The aim of this work is to discuss necessary and sufficient conditions for dispersiveness of linear control systems of the form \(\dot{x}=A\left( x\right) +B\left( u(t)\right)\) , \(x\in \mathbb {R}^{n}\) . It presents necessary conditions for the existence of control sets. Under the stability of the linear operator A, the dispersiveness occurs if and only if the trajectories through the origin 0 have no limit at infinity. The mean limit criterion ensures that the system is dispersive, if A does not reach the mean limit set of B. All the sufficient conditions for dispersiveness indicate necessary conditions for the existence of a control set. The results of the paper are applied to mechanical and electrical control systems.
