We show that input-to-state stabilizability (as defined by Sontag) is both necessary and sufficient for the solvability of a Hamilton-Jacobi-Isaacs equation associated with a meaningful differential game problem similar to, but more general than, the "nonlinear H-infinity" problem, The significance of the result stems from the fact that constructive solutions to the input-to-state stabilization problem are available (presented in the paper) and that, as shown here, inverse optimal controllers possess margins on input-to-state stability against a certain class of input unmodeled dynamics, Rather than completion of squares, the main tools in our analysis are Legendre-Fenchel transformations and the general form of Young＇s inequality.