Fixed-time bipartite synchronization with a pre-appointed settling time over directed cooperative–antagonistic networks

This paper mainly delves into the fixed-time bipartite synchronization problems of multi-agent networks embedded single- and double-integrator over directed graphs with cooperative–antagonistic interactions. Firstly, by introducing Pontryagin's maximum principle in distributed fashions, a new kind of distributed fixed-time bipartite synchronization protocols is developed. Then, through multi-step motion planning techniques, the designed protocols can ensure that the states of agents converge to values with the same module but different sign at a pre-appointed settling time. Compared with existing results, an important contribution of this paper is that it is over directed networks that fixed-time bipartite synchronization problems are solved. Meanwhile, the fixed settling time can be appointed in advance. Finally, the application to formation problems with a pre-appointed settling time is further considered to indicate the utility of proposed protocols.