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.