부산대학교 도서관

The authors consider the system $dx/dt=Ax+Bu$ with real matrices $A,B,x\inR^n$, $u\in\Omega\subset R^r$ under the restriction on $\Omega$ that thereexists a function $u_0\in\Omega$ such that $Bu_0=0$. They introduce thefollowing conditions: (i) If $y$ is a real eigenvector of the transposedmatrix $A^\ast$ then there exists a $u\in\Omega$ such that $(y,Bu)>0$; if$y$ is a complex eigenvector of $A$ then there exists a $u\in\Omega$ suchthat $(y,Bu)\neq 0$. (ii) If $y$ is an eigenvector of $A$ corresponding toan eigenvalue $\lambda$, $\text{Re}\,\lambda>0$, then there is a sequence$\{u_m\}\subset\Omega$ such that $|(y,Bu_m)|\rightarrow+\infty$ (withoutabsolute value if $\lambda$ is real). (iii) Condition (ii) holds and thereis an $\varepsilon>0$ such that $|(y,Bu_m)|\geq\varepsilon\|u_m\|$ (withoutabsolute value if $\lambda$ is real). The authors prove that (i) is anecessary and sufficient condition for local controllability (i.e., thedomain of controllability includes a neighborhood of the origin), (i) and(ii) are necessary, while (i) and (iii) are sufficient for globalcontrollability (i.e., the domain of controllability is $R^n$).