This paper is devoted to the stability analysis of nonlinear systems whose linear approximation exhibits a pair of purely imaginary eigenvalues. By using the center manifold approach and normalization procedure, we estimate the decay rate of solutions in the critical case considered. Such an estimate is applied for computing the cost of an optimal stabilization problem. As an example, the optimal stabilization problem by means of a smooth output feedback law is considered for a mechanical system with two degrees of freedoms.