An infinite-horizon optimal control problem subject to an infinite-dimensional state equation with state and control variables appearing in a bilinear form is investigated. A sensitivity analysis with respect to the initial condition is carried out. We show in particular that the value function is infinitely differentiable in the neighborhood of the steady state, under a stabilizability assumption. In a second part, we derive error estimates for controls generated by polynomial feedback laws, which are derived from Taylor expansions of the value function.