Computation of Lyapunov Functions for Nonlinear Differential Equations via a Massera-Type Construction

An approach for computing Lyapunov functions for nonlinear continuous-time differential equations is developed via a new, Massera-type construction. This construction is enabled by imposing a finite-time criterion on the integrated function. By means of this approach, we relax the assumptions of exponential stability on the system's equilibrium, while still allowing integration over a finite time interval. The resulting Lyapunov function can be computed based on any $\mathcal {K}_{\infty}$ -function of the norm of the solution of the system. In addition, we show how the developed converse theorem can be used to construct an estimate of the domain of attraction. Finally, a range of examples from the literature and biological applications, such as the genetic toggle switch, the repressilator, and the HPA axis, are worked out to demonstrate the efficiency and improvement in computation of the proposed approach.