contrôlables

Controllability, sometimes referred to in French as contrôlabilité, is a property of a dynamical system that indicates whether its state can be driven from any initial value to any final value in finite time using permissible inputs. It is a fundamental concept in control theory and underpins the design of state feedback and trajectory planning.

For linear time-invariant systems described by x' = Ax + Bu (continuous time) or x_{k+1} = Ax_k + Bu_k (discrete time), controllability is characterized by the Kalman rank condition. The controllability matrix is formed as [B AB A^2B ... A^{n-1}B], where n is the state dimension. If rank([B AB A^2B ... A^{n-1}B]) = n, the system is completely controllable; otherwise, only a subspace of states can be controlled. The same condition holds in discrete time with the appropriate matrices.

If a system is not controllable, its state space can be decomposed into controllable and uncontrollable subspaces. Modes associated with the uncontrollable subspace cannot be moved by the inputs, which affects achievable performance and stability. Designing actuators to expand the controllable subspace or reconfiguring the system are common remedies.

Applications include pole placement via state feedback, optimal control, and trajectory planning in engineering domains such as robotics, aerospace, and process control. For nonlinear systems, controllability concepts extend through Lie brackets and local controllability notions, offering a more nuanced assessment of what states can be reached from a given starting point.