  Our purpose of this paper is to study stochastic control problem for systems
driven by mean-field stochastic differential equations with elephant memory, in
the sense that the system (like the elephants) never forgets its history. We
study both the finite horizon case and the infinite time horizon case.
  - In the finite horizon case, results about existence and uniqueness of
solutions of such a system are given. Moreover, we prove sufficient as well as
necessary stochastic maximum principles for the optimal control of such
systems. We apply our results to solve a mean-field linear quadratic control
problem.
  - For infinite horizon, we derive sufficient and necessary maximum
principles.
  As an illustration, we solve an optimal consumption problem from a cash flow
modelled by an elephant memory mean-field system.
