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Discontinuity in dynamical system is caused by natural phenomenon or control actions engineered by control design such static friction, contacting collision, switching and variable structure control et al. The dynamical systems with the discontinuities are usually described by the differential equations with disconti8nous right hand side. That is, the vector field defining the dynamical system maybe a function which is discontinuous on the state or the time. For this kind of systems, to establish an analysis and synthesis framework, the most fundamental issue we must face is the notation of solution, and then the uniqueness and convergence. This talk will focus on these fundamental issues and on extending the basic results to control design. First, a brief review of the analysis of discontinuous dynamical systems will be addressed, and then the Filippov-framework for stability analysis of discontinuous systems will be surveyed shortly. Base on the fundamental results, two control design issues will be addressed. Finally, some challenging problems in control of mechanical systems and automotive powertrain systems with the discontinuous dynamical system theory will be introduced with physical background.