Max- and sum-separable Lyapunov functions for monotone systems and their level sets

For interconnected systems and systems of large size, aggregating information of subsystems studied individually is useful for addressing the overall stability. In the Lyapunov-based analysis, summation and maximization of separately constructed functions are two typical approaches in such a philosophy. This paper focuses on monotone systems which are common in control applications and elucidates some fundamental limitations of max-separable Lyapunov functions in estimating domains of attractions. This paper presents several methods of constructing sum- and max-separable Lyapunov functions for second order monotone systems, and some comparative discussions are given through illustrative examples.