  In this paper, the control ability with time attributy for the linear
continuous-time (LCT) systems are defined and analyzed by the volume computing
for the controllability region. Firstly, a relation theorem about the open-loop
control ability, the control strategy space (\textit{i.e.}, the solution space
of the input variable for control problems), and the some closed-loop
performance for the LCT systems is purposed and proven. This theorem shows us
the necessity to optimize the control ability for the practical engineering
problems. Secondly, recurssive volume-computing algorithms with the low
computing complexities for the finite-time controllability region are
discussed. Finally, two analytical volume computations of the infinite-time
controllability region for the systems with $n$ different and repeated real
eigenvalues are deduced, and then by deconstructing the volume computing
equations, 3 classes of the shape factors are constructed. These analytical
volume and shape factors can describe accurately the size and shape of the
controllability region. Because the time-attribute control ability for LCT
systems is directly related to the controllability region with the unit input
variables, based on these analytical expressions on the volume and shape
factors, the time-attribute control ability can be computed and optimized
conveniently.
