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As an illustration of controllability, my text presents a systems in which some states does not get affected by the input $u$ and hence is not of relevance in the input/output description.

It then proceeds to define a state as controllable if there is an insignal such that we end up at the state starting from the origin.

I have hard time putting the idea and the defintion togehter. How is our ability to "attain" a state connected with our ability to "steer" it?

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Controllability implies that you can go from any point (starting point) in the state space to another point (endpoint) in the state space in finite time by using a specific control input over the time interval (sometimes called control trajectory).

However, it does not imply that you can go there using arbitrary trajectories and that you will stay at the endpoint. You can only stay at the endpoint if it is an equilibrium point of the system. For a linear system

$$\dot{\boldsymbol{x}}=\boldsymbol{Ax}+\boldsymbol{Bu},$$

this works only for the nullspace (also called kernel) of the system matrix $\boldsymbol{A}$, which trivially always contains the origin. For nonlinear systems, you can have multiple discontinuous or continuous equilibrium points or no equilibrium point at all e.g. $\dot{x}=x^2+1$). A stronger version of controllability is called omnidirectional controllability, which implies that you can go from the starting point to the endpoint by using arbitrary trajectories.

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