On the Stochastic Flocking of the Cucker--Smale Flock with Randomly Switching Topologies

Abstract

We present an emergent stochastic flocking dynamics of the Cucker--Smale (CS) ensemble under randomly switching network topologies. The evolution of the CS ensemble with randomly switching topologies involves two random components (switching times and choices of network topologies at switching instants). First, we allow switching times for the network topology to be random so that the successive increments are i.i.d. processes following the common probability distribution. Second, at each switching instant, we choose a network topology randomly from a finite set of admissible network topologies whose union contains a spanning tree. Even for the fixed deterministic network topology, the CS ensemble may not exhibit a monocluster flocking, depending on the initial data and the decay mode of the communication weight functions measuring the degree of interactions between particles. For the flocking dynamics of the CS ensemble with these two random components, we first use a priori conditions on the network topologies and the uniform boundedness of the position diameter, and derive the flocking estimates via matrix theory together with a priori conditions, and then replace the a priori condition for the position diameter by some suitable condition on system parameters and communication weight. A priori conditions on the network topology will be guaranteed by the suitable spanning tree time-blocks with probability one.