Abstract
In this paper two state space related system properties, i.e., bounded connectedness and free state response completeness are introduced for linear dynamical systems, and a significance of the finiteness of a finite dimensional state space is explored by them. The results are applied to controllability concepts. The bounded connectedness requires of a linear system that every state reachable from the origin or able to be steered to the origin can be reached or steered uniformly within a time interval characteristic of the system. The free state response completeness requires that a free state response function should have the inverse. This paper shows that a time invariant linear dynamical system satisfies the bounded connectedness if its state space is finite and free state response completeness if its input-output response is analytic and its state space is finite. In particular, it is shown in the paper that a constant coefficient linear ordinary differential equation system possesses both properties.
There have been defined three kinds of controllability, that is, controllability from zero, controllability to zero and complete control-lability. In general, they are different concepts. This paper however shows that they are equivalent for the class of time invariant linear dynamic systems satisfying both properties.
This result imples that the three concepts of controllability are equivalent to each other for a system by a constant coefficient ordinary differential equation. This is well known but has not been proven by system theoretic arguments, as it is in this paper. Most of the results of linear control theory have not been sufficiently conceptualized so that the result can be helpful in understanding systems in an intuitive way. This paper, as well as another paper (Reference (4)), tries to make a contribution to the work of conceptualization of the linear control theory.
