Juhitavus

Juhitavus, or controllability, is a property of a dynamical system that describes whether it is possible to drive the system from any given initial state to any desired final state in finite time by applying appropriate inputs. It is a central concept in control theory and appears across engineering, physics, and applied mathematics.

In continuous-time linear time-invariant systems described by x' = Ax + Bu, the Kalman rank condition provides a practical test: the controllability matrix [B, AB, A^2B, ..., A^{n-1}B] must have full rank equal to the state dimension n. If this holds, the system can be steered to any state in finite time; if not, there exists a subspace of states that cannot be reached, regardless of the input. The controllability Gramian is another tool, with full rank over a given time interval indicating reachability and energy requirements.

For discrete-time systems x_{k+1} = Ax_k + Bu_k, the same rank condition applies to the reachability matrix [B, AB, ..., A^{n-1}B]. Nonlinear systems are more complex; notions such as local controllability and accessibility are used, with tools like Lie algebra rank conditions and Chow’s theorem providing sufficient criteria in certain classes of systems.

Juhitavus is distinct from observability, which concerns whether current states can be inferred from outputs. In practice, controllability informs design choices for actuators, control strategies, and safety analyses, and it underpins algorithms for trajectory planning, stabilization, and state estimation.