Lyapunov-based stability analysis

Abstract

This chapter presents generalizations of the direct Lyapunov method to TDSs. In the first section, for general TDSs, the stability notions are defined, and Lyapunov–Krasovskii and Lyapunov–Razumikhin stability theorems are stated. The second section gives a short introduction to linear matrix inequalities. Sections 3.3–3.7 and 3.10 are devoted to delay-independent and delay-dependent stability conditions for linear TDSs. Sufficient conditions are derived in terms of LMIs. Some of the presented ideas may be useful in the nonlinear case. Section 3.9 discusses Lyapunov-based necessary stability conditions for LTI RDEs.