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Journal of Optimization Theory and Applications

, Volume 178, Issue 2, pp 349–362 | Cite as

Solutions to Constrained Optimal Control Problems with Linear Systems

  • Shangrui Zhao
  • Jiani Zhou
Article
First Online:
Received:
Accepted:

Abstract

This paper is devoted to present solutions to constrained finite-horizon optimal control problems with linear systems, and the cost functional of the problem is in a general form. According to the Pontryagin’s maximum principle, the extremal control of such problem is a function of the costate trajectory, but an implicit function. We here develop the canonical backward differential flows method and then give the extremal control explicitly with the costate trajectory by canonical backward differential flows. Moreover, there exists an optimal control if and only if there exists a unique extremal control. We give the proof of the existence of the optimal solution for this optimal control problem with Green functions.

Keywords

Optimal control problems The Pontryagin’s maximum principle Canonical backward differential flows Linear systems 

Mathematics Subject Classification

49J15 
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References

  1. 1.
    Butenko, S., Murphey, R., Pardalos, P.M.: Recent Developments in Cooperative Control and Optimization. Springer, New York (2003)zbMATHGoogle Scholar
  2. 2.
    Hirsch, M.J., Commander, C.W., Pardalos, P.M., Murphey, R. (ed.): Optimization and Cooperative Control Strategies: Proceedings of the 8th International Conference on Cooperative Control and Optimization. Springer, New York (2009)Google Scholar
  3. 3.
    Pardalos, P.M., Tseveendorj, I., Enkhbat, R.: Optimization and Optimal Control. World Scientific, Singapore (2003)CrossRefzbMATHGoogle Scholar
  4. 4.
    Halkin, H.: Necessary conditions for optimal control problems with infinite horizons. Econom. J. Econ. Soc. 42, 267–272 (1974)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Kipka, R.J., Ledyaev, Y.S.: Pontryagin maximum principle for control systems on infinite dimensional manifolds. Set Valued Var. Anal. 23, 1–15 (2014)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Zhu, J.H., Wu, D., Gao, D.: Applying the canonical dual theory in optimal control problems. J. Glob. Optim. 54, 221–233 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Zhu, J.H., Zhao, S.R., Liu, G.H.: Solution to global minimization of polynomials by backward differential flow. J. Optim. Theor. Appl. 161, 828–836 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Russell, D.L.: Mathematics of Finite-Dimensional Control Systems: Theory and Design. M. Dekker, New York (1979)zbMATHGoogle Scholar
  9. 9.
    Sontag, E.D.: Mathematical Control Theory: Deterministic Finite Dimensional Systems. Springer, New York (1998)CrossRefzbMATHGoogle Scholar
  10. 10.
    Moylan, P.J., Moore, J.: Generalizations of singular optimal control theory. Automatica 7, 591–598 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Ge, W., Li, C., Wang, H.: Ordinary Differential Equations and Boundary Valued Problems. Science Press, Beijing (2008)Google Scholar
  12. 12.
    Pontryagin, L.S.: Mathematical Theory of Optimal Processes. CRC Press, Boca Raton (1987)Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Tongji UniversityShanghaiChina
  2. 2.Shanghai Lixin University of Accounting and FinanceShanghaiChina

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