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A Variation Evolving Method for Optimal Control Computation

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Abstract

A new method, which originates from the continuous-time dynamics stability theory in the control field, is proposed for the optimal control computation. By introducing a virtual dimension, the variation time, an infinite-dimensional dynamic system that describes the variation motion of variables is derived from the optimal control problem based on the Lyapunov principle. The optimal solution is its stable equilibrium point and will be obtained in an asymptotically evolving way. Through this method, the intractable optimal control problems are transformed to the initial-value problems and they may be solved with mature ordinary differential equation numerical integration methods. Especially, the deduced dynamic system is globally stable, so any initial value may evolve to an extremal solution ultimately.

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References

  1. Pesch, H.J., Plail, M.: The maximum principle of optimal control: A history of ingenious ideas and missed opportunities. Control Cybern. 38(4), 973–995 (2009)

    MathSciNet  MATH  Google Scholar 

  2. Betts, J.T.: Survey of numerical methods for trajectory optimization. J. Guid. Control Dyn. 21(2), 193–206 (1998)

    Article  MATH  Google Scholar 

  3. Lin, Q., Loxton, R., Teo, K.L.: The control parameterization method for nonlinear optimal control: a survey. J. Ind. Manag. Optim. 10(1), 275–309 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  4. Hargraves, C., Paris, W.: Direct trajectory optimization using nonlinear programming and collocation. J. Guid. Control Dynam. 10(4), 338–342 (1987)

    Article  MATH  Google Scholar 

  5. Stryk, O.V., Bulirsch, R.: Direct and indirect methods for trajectory optimization. Ann. Oper. Res. 37(1), 357–373 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  6. Pesch, H.J.: A practical guide to the solution of real-life optimal control problems. Control Cybern. 23(1/2), 7–60 (1994)

    MathSciNet  MATH  Google Scholar 

  7. Peng, H.J., Gao, Q., Wu, Z.G., Zhong, W.X.: Symplectic approaches for solving two-point boundary-value problems. J. Guid. Control Dyn. 35(2), 653–658 (2012)

    Article  Google Scholar 

  8. Rao, A.V.: A survey of numerical methods for optimal control. In Proceedings of AAS/AIAA Astrodynamics Specialist Conference, Pittsburgh, PA, 2009, AAS Paper 09-334 (2009)

  9. Conway, B.A.: A survey of methods available for the numerical optimization of continuous dynamic systems. J. Optim. Theory Appl. 152, 271–306 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  10. Bertrand, R., Epenoy, R.: New smoothing techniques for solving bang-bang optimal control problems—numerical results and statistical interpretation. Optim. Control Appl. Meth. 23(4), 171–197 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  11. Pan, B.F., Lu, P., Pan, X., Ma, Y.Y.: Double-homotopy method for solving optimal control problems. J. Guid. Control Dynam. 39(8), 1706–1720 (2016)

    Article  Google Scholar 

  12. Garg, D., Patterson, M.A., Hager, W.W., Rao, A.V., et al.: A Unified framework for the numerical solution of optimal control problems using pseudospectral methods. Automatica 46(11), 1843–1851 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  13. Ross, I.M., Fahroo, F.: A perspective on methods for trajectory optimization. In: Proceedings of AIAA/AAS Astrodynamics Conference, Monterey, CA, 2002, AIAA Paper No. 2002-4727 (2002)

  14. Ross, I.M., Fahroo, F.: Pseudospectral methods for optimal motion planning of differentially flat systems. IEEE Trans. Autom. Control 49(8), 1410–1413 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  15. Courant, R.: Variational methods for the solution of problems of equilibrium and vibrations. Bull. Am. Math. Soc. 49, 1–23 (1943)

    Article  MathSciNet  MATH  Google Scholar 

  16. Kelley, H.J.: Methods of gradients. In: Leitmann, G. (ed.) Optimization Techniques, pp. 216–232. Academic Press, London (1962)

    Google Scholar 

  17. Khalil, H.K.: Nonlinear Systems, 3rd edn. Prentice Hall, Upper Saddle River (2002)

    MATH  Google Scholar 

  18. Chiou, J.P., Na, T.Y.: On the solution of Troesch’s nonlinear two-point boundary value problem using an initial value method. J. Comput. Phys. 19(3), 311–316 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  19. Fazio, R., Iacono, S.: On the translation groups and non-iterative transformation methods. In: Bernardis, E., De Spligher, R., Valenti, V. (eds.) Applied and Industrial Mathematics in Italy III, pp. 331–340. World Scientific, Singapore (2010)

    Google Scholar 

  20. Hartl, R.F., Sethi, S.P., Vickson, R.G.: A survey of the maximum principles for optimal control problems with state constraint. SIAM Rev. 37(2), 181–218 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  21. Cassel, K.W.: Variational Methods with Applications in Science and Engineering. Cambridge University Press, Cambridge (2013)

    Book  MATH  Google Scholar 

  22. Wu, D.G.: Variation Method. Higher Education Press, Beijing (1987)

    Google Scholar 

  23. Zhang, H.X., Shen, M.Y.: Computational Fluid Dynamics—Fundamentals and Applications of Finite Difference Methods. National Defense Industry Press, Beijing (2003)

    Google Scholar 

  24. Bryson, A.E., Ho, Y.C.: Applied Optimal Control: Optimization, Estimation, and Control. Hemisphere, Washington, DC (1975)

    Google Scholar 

  25. Xie, X.S.: Optimal Control Theory and Application. Tsinghua University Press, Beijing (1986)

    Google Scholar 

  26. Sussmann, H.J., Willems, J.C.: The Brachistochrone problem and modern control theory. In: Anzaldo-Meneses, A., Bonnard, B., Gauthier, J.P., Monroy-Perez, F. (eds.) Contemporary Trends in Nonlinear Geometric Control Theory and Its Applications, pp. 113–166. World Scientific, Singapore (2000)

    Google Scholar 

  27. Patterson, M.A., Rao, A.V.: GPOPS-II: AMATLAB software for solving multiple-phase optimal control problems using hp-adaptive Gaussian quadrature collocation methods and sparse nonlinear programming. ACM Trans. Math. Software 41(1), 1–37 (2014)

    Article  Google Scholar 

  28. Hull, D.G.: Optimal Control Theory for Applications. Springer, New York (2003)

    Book  MATH  Google Scholar 

  29. Teo, K.L., Goh, C.J., Wong, K.H.: A Unified Computational Approach to Optimal Control Problems. Longman Scientific and Technical, New York (1991)

    MATH  Google Scholar 

  30. Zhang, G.C.: Optimal Control Computation Methods. Chengdu Science and Technology University Press, Chengdu (1991)

    Google Scholar 

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Acknowledgements

In refining the paper, the authors get the help from Prof. K. L. Teo from Curtin University and Prof. H. J. Pesch from University of Bayreuth. Sincere thanks to them. The authors would also like to thank the Editor in Chief, Prof. B. A. Conway, and the anonymous referees for their help that further improves the paper.

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Authors and Affiliations

  1. China Aerodynamics Research and Development Center, Mianyang, 621000, China

    Sheng Zhang, En-Mi Yong, Wei-Qi Qian & Kai-Feng He

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  1. Sheng Zhang
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  2. En-Mi Yong
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Correspondence to Sheng Zhang.

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Communicated by Bruce A. Conway.

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Research supported by the National Defense Key Laboratory Fund of China, Grant Number: 614222003060717.

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Zhang, S., Yong, EM., Qian, WQ. et al. A Variation Evolving Method for Optimal Control Computation. J Optim Theory Appl 183, 246–270 (2019). https://doi.org/10.1007/s10957-019-01537-4

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