A two-person zero-sum differential game with unbounded controls is considered. Under
proper coercivity conditions, the upper and lower value functions are characterized as the
unique viscosity solutions to the corresponding upper and lower Hamilton–Jacobi–Isaacs
equations, respectively. Consequently, when the Isaacs’ condition is satisfied, the upper
and lower value functions coincide, leading to the existence of the value function of the
differential game. Due to the unboundedness of the controls, the corresponding upper and
lower Hamiltonians grow super linearly in the gradient of the upper and lower value
functions, respectively. A uniqueness theorem of viscosity solution to Hamilton–Jacobi
equations involving such kind of Hamiltonian is proved, without relying on the
convexity/concavity of the Hamiltonian. Also, it is shown that the assumed coercivity
conditions guaranteeing the finiteness of the upper and lower value functions are sharp in
some sense.