Indefinite derivative Lyapunov–Krasovskii functional method for input to state stability of nonlinear systems with time-delay

In this paper, the problem of the input to state stability (ISS) for nonlinear systems with time-delay is investigated. A continuously differentiable Lyapunov–Krasovskii functional with indefinite derivative is introduced to derive the ISS of the systems, which generalizes the classic Lyapunov–Krasovskii functional with positive definite derivative. As a result, the uniform stability and uniform asymptotic stability criteria of nonlinear systems with time-delay are also established by employing the proposed Lyapunov–Krasovskii functional.