Arthur Krener - Naval Postgraduate School Al'brekht's method is a way optimally stabilize a finite dimensional nonlinear control system to an operating point. We generalize it to infinite dimensional nonlinear control systems. The systems are described in Fredholm form. By completing the square we derive the Riccati PDE for the linear part of the problem. It is a PDE is of elliptic type with a quadratic nonlinearity. We also derive linear elliptic PDEs for the higher degree terms in Taylor polynomial of the optimal cost. This yields linear equations for the higher degree terms in the Taylor polynomial of the optimal feedback.