The controllability of the equation $\dot x=ux$

Yu. M. Semenov

Matematičeskie zametki, Tome 23 (1978) no. 2, pp. 253-259.

Résumé

The equation $\dot x=ux$, where $x\in R^n$ and $u\in G\subset M_n$ ($M_n$ is the ring of all $n\times n$ real matrices), is considered. The equation is called weakly controllable if for arbitrary points $a,b\in R^n$ these exist points $a'$ and $b'$ as near to $a$ 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.