Fixed/predefined-time control of social network-based delayed interconnected system with discontinuity

This article aims to study the fixed/predefined-time control problem of a social network-based delayed interconnected system with a discontinuous interconnection. First, unlike the fractional power function, hyperbolic tangent function and logistic function, a particular exponential function with an indefinite coefficient is utilized to establish fixed/predefined-time stability criteria for discontinuous delayed differential equations based on delayed differential inclusion. The settling time of fixed-time stability is estimated, and the estimation formula is independent of the system's initial states. The Lyapunov energy function method in this article extends the existing fixed/predefined time stability to the case of indefinite derivatives. Second, taking time-varying coefficients, discontinuity, and the time-delay effect, a social network-based interconnected model is established. With the help of the established fixed/predefined-time theorems and the design of the state feedback control schemes, the fixed/predefined-time stabilization is implemented on the social network-based delayed interconnected system with discontinuity. Finally, a numerical example is given to verify the effectiveness of the fixed/predefined-time control. Numerical simulation shows that, under the state feedback control scheme, the social network-based delayed interconnected system with two subsystems is stabilized in a fixed time at zero within 0.69 seconds. Under the predefined time control scheme, the stabilization at zero for the predefined time can be ensured within any set predefined time $T_{\mathrm{p}}=0.25$. It is of practical significance to study the social network-based delayed interconnected system and its fixed/predefined-time control for the rapid formation of stable relationships and the implementation of accurate social governance.