Differential Games with Unbounded Versus Bounded Controls

A zero-sum differential game with an unbounded control and no coercivity assumptions is investigated. By means of suitable reparameterization techniques one is able to avoid the serious drawbacks which derive from the lack of a sufficiently fast growth hypothesis. In particular, one achieves a remarkable regularization of the two related Hamilton-Jacobi boundary value problems. One also proves that the latter admit unique continuous solutions, which coincide necessarily with the upper and lower values of the game.