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 the 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, the last being a minimum of an

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 (adjoint pairs of conditions).

Key words:

singular extremal, weak minimum, Pontryagin minimum, strong minimum,

quadratic order of estimation, necessary and sufficient conditions,

critical cone, second variations, Legendre conditions.