Controllability is an important property of a control system, and the controllability property plays a crucial role in many control problems, such as stabilization of unstable systems by feedback, or optimal control.
Controllability and observability are dual aspects of the same problem.
Roughly, the concept of controllability denotes the ability to move a system around in its entire configuration space using only certain admissible manipulations. The exact definition varies slightly within the framework or the type of models applied.
The following are examples of variations of controllability notions which have been introduced in the systems and control literature:
 State controllability
 Output controllability
 Controllability in the behavioural framework
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State controllability
The state of a system, which is a collection of system's variables values, completely describes the system at any given time. In particular, no information on the past of a system will help in predicting the future, if the states at the present time are known.
Complete state controllability (or simply controllability if no other context is given) describes the ability of an external input to move the internal state of a system from any initial state to any other final state in a finite time interval.^{[1]}^{:737}
Continuous Linear TimeInvariant (LTI) Systems
Consider the continuous linear timeinvariant system
where
The controllability matrix is given by
The system is controllable if the controllability matrix has full rank (i.e. ).
Discrete Linear TimeInvariant (LTI) Systems
For a discretetime linear statespace system (i.e. time variable ) the state equation is
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