Asymptotic Stability of Time-Varying Nonlinear Cascade Systems with Delay via Lyapunov–Razumikhin Approach

This paper addresses nonlinear time-varying cascade systems governed by differential equations with finite delay. Several sufficient conditions for asymptotic stability are derived, based on differing assumptions regarding the isolated subsystems and their interconnection. The cascade structure enables the treatment of a broad class of systems while simplifying stability analysis compared to conventional approaches. Moreover, it allows the stabilization problem to be decoupled: under suitable conditions, the asymptotic stability of the overall cascade system follows from the stability properties of its individual subsystems. These properties are typically verified using the direct Lyapunov method. In contrast to existing results, the theorems presented herein apply to an extended class of systems and impose relaxed conditions on the Lyapunov functions employed to establish uniform asymptotic stability. Additionally, new results are provided on semiglobal exponential stability and (non-uniform) asymptotic stability for time-varying cascade systems with delay. Collectively, these contributions broaden the applicability of the direct Lyapunov method to delayed cascade systems.