BarbalatLemma

BarbalatLemma, commonly written as Barbalat’s lemma, is a result in real analysis and control theory named after Joseph Barbalat. It provides a simple convergence condition for functions on [0, ∞) and is a standard tool in Lyapunov-based stability analysis. The lemma links integrability and regularity properties of a signal to its long-run behavior, enabling conclusions about asymptotic convergence without requiring explicit solutions.

One standard formulation states: if f: [0, ∞) → R is uniformly continuous on [0, ∞) and the improper integral ∫0^∞ f(t) dt exists (is finite), then lim_{t→∞} f(t) = 0. A closely related variant used in control theory assumes f is continuously differentiable on [0, ∞) and f' is uniformly continuous, with the same integrability condition on f; under these conditions f(t) → 0 as t → ∞. The uniform continuity assumption is important to rule out persistent oscillations that could prevent convergence.

Applications: Barbalat’s lemma is frequently employed in Lyapunov analyses to deduce convergence of signals. For example, if dV/dt ≤ -W(x) with W(x) ≥ 0, then ∫0^∞ W(x(t)) dt < ∞, and Barbalat implies W(x(t)) → 0 as t → ∞. If W is positive definite, this often yields convergence of the state x(t) to an equilibrium or to the largest invariant set where W(x) = 0, contributing to proofs of asymptotic stability.

See also: Lyapunov stability, asymptotic stability, stability analysis.