A PDE approach to deployment of mobile agents under leader relative position measurements

We study the deployment of a first-order multi-agent system over a desired smooth curve in 2D or 3D space. We assume that the agents have access to the local information of the desired curve and their relative positions with respect to their closest neighbors, whereas in addition a leader is able to measure his relative position with respect to the desired curve. For the case of an open C2 curve, we consider two boundary leaders that use boundary instantaneous static output-feedback controllers. For the case of a closed C2 curve we assume that the leader transmits his measurement to other agents through a communication network. The resulting closed-loop system is modeled as a heat equation with a delayed (due to the communication) boundary state, where the state is the relative position of the agents with respect to the desired curve. By choosing appropriate controller gains (the diffusion coefficient and the gain multiplying the leader state), we can achieve any desired decay rate provided the delay is small enough. The advantage of our approach is in the simplicity of the control law and the conditions. Numerical example illustrates the efficiency of the method.