The Mandalay Derivative for Nonsmooth Systems: Applications to Nonsmooth Control Barrier Functions

One of the most challenging aspects of nonsmooth analysis is to overcome nondifferentiability. A possible approach is to use the generalized notions of the classical gradient and directional derivatives. In this letter we define a generalized directional derivative, the Mandalay derivative, based on set-valued Lie derivatives. For this operator, we derive the analogues to the classical chain rule, superposition rule (for linear combinations of functions), product rule, and quotient rule in the form of inequalities, which facilitate the computation of the Mandalay derivative in the context of nonsmooth system analysis and design. Moreover, we demonstrate the application of the Mandalay derivative for both first and high-order nonsmooth Control Barrier Functions in multiple examples.