Construction of functions with semidefinite derivative along trajectories of autonomous systems with arbitrary order of smoothness

For autonomous systems of ordinary differential equations the theorem of existence of function having semidefinite derivative is proved. It is shown that for any m such function can be chosen from the class of continuously differentiable up to order m inclusively. The proof is constructive and gives some approach for constructing a function with semidefinite derivative. A class of systems generalizing Artstein’s circle is considered as an example.