Certain properties of controllers designed using the non-linear H-infinity technique are studied. It is well known that the explicit solution of the Hamilton-Jacobi-Isaacs (HJI) inequality is generally not feasible. In this paper by applying the polynomial approximation method, approximate expressions of the co-state and the two players of the game are considered. Using Lyapunov techniques, we prove a property related to the conjecture by van der Schaft (1993), which states that the non-linear feedback controller always results in a larger domain of validity than its linearized controller. Specifically, it is shown that the estimate of the domain of validity grows proportionally to the order of the control approximation. Effects of attenuation level and weighting the controlled output on the estimate of the domain of validity are also discussed. In this connection, a fictitious autonomous system derived from the original system and its HJI inequality is first introduced. The effect of approximation is then represented by introducing a perturbation term. It is shown that the estimate of the domain of validity for the HJI inequality may be related to the estimate of the domain of attraction of the equilibrium point of the fictitious system.