A Fusion of Max- and Sum-Separable Lyapunov Functions Capable of Addressing iISS in Networks

This paper initiates a geometric approach to construction of Lyapunov functions for networks of integral input-to-state stable (iISS) systems. For networks consisting of input-to-state stable (ISS) systems, a geometric construction called the max-separable Lyapunov function has been popular. However, the iISS property is too weak to admit it. In the literature, iISS networks have been addressed by the sum-separable construction, which is algebraic so that a Lyapunov function is given explicitly. Since the Lyapunov function contains all combinations of gain-related functions in a complete graph regardless of the original network structure, the complexity grows very rapidly. The sum-separable Lyapunov function also involves exponents which explode extremely as stability margins decrease. This paper introduces a fusion between the sum- and max-separable functions to process necessary complexity geometrically, and maintain the simplicity of the structure of a constructed Lyapunov function. The proposed framework aims to significantly facilitate the use of Lyapunov functions in analysis and controller design for iISS networks.