For finite-dimensional, autonomous, nonlinear systems of the form \[ \dot x= f(x, u), \] the authors introduce a notion of input-to-state stability which takes into account not only the norm of the exogenous input \(u(t)\), but also the norm of its derivatives up to some finite order \(k\). More precisely, the system is said to be \(D^k iss\) if \[ | x(t,\xi,u)|\leq \beta(|\xi|, t)+ \gamma_0(\| u\|)+ \gamma_1(\|\dot u\|)+\cdots+ \gamma_k(\| u^{(k)}\|),\quad\forall t\geq 0 \] for some gain functions \(\beta,\gamma_0,\dots,\gamma_k\). This notion is especially suited for systems whose inputs can be reasonably considered smooth enough.
The authors prove Lyapunov characterizations of the \(D^k iss\) property, and compare it with other well-known notions. Moreover, they investigate several applications to cascaded nonlinear systems.