Separable Lyapunov functions for monotone systems: constructions and limitations (Q258406)

The authors investigate the existence and construction of two types of Lyapunov functions for monotone dynamical systems defined on \(\mathbb{R}^n_+\) via the differential equation \(\dot{x} = f(x)\): sum-separable and max-separable Lyapunov functions. It is shown that both max-separable and sum-separable Lyapunov functions exist in a neighborhood of the origin if each eigenvalue of the Jacobian \(Df(0)\) has a negative real part. Results are also obtained for construction of such Lyapunov functions locally or globally, with some limitations. Various examples are provided to illustrate the constructive methods or to show the nonexistence of such functions.