부산대학교 도서관

The authors consider the system $dx/dt=Ax+Bu$ with real matrices $A,B,x\in R^n$, $u\in\Omega\subset R^r$ under the restriction on $\Omega$ that there exists a function $u_0\in\Omega$ such that $Bu_0=0$. They introduce the following conditions: (i) If $y$ is a real eigenvector of the transposed matrix $A^\ast$ then there exists a $u\in\Omega$ such that $(y,Bu)>0$; if $y$ is a complex eigenvector of $A$ then there exists a $u\in\Omega$ such that $(y,Bu)\neq 0$. (ii) If $y$ is an eigenvector of $A$ corresponding to an eigenvalue $\lambda$, $\text{Re}\,\lambda>0$, then there is a sequence $\{u_m\}\subset\Omega$ such that $|(y,Bu_m)|\rightarrow+\infty$ (without absolute value if $\lambda$ is real). (iii) Condition (ii) holds and there is an $\varepsilon>0$ such that $|(y,Bu_m)|\geq\varepsilon\|u_m\|$ (without absolute value if $\lambda$ is real). The authors prove that (i) is a necessary and sufficient condition for local controllability (i.e., the domain of controllability includes a neighborhood of the origin), (i) and (ii) are necessary, while (i) and (iii) are sufficient for global controllability (i.e., the domain of controllability is $R^n$).