The design of a nonlinear state feedback suboptimal control using parameter expansion techniques and the minimum principle is developed with a quadratic performance index to be minimized. The design is systematic when the nonlinearities of the system can be written as polynomial functions of the state variables. Then, the linear portion of the control is designed using the Riccati equation in the same way as in linear systems, and the nonlinear part is assumed to be homogeneous polynomials of the state variables with unknown coefficients that are in general function of t. A computer routine is developed to aid the determination of these coefficients. The method is applied to two examples. The first illustrates the design procedure and some features of the control, and the second is a study of a physical system.