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 ZhouEmail author


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.


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

Mathematics Subject Classification



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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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