Lyapunov stability tests for integral delay systems

An overview of stability conditions in terms of the Lyapunov matrix for linear integral delay equations is presented. Several examples in the analysis, control and modeling motivate their study. In the framework of Lyapunov–Krasovskii functionals with prescribed derivatives, we review the stability theorems for these functionals and prove a stability criterion (necessary and sufficient condition) in terms of the system delay Lyapunov matrix. The organization of the paper and the detailed developments have the purpose of serving as a tutorial. As a new result, we prove that the stability criterion can be tested in a finite number of operations. Finally, we suggest future directions of research in the field, in particular, the reduction of the bound for which sufficiency is guaranteed and the extension to more general classes of systems.