Controlling flocking in a Cucker-Smale type model with pattern formation and velocity regulating

We consider the Cucker-Smale type flocking model with pattern formation and velocity regulation. This is a multi-agent model consisting of three terms, taking effects for network coupling, spatial pattern formation control, and pinning control for velocity, respectively. By means of energy function and gradient theory, we show that the relative positions evolve into a target pattern formation, and a target common velocity is achieved by imposing the pinning control on a few agents only. Furthermore, we formulate an optimal control problem of the flocking model by controlling the coupling strength of network, for which the existence of optimal control is analyzed. To minimize the cost function, we utilize the standard gradient descent approach to identify the optimal control. Finally, we carry out numerical simulations to validate the pinning control and optimal control strategy, and investigate the influence of various environments on flocking dynamics.