A convex sweeping process is considered in a separable Hilbert space. The majority of works on sweeping processes use the Hausdorff distance to describe the movement of the convex set generating the process. However, for unbounded sets the use of the Hausdorff distance does not always guarantee the fulfilment of conditions under which a solution exists. In the present work, instead of the Hausdorff distance we use the \(\rho \)-excesses of sets. These excesses are subjected to positive Radon measures depending on \(\rho \). We prove the existence of right continuous BV solutions and establish their dependence on single-valued perturbations depending only on time. The results we obtain are applied to prove the theorems on existence and relaxation of extremal right continuous BV solutions of a sweeping process with multivalued perturbations. Some results on absolutely continuous solutions are derived as corollaries.

在可分离的希尔伯特空间中考虑了凸扫描过程。关于清扫过程的大多数工作都使用Hausdorff距离来描述产生过程的凸集的运动。但是，对于无界集合，使用Hausdorff距离并不总是保证满足存在解的条件。在当前的工作中，我们使用\（\ rho \）-超出集合来代替Hausdorff距离。这些过量项将根据\（\ rho \）受到正Radon量的影响。我们证明了正确的连续BV解的存在，并建立了它们对仅依赖于时间的单值扰动的依赖。我们获得的结果被用来证明关于多值摄动过程的极右连续BV解的存在性和松弛定理。关于绝对连续解的一些结果作为推论得出。