  In this paper, we propose necessary and sufficient conditions for a scalar
function to be nonincreasing along solutions to general differential inclusions
with state constraints. The problem of determining if a function is
nonincreasing appears in the study of stability and safety, typically using
Lyapunov and barrier functions, respectively. The results in this paper present
infinitesimal conditions that do not require any knowledge about the solutions
to the system. Results under different regularity properties of the considered
scalar function are provided. This includes when the scalar function is lower
semicontinuous, locally Lipschitz and regular, or continuously differentiable.

Optimization and Control