Smaller and negative exponents in Lyapunov functions for interconnected iISS systems

For integral input-to-state stable (iISS) systems, stability of their interconnections can be established through a small-gain condition. Unlike input-to-state stable (ISS) systems, iISS systems admit gain functions only in limited ranges. Thus, composing a Lyapunov function which is valid globally is instrumental for addressing iISS. A Lyapunov function which is popular in the iISS framework proves the stability of interconnected systems whenever the small-gain condition is satisfied. However, it is hardly practical since its nonlinearities are often artificial and involve astonishingly large exponents. This paper drastically reduces the exponents analytically and numerically. This paper also extends the exponents to negative numbers, and demonstrates that two-sided exponents allow one to avoid unnecessary complicated Lyapunov functions. Copyright (C) 2020 The Authors.