  This paper considers the problem of steering a colossal group of agents of
which the dynamics are governed by a discrete-time first-order linear system,
which is a very preliminary version. The group of agents are characterized as a
probability density function and an occupation measure respectively in the
paper and two corresponding treatments are given. We propose to use the power
moments to characterize the density function/occupation measure of the agents.
A moment system representation of the original system is put forward for
control and an empirical control scheme corresponding to it is proposed. By the
designed control law, the moment sequence of the control at each time step is
positive, which ensures the existence of the control for the moment system. We
then realize the control as an analytic form of function by a convex
optimization scheme of which the existence and uniqueness of the solution have
been proved in our previous paper. An error analysis of the terminal density
from the specified one is also provided. For the problem where the group of
agents is characterized as an occupation measure, the control for each agent is
determined by drawing independent and identically-distributed(i.i.d) samples
from the realized analytic function. Finally we simulate both unconstrained and
constrained controls of a colossal group of agents, which validate our proposed
algorithms.

Optimization and Control