Differential games with a given value function

The set of solutions of a differential game with a terminal payoff functional is investigated. A method is obtained that allows to establish whether a given function is a value of some differential game with a terminal payoff functional. The condition obtained is in fact a condition for the given function to be a minimax (viscosity) solution of some Hamilton–Jacobi equation with Hamiltonian homogeneous in the third variable. We also obtain a sufficient condition for a function to belong to the set of values of differential games with a terminal payoff function.