Distributed optimisation for multi-agent systems with the first-order integrals under Markovian switching topologies

This paper studies the distributed optimisation problem for multi-agent systems with the first-order dynamics over Markovian switching topologies. The interaction topology among agents’ switches following a Markov process and each topology is modelled as a state of the Markov process. The aim is to minimise the global cost functions and make the agents converge to the optimal point through the network communication between the agents, where each agent has a local convex cost function only known by itself. Utilising the knowledge of convex analysis and graph theory, we establish a distributed algorithm for the optimisation problem with randomly switching topologies. A sufficient condition for the existence of such algorithm is obtained by using the Lyapunov method. Besides, the result is also extended to the cases of a Markov process with partially unknown transition rates. Finally, numerical simulations are given to validate the proposed algorithm.