Input-to-state stability

Definition

Let $f \colon \R^n \times \R^m \to \R^n$ be locally Lipschitz with $f(0, 0) = 0$. Consider the control system $\dot x(t) = f(x(t), u(t))$, where the inputs $u \colon [0, \infty) \to \R^m$ are measurable and essentially bounded, with norm $\norm{u}_\infty = \operatorname{ess\,sup}_{t \ge 0} \abs{u(t)}$. A function $\gamma \colon [0, \infty) \to [0, \infty)$ is of class $\mathcal{K}$ if it is continuous, strictly increasing and $\gamma(0) = 0$. A function $\beta \colon [0, \infty) \times [0, \infty) \to [0, \infty)$ is of class $\mathcal{KL}$ if two conditions hold. For each $t$, $\beta(\cdot, t)$ is of class $\mathcal{K}$; for each $r$, $\beta(r, \cdot)$ is nonincreasing with limit $0$ at infinity. The system is input-to-state stable (ISS) if there are $\beta$ of class $\mathcal{KL}$ and $\gamma$ of class $\mathcal{K}$ with this property. For every $x_0 \in \R^n$ and every input $u$, the solution with $x(0) = x_0$ exists for all $t \ge 0$ and satisfies $$\abs{x(t)} \le \beta(\abs{x_0}, t) + \gamma(\norm{u}_\infty) \quad \text{for all } t \ge 0.$$