  Control of multi-agent systems via game theory is investigated. Assume a
system level object is given, the utility functions for individual agents are
designed to convert a multi-agent system into a potential game. First, for
fixed topology, a necessary and sufficient condition is given to assure the
existence of local information based utility functions. Then using local
information the system can converge to a maximum point of the system object,
which is a Nash equilibrium. It is also proved that a networked evolutionary
potential game is a special case of this multi-agent system. Second, for
time-varying topology, the state based potential game is utilized to design the
optimal control. A strategy based Markov state transition process is proposed
to assure the existence of state based potential function. As an extension of
the fixed topology case, a necessary and sufficient condition for the existence
of state depending utility functions using local information is also presented.
It is also proved that using better reply with inertia strategy, the system
converges to a maximum strategy of the state based system object, which is
called the recurrent state equilibrium.
