11. Input-to-State Stability

Abstract

Let us consider a system described by the following equation in a Euclidean space Cⁿ:

$\displaystyle \dot x (t) = \Phi(x(t), u(t),t) \quad (t\geq 0). \quad\quad (1.1)$

where $x:R_{+} \rightarrow \mbox{C}^n$ is the state, $u:R_{+}\rightarrow \mbox{C}^m$ is the input and Φ maps C$^n \times \mbox{C}^m \times R_{+}$ into Cⁿ with the property Φ (0,0,t)≡ 0. Here R₊ = [0,∞).