It is shown in this paper that the successive approximation procedure simplifies computations of the optimal solution of a bilinear-quadratic optimal control problem. On the contrary of the results of Hofer and Tibken where the optimal solution has been obtained in terms of a sequence of the differential Riccati equations, in the presented method only solutions of a sequence of the differential Lyapunov equations are required. A chemical reactor example is used to demonstrate the efficiency of the new method.