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We establish necessary and sufficient conditions that characterize the Markov Perfect Nash Equilibrium (MPNE) in differential games, specifically when the equilibrium may be only piecewise smooth with respect to the state variable. Our approach follows the classical framework presented in [1] and its references, which is based on constructing an admissible set of discontinuities for the strategies. We connect this concept with the well-known Rankine-Hugoniot jump conditions, which are commonly used in fluid dynamics to study weak solutions of the equations that define the problem. Under appropriate strict concavity assumptions, the entropy condition, as applied to the weak solutions of the fluid dynamic equations, allows us to identify the unique MPNE from a large number of candidate profiles that are also weak solutions. We illustrate our findings using a finite-horizon differential game involving the non-cooperative management of a non-renewable resource. We demonstrate how imposing a non-concave utility function at the final time leads to a discontinuous MPNE.

[1] W.H. Fleming and R.W. Rishel. Deterministic and Stochastic Optimal Control. Springer, New York, 1975.