  In this paper, the analytical volume computations of the zonotopes generated
by the matrix pair with $n$ different or repeated real eigenvalues are
discussed firstly, 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 zonotopes. Because
the control ability for LDT systems with the unit input variables (i.e. the
input variable is with bounded value as 1) is directly related to the
controllability region \cite{zhaomw202003}, based on these analytical
expressions on the volume and shape factors, the control ability can be
quantized conveniently. By choosing these analytical expressions as the
objective function or constraint conditions, a novel optimizing problems and
solving method for the control ability can be founded. Based on the
optimization, not only the open-loop control ability, and but also the some
closed-loop control performance, such as the optimal time waste, robustness of
the control strategy, etc, can be promoted, according to the conclusions in
paper \cite{zhaomw202003}.
