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  "body": "\n## Summary\n\nA [[control system|ZPDTVZQN5K]] is controllable if every state can be steered to every other state by a suitable choice of the control.\nIt is the first question of control theory: before asking how to steer well, one asks whether steering is possible at all.\nFor linear systems $x' = A x + B u$ it is decided by the [[Kalman rank condition|9NSCZ2KBEB]], and for systems without drift by the [[Chow–Rashevskii theorem|VQPN24VDH1]].\n\n## Explanations\n\n### L1 human\n\nImagine a cart on a rail that can be pushed forward or backward.\nBy pushing at the right times, one can bring it to any position with any velocity, so the cart is controllable.\nIf instead the push only acts on a second cart that is not attached to the first, the first cart cannot be moved, and the system is not controllable.\nControllability asks whether the inputs reach every direction of the state, directly or through the dynamics.\n",
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