In this study of the nonlinear H-infinity-optimal control design for strict-feedback nonlinear systems, our objective is to construct globally stabilizing control laws to match the optimal control law up to any desired order and to be inverse optimal with respect to some computable cost functional. Our recursive construction of a cost functional and the corresponding solution to the Hamilton-Jacobi-Isaacs (HJI) equation employs a new concept of nonlinear Cholesky factorization, When the value function for the system has a nonlinear Cholesky factorization, we show that the backstepping design procedure can be tuned to yield the optimal control law.