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This paper presents a study of the Koopman operator theory and itsapplication to optimal control of a multi-robot system. The Koopman operator,while operating on a set of observation functions of the state vector of anonlinear system, produces a set of dynamic equations which, through a dynamictransformation, form a new dynamic system. As an operator, it has a richspectrum of mathematical properties, and as a tool for dynamic system analysisand control design, it has a unique collection of practical meanings.  The Koopman system technique is then applied to the development of a linearor bilinear model approximation of nonlinear utility functions for optimalcontrol of a system of multiple (mobile) robots, by selecting the utilityfunctions as the Koopman system state variables and expressing the set ofKoopman variables as the state variables of a linear or bilinear system whoseparameters are determined through optimization. An iterative (online) algorithmis developed for adaptively estimating the parameters of the approximationmodel of the robot system with nonlinear utility functions.  Finally, the control problems based on a linear or bilinear model for thenonlinear utility functions are formulated for optimal control of themulti-robot system, by transforming the nonlinear programming problem to alinear programming problem.

Koopman System Approximation Based Optimal Control of Multiple Robots -- Part I: Concepts and Formulations