Existence of a function with non-positive derivative along trajectories of a non-autonomous system of differential equations

The paper provides a proof of existence of a function with non-positive derivative along trajectories of a non-autonomous system of differential equations. The function is built to be differentiable and to allow arbitrary small time-independent upper bound. The function is also proven to be periodic in the case of periodic system. The construction of the function is illustrated for a simple third-order system.