  In this paper we study mean-field type control problems with risk-sensitive
performance functionals. We establish a stochastic maximum principle (SMP) for
optimal control of stochastic differential equations (SDEs) of mean-field type,
in which the drift and the diffusion coefficients as well as the performance
functional depend not only on the state and the control but also on the mean of
the distribution of the state. Our result extends the risk-sensitive SMP
(without mean-field coupling) of Lim and Zhou (2005), derived for feedback (or
Markov) type optimal controls, to optimal control problems for non-Markovian
dynamics which may be time-inconsistent in the sense that the Bellman
optimality principle does not hold. In our approach to the risk-sensitive SMP,
the smoothness assumption on the value-function imposed in Lim and Zhou (2005)
need not to be satisfied. For a general action space a Peng's type SMP is
derived, specifying the necessary conditions for optimality. Two examples are
carried out to illustrate the proposed risk-sensitive mean-field type SMP under
linear stochastic dynamics with exponential quadratic cost function. Explicit
solutions are given for both mean-field free and mean-field models.
