Fixed-time distributed optimization for multi-agent systems with external disturbances over directed networks

This article considers the fixed-time distributed optimization problem of multi-agent systems with external disturbances, in which the global optimization objective is a convex combination of local objective functions. To solve this issue, a directed communication network is carefully designed, and an integral sliding mode control protocol is proposed based on the gradient of global objective function first. Moreover, two distributed optimal protocols are designed by using the gradient and the Hessian matrix of local objective function, respectively. By employing Lyapunov stability theory, graph theory, convex analysis, and inequality techniques, we prove that all proposed protocols can make agents achieve consensus and converge accurately to the optimal solution of the considered problem in some fixed-time intervals. Finally, some numerical simulations are given to verify the feasibility of the theoretical results.