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Numerical solution of minimax problems of optimal control, part 2

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Abstract

In a previous paper (Part 1), we presented general transformation techniques useful to convert minimax problems of optimal control into the Mayer-Bolza problem of the calculus of variations [Problem (P)]. We considered two types of minimax problems: minimax problems of Type (Q), in which the minimax function depends on the state and does not depend on the control; and minimax problems of Type (R), in which the minimax function depends on both the state and the control. Both Problem (Q) and Problem (R) can be reduced to Problem (P).

In this paper, the transformation techniques presented in Part 1 are employed in conjunction with the sequential gradient-restoration algorithm for solving optimal control problems on a digital computer. Both the single-subarc approach and the multiple-subarc approach are employed. Three test problems characterized by known analytical solutions are solved numerically.

It is found that the combination of transformation techniques and sequential gradient-restoration algorithm yields numerical solutions which are quite close to the analytical solutions from the point of view of the minimax performance index. The relative differences between the numerical values and the analytical values of the minimax performance index are of order 10−3 if the single-subarc approach is employed. These relative differences are of order 10−4 or better if the multiple-subarc approach is employed.

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References

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Author notes

  1. B. P. Mohanty (Senior Research Associate)

    Present address: School of Systems Science, Arkansas Technical University, Russellville, Arkansas

Authors and Affiliations

  1. Department of Mechanical Engineering, Rice University, Houston, Texas

    A. Miele (Professor of Astronautics and Mathematical Sciences), B. P. Mohanty (Senior Research Associate), P. Venkataraman (Graduate Student) & Y. M. Kuo (Graduate Student)

Authors
  1. A. Miele
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  2. B. P. Mohanty
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  3. P. Venkataraman
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  4. Y. M. Kuo
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Additional information

This research was supported by the National Science Foundation, Grant No. ENG-79-18667, and by Wright-Patterson Air Force Base, Contract No. F33615-80-C3000. This paper is a condensation of the investigations reported in Refs. 1–7. The authors are indebted to E. M. Coker and E. M. Sims for analytical and computational assistance.

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Miele, A., Mohanty, B.P., Venkataraman, P. et al. Numerical solution of minimax problems of optimal control, part 2. J Optim Theory Appl 38, 111–135 (1982). https://doi.org/10.1007/BF00934326

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