Lyapunov second method is ubiquitous in stability analysis of nonlinear systems. Asymptotic stability of the origin can be checked via this method if one can find a positive definite function whose derivative along solutions of the system is negative definite (strong Lyapunov function). Unfortunately, however, the most natural functions, such as the total energy function in mechanical systems, typically do not satisfy these conditions but only weaker conditions. For example, the derivative of the Lyapunov function may be only negative semi-definite along solutions of the system (i.e. weak Lyapunov function). It is highly desirable to obtain strong Lyapunov functions because one can then analyze stability robustness or use the Lyapunov functions in controller redesign for robustness enhancement. We have provided several constructions of strong Lyapunov functions in situations when only a weak Lyapunov function is available.