L1-input-to-state Stability for Nonlinear Systems

In this paper, we consider the L
¹
-input-to-state stability (L
¹
-ISS) for nonlinear systems with L
¹
-inputs. The work is twofold: (i) we provide Lyapunov characterizations of L
¹
-ISS by using the classical Lyapunov method (CLM); (ii) we introduce a novel method, namely, the approximative Lyapunov method (ALM), for the L
¹
-ISS analysis of nonlinear systems. In particular, unlike the CLM that mainly focuses on the construction of Lyapunov functions (LFs), the ALM amounts to constructing an approximation of LFs by using a convex function, which provides a new methodology for stability analysis of nonlinear systems. In order to illustrate the difference of the two methods, we provide an example of the L
¹
-ISS analysis for a class of nonlinear systems with L
¹
inputs.