Linear problem of tracking a given motion under an integral constraint on control

We consider the problem of optimally tracking a given vector function by means of a generalized projection of the trajectory of a linear controlled object with an integral constraint on the control. The deviation from a given motion is measured in the metric of the space $C^m[0,T]$ of continuous vector functions of appropriate dimension $m$. We describe a constructive method for solving this optimization problem with a given accuracy.