Prescribed-time consensus of time-varying open multi-agent systems with delays on time scales

In this study, we focus on prescribed-time leader-following consensus problems for a class of time-varying open multi-agent systems (OMASs) with time delays on time scales. To achieve this objective, we propose a novel segmented state feedback control protocol which contains a time-varying scalar function. It ensures prescribed-time consensus can be achieved while the monotonicity of Lyapunov function is unknown. On this basis, impulsive signals dependent on the leader are further introduced at each opening instant to avoid the possibility of the controller norm being too large. During the process of theoretical analysis, we further innovatively extend Halanay-like inequalities on time scales to resolve theoretical analysis difficulties caused by time delays. Based on time scale theory and Lyapunov stability theory, sufficient conditions depending on system parameters and controller parameters for achieving prescribed-time consensus can be derived. Besides, considering absolute information of each agent cannot be completely obtained, we further design one observer system to reconstruct information of the original system, guaranteeing it can still reach consensus within the pre-set time. Finally, two simulation examples are used to illustrate the validity of our proposed theory.