A general bilinear optimal control problem subject to an infinitedimensional state equation is considered. Polynomial approximations of the associated value function are derived around the steady state by repeated forma ldifferentiation of the Hamilton–Jacobi-Bellman equation. The terms of the approximations are described by multilinear forms, which can be obtained as solutions to generalized Lyapunov equations with recursively defined right-handsides.They form the basis for defining a sub-optimal feedback law.The approximation properties of this feedback law are investigated. An application to the optimal control of a Fokker-Planck equation is also provided.