stabilizability

Stabilizability is a property of a linear time-invariant system described by x' = A x + B u in continuous time or x_{k+1} = A x_k + B u_k in discrete time. The system is stabilizable if there exists a state-feedback law u = K x such that the closed-loop dynamics (A + B K) x are asymptotically stable (continuous time) or Schur stable (discrete time). In other words, one can stabilize the system by appropriate choice of a state feedback, even if not every state can be directly influenced.

Stabilizability is weaker than controllability: controllability implies stabilizability, but stabilizability does not require full controllability. The uncontrollable modes must be stable by themselves. For continuous-time systems, a standard criterion is that every eigenvalue λ of A with Re(λ) ≥ 0 must be controllable, equivalently the uncontrollable eigenvalues of A lie in the open left-half plane. A practical algebraic test is the rank condition: for all λ with Re(λ) ≥ 0, rank([λ I − A, B]) = n, where n is the state dimension.

For discrete-time systems, the analogous condition states that all eigenvalues λ of A with |λ| ≥ 1 must be controllable; equivalently there exists K such that A + B K has spectral radius less than 1.

Stabilizability interacts with detectability (the dual notion related to observer design) and underpins many design methods, including state feedback, optimal control (e.g., LQR), and observer-based controllers.