Second Order Optimality Conditions for Singular Extremals

  • A. V. Dmitruk
Part of the ISNM International Series of Numerical Mathematics book series (ISNM, volume 115)


We consider the class of optimal control problems, linear in the control, with control bounded by linear inequalities, and with terminal equality and inequality constraints. Both control and state variables are multidimensional, and the examined control is totally singular.

For such problems we suggest quadratic-order necessary and sufficient conditions for a weak and so-called Pontryagin minimum, which is a minimum of intermediate type between classic weak and strong minima. Necessary conditions transform into sufficient ones only by strengthening an inequality, what is similar to conditions in the classical analysis and calculus of variations (close pairs of conditions).


Singular extremal weak and Pontryagin minimum quadratic order of estimation necessary and sufficient conditions third variation of Lagrange function 


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Copyright information

© Birkhäuser Verlag Basel 1994

Authors and Affiliations

  • A. V. Dmitruk
    • 1
  1. 1.Central Economic and Mathematical InstituteRussian Academy of SciencesMoscowRussia

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