Kontrollierbares

Kontrollierbares is a German term used primarily in control theory to denote something that can be controlled. Grammatically, it is the nominalized form of the adjective kontrollierbar, and it may refer to a system, a state, or a variable that can be driven to a desired value by applying suitable inputs. In practice, the term highlights the existence of control pathways that can steer the system within its state space.

In linear time-invariant systems described by ẋ = A x + B u, the system is controllable if the controllability matrix [B, AB, A^2B, ..., A^{n-1}B] has full rank equal to the state dimension. If this condition holds, any initial state can be moved to any final state in finite time using an appropriate control signal; if not, only a controllable subspace exists and some states are unreachable. The discrete-time analogue uses the corresponding reachability criteria.

Nonlinear and geometric control theory extend the idea beyond linear models. Controllability is analyzed via concepts such as the Lie algebra rank condition, which involves the Lie brackets of control vector fields to determine whether the system is controllable or accessible in a local or global sense. Practical considerations include actuator placement, energy efficiency, and robustness of control under disturbances.

Terminology notes: das Kontrollierbare can be used as a noun to refer to the controllable part or set of states, though more common is the use of Kontrollierbarkeit (the property of controllability) and kontrollierbar (the adjective). The concept is central in engineering disciplines such as robotics, aircraft guidance, process control, and automated systems design.