  In this study, we investigate optimal control problems that involve sweeping
processes with a drift term and mixed inequality constraints. Our goal is to
establish necessary optimality conditions for these problems. We address the
challenges that arise due to the combination of sweeping processes and
inequality mixed constraints in two contexts: regular and non-regular. This
requires working with different types of multipliers, such as finite positive
Radon measures for the sweeping term and integrable functions for regular mixed
constraints. For non-regular mixed constraints, the multipliers correspond to
purely finitely additive set functions.
