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In this paper, we consider the H∞ nonlinear state feedback control of constrained input systems. The input constraints are encoded via a quasi-norm that enables applying quasi L2-gain analysis of the corresponding closed-loop nonlinear system. The quasi-norm allows using nonquadratic supply rates along with the theory of dissipative systems to formulate the robust optimal control problem for constrained input systems using the Hamilton-Jacobi-Isaacs (HJI) equation. Hence, the constrained optimal control problem is formulated as a closely related unconstrained problem. The saddle point strategy corresponding to the related zero-sum differential game is derived, and shown to be unique. Finally, an iterative solution technique based on the game theoretic interpretation of the HJI equation is presented. This iterative approach allows a deeper insight on the relation between the attenuation gain and the region of asymptotic stability of the H∞ controller for constrained input systems.