Barbalat's lemma

Remarks

Suppose $f'(t) \not\to 0$. Then there are $\varepsilon > 0$ and times $t_n \to \infty$ with $\abs{f'(t_n)} \ge \varepsilon$. Uniform continuity gives $\delta > 0$, independent of $n$, with $\abs{f'(t)} \ge \varepsilon / 2$ and constant sign on $[t_n, t_n + \delta]$. Then $\abs{f(t_n + \delta) - f(t_n)} \ge \varepsilon \delta / 2$ for all $n$, which contradicts the Cauchy criterion for the limit of $f$.

The hypothesis on $f'$ is necessary. For $f(t) = \sin(t^2) / (t + 1)$, the limit of $f$ is $0$, but $f'(t) = 2t \cos(t^2) / (t + 1) - \sin(t^2) / (t + 1)^2$ does not tend to $0$. Here $f'$ is not uniformly continuous.

An equivalent form: if $g \colon [0, \infty) \to \R$ is uniformly continuous and $\int_0^\infty g(t) \, dt$ converges, then $g(t) \to 0$. In stability theory, it is applied to $f(t) = V(t, x(t))$ for a Lyapunov function $V$ along a solution $x$. If $V$ is bounded below, $\dot V \le 0$ and $\dot V$ is uniformly continuous in $t$, then $\dot V(t, x(t)) \to 0$.

Going deeper

For a student

For time-varying systems, LaSalle's invariance principle does not apply directly. Barbalat's lemma replaces it: from a Lyapunov function with $\dot V \le 0$ one gets $\dot V \to 0$ along solutions, which often forces some components of the state to tend to $0$. This is a standard step in adaptive control. A common sufficient condition for uniform continuity of $\dot V$ is that $\ddot V$ is bounded.