  We study the deployment of a first-order multi-agent system over a desired
smooth curve in 3D space. We assume that the agents have access to the local
information of the desired curve and their displacements with respect to their
closest neighbors, whereas in addition a leader is able to measure his absolute
displacement with respect to the desired curve. In this paper we consider the
case that the desired curve is a closed C^2 curve and we assume that the leader
transmit his measurement to other agents through a communication network. We
start the algorithm with displacement-based formation control protocol.
Connections from this ODE model to a PDE model (heat equation), which can be
seen as a reduced model, are then established. The resulting closed-loop system
is modeled as a heat equation with delay (due to the communication). The
boundary condition is periodic since the desired curve is closed. 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.
