Об оптимальности квазиособых управлений в одной задаче оптимального управления, описываемой обыкновенным дифференциальным уравнением с нетиповым функционалом

A non-standard optimal control problem governed by a system of ordinary differential equations with a general multi-point quality criterion is analyzed. The control domain represents a convex and bounded set. We derive a second-order formula for the increment of the functional associated with two admissible controls. This result allows us to prove an analogue of the linearized maximum principle introduced by L. S. Pontryagin. Furthermore, we investigate the case when this principle degenerates into what is known as the quasi-singular scenario. Finally, integral pointwise necessary conditions ensuring the optimality of such quasi-singular controls are formulated in a constructively applicable form.