Abstract: Practical industrial systems have actuator constraints and restricted availability of measurements.  Unfortunately, most results on robust H-infinity control are presented for full state feedback and unconstrained control inputs.  Neural networks can be used with H-infinity control techniques to overcome these limitations.   In this paper we consider H-infinity nonlinear output feedback control for constrained input systems.  The constraints on the input to the system are encoded via a quasi-norm that allows non-quadratic supply rates along with dissipativity theory to formulate the robust output feedback control problem using Hamilton-Jacobi-Isaac (HJI) equations.  A rigorous approach is presented that allows for formal mathematical proofs of convergence and closed-loop stability.  An iterative solution technique based on a game theoretic interpretation is presented.  To provide a computationally tractable controller design method, the solution is approximated at each iteration with a neural network.  The result is a closed-loop control based on a neural net that has been tuned a priori off-line.  This neural network controller only depends on the available output measurements and also guarantees bounded control inputs.