In this paper a nonlinear feedback control law will be constructed for a class of linear systems, To this end a. very particular criterion functional is introduced, involving a certainS-norm of the state ||x: ||x||s. When ||x:||s= 0, singular phenomena occur which are considered in detail. It is shown that optimal arcs may end up with a singular trajectory. This problem is considered for n -dimensional linear time-dependent systems with vector control.