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In this paper, the works on the analytical volume analysis for thecontrollable regions of the linear discrete-time (LDT) systems in papers\cite{zhaomw202001} and \cite {zhaomw202004} are discussed further and a newtheorem on the analytical computing for the finite-time controllabilityzonotope (controllable region) of LDT systems are proven. And then, threeanalytical factors describing the control capability of the systems aredeconstructed successfully from the analytical volume expression of thecontrollable region. Finally, the theorem is generalized to three cases: thenarrow controllable region, the matrix $A$ with $n$ negative eigenvalues, thelinear continuous-time systems.

Analytical Volume Analysis for the Finite-time Controllable Region of the Linear Discrete-time Systems