Algebraic structure of a set of states of a dynamic system

Assuming that the input-to-output mapping is linear an algebraical structure is obtained over a dynamic system state set. In a general case the algebraical structure varies in time. A commutatipe diagram is obtained which related the state transfer function with the input-to-output mapping. The linearity of the state transfer function with respect to the structure is used to obtain the necessary and sufficient conditions for controllability of the system.