameter. The resulting expansion is essentially a harmonic series in which, for small values of the expansion parameter and a few terms of the series, excellent agreement with the numerical solution is obtained. A reasonable approximation of the solution is achieved for a relatively large value of the expansion parameter.
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This work was sponsored partially by the National Science Foundation, Grant No. ECS-84-13745.
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Evans, R.T., Speyer, J.L. & Chuang, C.H. Solution of a periodic optimal control problem by asymptotic series. J Optim Theory Appl 52, 343–364 (1987). https://doi.org/10.1007/BF00938212
- Periodic optimal control problems
- asymptotic series expansions
- Lindstedt and Poincaré expansions
- two-point boundary-value problems