Existence and Lipschitzian Regularity for Relaxed Minimizers

  • Manuel Guerra
  • Andrey Sarychev


In this contribution we follow two main goals: to reconstruct a result announced in [4] about existence of relaxed minimizers for (nonconvex) Lagrange problems of optimal control (Theorem 1); to derive conditions for Lipschitzian regularity of trajectories corresponding to relaxed minimizers (Theorem 3). In passing, elaborating on the approach used in [10], we provide a condition for Lipschitzian regularity of non relaxed minimizers (Theorem 2).


Maximum Principle Optimal Control Problem North Pole Pontryagin Maximum Principle Growth Assumption 
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  1. 1.
    . Agrachev AA, Sachkov YuL (2004) Control Theory from the Geometric Viewpoint. Springer.Google Scholar
  2. 2.
    Alexeev VM, Tikhomirov VM, Fomin SV (1987) Optimal Control. Consultants Bureau, N.Y.Google Scholar
  3. 3.
    Cellina A, Colombo G (1990) On a classical problem of the clalculus of variations without convexity assumptions. Annales Inst. Henri Poincare, Analyse Non Lineaire, 7:97–106.zbMATHMathSciNetGoogle Scholar
  4. 4.
    Cesari L (1983) Optimization-theory and applications. Problems with ordinary differential equations. Springer-Verlag, New YorkzbMATHGoogle Scholar
  5. 5.
    . Clarke FH (1983) Optimization and Nonsmooth Analysis. Wiley-Interscience Publication.Google Scholar
  6. 6.
    Clarke FH (1989) Methods of dynamic and nonsmooth optimization. CBMS-NSF Regional Conference Series in Applied Mathematics, 57. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA.Google Scholar
  7. 7.
    Clarke FH, Vinter R (1990) Regularity properties of optimal controls. SIAM J. Control Optim., 28(4):980–997.zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    . Gamkrelidze RV (1978) Foundations of Control Theory. Plenum Press.Google Scholar
  9. 9.
    Loeb P (1967) A Minimal Compactification for Extending Continuous Functions. Proc. Amer. Math. Soc., 18:282–283.zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Sarychev A, Torres DFM (2000) Lipschitzian regularity, Applied Mathematics and Optimization, 41:237–254.zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Sussmann HJ (2007) Set separation, approximating multicones, and the Lipschitz maximum principle. J. Differential Equations, 243:446–488.CrossRefMathSciNetGoogle Scholar
  12. 12.
    Torres DFM (2003) Lipschitzian Regularity of the Minimizing Trajectories for Nonlinear Optimal Control Problems. Mathematics of Control, Signals and Systems, 16:158–174.zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Vinter R (2000) Optimal Control. Birkhäuser, Boston.zbMATHGoogle Scholar
  14. 14.
    Whyburn GT (1953) A Unified Space for Mappings. Trans. Amer. Mathem. Society, 74:344–350.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Manuel Guerra
    • 1
  • Andrey Sarychev
    • 2
  1. 1.CEOC and ISEG-T.U.LisbonLisboaPortugal
  2. 2.DiMaDUniversity of FlorenceFirenzeItalia

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