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Controllability Gramian characterizes the minimum energy input to the linear system 
that steers the initial zero state at infinite past to a point on the unit ball at t = 0 (Top). Moreover, this Gramian defines how inputs on the unit sphere (Bottom Left), such as noise or external signals, map to system states: Directions that are more controllable are characterized by elongated ellipsoidal axes, whereas the shortened axes are less controllable directions (Bottom Right).
