ControlLyapunov

ControlLyapunov, in the standard literature usually written as control Lyapunov function (CLF), is a concept from nonlinear control theory used to certify and construct stabilizing feedback for dynamical systems. It extends the classical Lyapunov stability method to systems where the control input can be chosen to influence the state trajectory. A CLF provides a scalar measure of the system’s energy or error that can be actively driven downward by an admissible control.

For a control-affine system ẋ = f(x) + G(x)u with u in a set U, a function V: R^n → R is a CLF if V is positive definite, radially unbounded, and for every nonzero x there exists a control u ∈ U such that the derivative of V along system trajectories satisfies L_f V(x) + L_G V(x) u ≤ -α(|x|) for some positive definite α. Here L_f V and L_G V denote the Lie derivatives of V along f and G, respectively. If such a V exists, the origin is stabilizable by a feedback law; in particular, a feedback that enforces V̇ ≤ -α(|x|) yields convergence to the origin.

Controller synthesis often proceeds by choosing u to minimize a measure of V̇, for example solving a real-time quadratic program: minimize ||u||^2 subject to L_f V(x) + L_G V(x) u ≤ -α(|x|). When L_G V ≠ 0, Sontag’s universal formula provides a continuous stabilizing control law based on V and its derivatives.

CLFs are a sufficient condition for stabilizability, widely used in nonlinear control, robotics, and motion planning. Constructing a CLF can be challenging, and its applicability depends on system structure and admissible controls. Synonymous with control-Lyapunov function, the concept remains central to Lyapunov-based control design.