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            Abstract
  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 a 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).


  Key words:
  singular extremal, weak and Pontryagin minimum, quadratic order of estimation,
  necessary and sufficient conditions, third variation of Lagrange function.
 

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