This work addresses the optimal stabilization problem of a nonlinear control system by using a smooth output feedback. The optimality criterion is the maximization of the decay rate of solutions in a neighborhood of the origin. We formulate this criterion as a minimax problem with respect to non-integral functional. An explicit construction of a Lyapunov function is proposed to evaluate the optimal cost. This design methodology is justified for nonlinear systems in a critical case of stability with a pair of purely imaginary eigenvalues. As an example, a minimax optimal controller is obtained for a spring-pendulum system with partial measurements of the state vector.