Filipov Approach in One Sided Lipschiz Continuous Impulsive Quantum Stochastic Differential Inclusion

Abstract

This paper presents a novel application of the Filippov theorem to impulsive quantum stochastic differential inclusions of Hudson and Parthasarathy formulation of quantum stochastic calculus. The study extends classical differential inclusion methods to quantum systems subject to stochastic noise and impulsive effects, capturing the nonsmooth dynamics often encountered in quantum control and quantum information systems. By using the Filippov-Wazewski theorem,we relax the Lipschitz continuity to a one-sided Lipschitz continuity, we established the existence and uniqueness of solutions for systems where the dynamics are not fully deterministic and undergo abrupt, discrete changes using continuous selection. One-sided Lipschitz continuity conditions ensure the well-posedness of these systems, offering a robust mathematical foundation for modeling impulsive behavior in quantum stochastic environments. This research provides new insights into quantum systems with complex dynamics, laying groundwork for future applications in quantum computing, communication, and control theory.