A novel Lyapunov theorem on finite/fixed-time stability of discontinuous impulsive systems

This paper deals with the Finite/Fixed-Time Stability (FTS) problem of the discontinuous impulsive differential equation. Under the framework on differential inclusion, this problem can be transformed into the FTS problem of impulsive differential inclusion. A uniform criterion on FTS of nonlinear discontinuous impulsive differential systems with pre-given finite impulse instances is established, which is effective for both stabilizing impulses and destabilizing impulses. During this process, we propose an improved Lyapunov method, where the derivative of the Lyapunov Function (LF) may not exist in some instances. Moreover, the upper-bound estimation for the derivative of LF is allowed to be a time-varying function and takes both positive and negative values. Finally, the proposed criterion is supported by two numerical examples.