Novel finite and fixed-time stability theorems for fractional-order impulsive discontinuous systems and their application to multi-agent systems

This article studies the finite-time (FNT) and fixed-time (FXT) stability theorems of general fractional-order impulsive discontinuous systems (FOIDSs) through an indefinite Lyapunov functional (LF) approach. Using differential inclusion theory and set-valued map concept, the general FOIDSs can be transformed into fractional-order impulsive differential inclusions (FOIDIs). Two new stability theorems on finite and FXT stability of FOIDSs with finite impulsive instants are derived using an indefinite LF and fractional derivatives of the LF on some impulsive instants, respectively. The settling time is explicitly calculated. The obtained theoretical findings are applied to fractional-order discontinuous multi-agent systems. Discrete control schemes and indefinite Lyapunov-Krasovskii functions (LKFs) are also developed for ensuring FXT synchronization in fractional-order discontinuous multiagent systems. To demonstrate the efficiency and feasibility of the proposed method, some numerical simulations are conducted.