The controllability of the equation x = ux

The equation x=uv, wherex R_n andu GM _n (M_n is the ring of all n × n real matrices), is considered. The equation is called weakly controllable if for arbitrary pointsa, b R ⁿ these exist pointsa and b' as near toa and b, respectively, as we like and a control transforming a into b. In this note algebraic criteria are given for the complete and the weak controllability of such equations in the case where the limiting set G is closed with respect to the operation of matrix multiplication and the G-module R_n is semisimple.Translated from Matematicheskie Zametki, Vol. 23, No. 2, pp. 253–259, February, 1978.