Abstract We consider the solution of the speed problem for linear time-varying discrete-time systems with convex control constraints. A method is proposed for reducing the general case of the speed problem to the case of linear control constraints using polyhedral approximation algorithms. Sufficient optimality conditions for the guaranteed solution are stated and proved. Examples are given. Based on the methods obtained, the speed-optimal damping problem for a high-rise structure located in a seismic activity zone is solved.