The control problem for stage-by-stage changing linear systems of loaded differential equations

We consider the control problem for stage-by-stage changing linear differential equations and the optimal control problem with a quality criterion defined for the entire time interval. We formulate necessary and sufficient conditions for complete controllability and the existence of a program control and motion. We construct an explicit form of the control action for the control problem and propose a method for solving the optimal control problem. We give a solution of the control problem for a specific loaded system.