Thanks for the question. I agree with George. The initial conditions depend on - 'How you define your system'. There is no specific condition, but you have to follow the basic scientific and mathematical rules.

I agree with George and Hasi that in most cases the i.c. come from the system you are trying to model.  However, there can be cases where one knows the state of the system at points in time other than the initial time -- either the final time, or at intermediate times between beginning and end, but you are still interested in estimates of the initial conditions.  These are 'boundary value problems', and some years back received a lot of attention as methods for numerical solution of multi-point boundary value problems became widespread.  My own experience was with using variants of the quasilinearization strategy to solve multi-point boundary value problems in ordinary differential equations.  Richard Bellman and Robert Kalaba, Rand Corp. were early pioneers in this area.