  We consider a general type of non-Markovian impulse control problems under
adverse non-linear expectation or, more specifically, the zero-sum game problem
where the adversary player decides the probability measure. We show that the
upper and lower value functions satisfy a dynamic programming principle (DPP).
We first prove the dynamic programming principle (DPP) for a truncated version
of the upper value function in a straightforward manner. Relying on a uniform
convergence argument then enables us to show the DPP for the general setting.
Following this, we use an approximation based on a combination of truncation
and discretization to show that the upper and lower value functions coincide,
thus establishing that the game has a value and that the DPP holds for the
lower value function as well. Finally, we show that the DPP admits a unique
solution and give conditions under which a saddle-point for the game exists.
  As an example, we consider a stochastic differential game (SDG) of impulse
versus classical control of path-dependent stochastic differential equations
(SDEs).
