  Complete Lyapunov functions for a dynamical system, given by an autonomous
ordinary differential equation, are scalar-valued functions that are strictly
decreasing along orbits outside the chain-recurrent set. In this paper we show
that we can prescribe the (negative) values of the derivative along orbits in
any compact set, which is contained in the complement of the chain-recurrent
set. Further, the complete Lyapunov function is as smooth as the vector field
defining the dynamics. This delivers a theoretical foundation for numerical
methods to construct complete Lyapunov functions and renders them accessible
for further theoretical analysis and development.
