Lyapunov functionals for linear continuous-time difference systems: A single delay case

Abstract

This paper focuses on the Lyapunov stability problem of linear continuous-time difference systems. We show that although the stability of these systems has been extensively investigated, the problem of determining suitable quadratic Lyapunov functionals is not completely solved yet, even for the simplest system with a single delay. Herein, using the single delay case as an example, we construct new quadratic Lyapunov functionals. The functionals, which depend on a matrix-valued function strongly related to the discrete-time Lyapunov matrix equation, allow deriving a converse Lyapunov functional result. We show how the functionals can be used to solve problems such as the computation of exponential estimates for solutions and robustness bounds of perturbed systems.