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The article presents the problem of operation speed for a linear discrete system with bounded control. For the case when the minimum number of steps necessary for the system to reach zero significantly exceeds the dimension of the phase space, a method of decomposition into scalar and two-dimensional subsystems is developed, based on the reduction of the state matrix to normal Jordan form. Moreover, due to the developed algorithm for adding two polyhedrons with linear complexity, it is possible to construct sets of 0-controllability for two-dimensional subsystems in an explicit form. A description of the main tools for solving the problem of operation speed is also presented, as well as the statement of the decomposition problem. Further, some properties of polyhedrons in the plane are formulated and proved, on the basis of which an algorithm for calculating the set of vertices of the sum of two polyhedrons in R2 in explicit form is developed. In conclusion, the main decomposition theorem is formulated and proved. And on the basis of the developed methods, the solution to the problem of the optimal damping speed of a high-rise structure located in the zone of seismic activity was constructed.

The method that allows one to reduce a stationary system with control of incomplete dimension to a non-stationary periodic system with control of full dimension is considered in this article. The paper proves the equivalence of these systems, and also that the optimal in terms of speed for a non-stationary system is also optimal for the original stationary system. A satellite attitude control system is considered as an example.