The cooperatability of the first-order multi-agent systems consisting of a leader and a follower with multiplicative noises under Markov switching topologies

We investigate the cooperatability of the first-order leader-following multi-agent systems consisting of a leader and a follower with multiplicative noises under Markov switching topologies. Each agent exhibits first-order linear dy-namics, and there are multiplicative noises along with information exchange among the agents. What is more, the communication topologies are Markov switching topologies. By utilizing the stability theory of the stochastic differential equations with Markovian switching and the Markov chain theory, we establish the necessary and sufficient conditions for the cooperatability of the leader-following multi-agent systems. The conditions are outlined below: (i) The product of the system parameter and the square of multiplicative noise intensities should be less than 1/2; (ii) The transition rate from the unconnected graph to the connected graph should be twice the system parameter; (iii) The transition rate from the connected graph to the unconnected graph should be less than a constant that is related to the system parameter, the intensities of multiplicative noises, and the transition rate from the unconnected graph to the connected graph. Finally, the effectiveness of our control strategy is demonstrated by the population growth systems.