Well-posedness of Optimal Control Problems Without Convexity Assumptions

  • Alexander J. Zaslavski
Part of the Springer Optimization and Its Applications book series (SOIA, volume 82)


In this chapter we prove generic existence results for classes of optimal control problems in which constraint maps are also subject to variations as well as the cost functions. These results were obtained in [87, 90]. More precisely, we establish generic existence results for classes of optimal control problems (with the same system of differential equations, the same boundary conditions and without convexity assumptions) which are identified with the corresponding complete metric spaces of pairs (f, U) (where f is an integrand satisfying a certain growth condition and U is a constraint map) endowed with some natural topology. We will show that for a generic pair (f, U) the corresponding optimal control problem has a unique solution.


Optimal Control Problem Variational Principle Closed Subset Lower Semicontinuous Weak Topology 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 12.
    Berkovitz LD (1974) Optimal control theory. Springer, New YorkMATHCrossRefGoogle Scholar
  2. 13.
    Berkovitz LD (1974) Lower semicontinuity of integral functionals. Trans Amer Math Soc 192:51–57MathSciNetMATHCrossRefGoogle Scholar
  3. 17.
    Carlson DA, Haurie A, Leizarowitz A (1991) Infinite horizon optimal control. Springer, BerlinMATHCrossRefGoogle Scholar
  4. 21.
    Cesari L (1983) Optimization - theory and applications. Springer, New YorkMATHCrossRefGoogle Scholar
  5. 22.
    Cinquini S (1936) Sopra l’esistenza della solusione nei problemi di calcolo delle variazioni di ordine n. Ann Scuola Norm Sup Pisa 5:169–190MathSciNetGoogle Scholar
  6. 32.
    Diestel J, Uhl JJ (1977) Vector measures. American Mathematical Society, Providence, RIMATHGoogle Scholar
  7. 33.
    Doob JL (1994) Measure theory. Springer, New YorkMATHCrossRefGoogle Scholar
  8. 41.
    Ioffe AD, Tikhomirov VM (1979) Theory of extremal problems. North-Holland, New YorkMATHGoogle Scholar
  9. 42.
    Ioffe AD, Zaslavski AJ (2000) Variational principles and well-posedness in optimization and calculus of variations. SIAM J Contr Optim 38:566–581MathSciNetMATHCrossRefGoogle Scholar
  10. 44.
    Kelley JL (1955) General topology. Van Nostrand, Princeton, NJMATHGoogle Scholar
  11. 57.
    Maz’ja VG (1985) Sobolev spaces. Springer, BerlinMATHGoogle Scholar
  12. 86.
    Zaslavski AJ (1996) Generic existence of solutions of optimal control problems without convexity assumptions, preprintGoogle Scholar
  13. 87.
    Zaslavski AJ (2000) Generic well-posedness of optimal control problems without convexity assumptions. SIAM J Contr Optim 39:250–280MathSciNetMATHCrossRefGoogle Scholar
  14. 88.
    Zaslavski AJ (2001) Existence of solutions of optimal control problems without convexity assumptions. Nonlinear Anal 43:339–361MathSciNetMATHCrossRefGoogle Scholar
  15. 90.
    Zaslavski AJ (2001) Generic well-posedness for a class of optimal control problems. J Nonlinear Convex Anal 2:249–263MathSciNetMATHGoogle Scholar
  16. 108.
    Zeimer WP (1989) Weakly differentiable functions. Springer, New YorkCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Alexander J. Zaslavski
    • 1
  1. 1.Department of MathematicsTechnion - Israel Institute of TechnologyHaifaIsrael

Personalised recommendations