  This paper presents a study of the Koopman operator theory and its
application to optimal control of a multi-robot system. The Koopman operator,
while operating on a set of observation functions of the state vector of a
nonlinear system, produces a set of dynamic equations which, through a dynamic
transformation, form a new dynamic system. As an operator, it has a rich
spectrum of mathematical properties, and as a tool for dynamic system analysis
and control design, it has a unique collection of practical meanings.
  The Koopman system technique is then applied to the development of a linear
or bilinear model approximation of nonlinear utility functions for optimal
control of a system of multiple (mobile) robots, by selecting the utility
functions as the Koopman system state variables and expressing the set of
Koopman variables as the state variables of a linear or bilinear system whose
parameters are determined through optimization. An iterative (online) algorithm
is developed for adaptively estimating the parameters of the approximation
model of the robot system with nonlinear utility functions.
  Finally, the control problems based on a linear or bilinear model for the
nonlinear utility functions are formulated for optimal control of the
multi-robot system, by transforming the nonlinear programming problem to a
linear programming problem.
