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Robust control problems for nonlinear systems are usually formulated as L2-induced norm minimization problems and it is known that those problems are reduced to the solvability of Hamilton-Jacobi equations. However, in the case of bilinear systems the usual L2-induced norm minimization problem leads to an obvious solution (the zero input). To avoid the obvious solution Simizu et al. (1997) introduced nonlinear weights on the evaluated signal and proposed a design method using linearization of the state-dependent matrix Riccati inequality derived from the Hamilton-Jacobi equation. In contrast to this, the purpose of the present paper is to propose a new design method using SOS (Sum-of-Squares) optimization technique without linearization.