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In this paper, we propose necessary and sufficient conditions for a scalarfunction to be nonincreasing along solutions to general differential inclusionswith state constraints. The problem of determining if a function isnonincreasing appears in the study of stability and safety, typically usingLyapunov and barrier functions, respectively. The results in this paper presentinfinitesimal conditions that do not require any knowledge about the solutionsto the system. Results under different regularity properties of the consideredscalar function are provided. This includes when the scalar function is lowersemicontinuous, locally Lipschitz and regular, or continuously differentiable.

Necessary and Sufficient Conditions for the Nonincrease of Scalar Functions Along Solutions to Constrained Differential Inclusions