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A functional differential inclusion-based approach to L2-gain analysis and feedback control problems is presented for a class of discontinuous time-delay systems. Motivated by Filippov solution in the differential equations with discontinuous right-hand side, definition of the discontinuous time-delay systems forced by external signals is introduced, and a description of L2-gain property in the sense of the solution concept is given. Then, a condition such that a given discontinuous time-delay system has L2-gain less than a given level and the unforced system is stable in the sense of Filippov solution is presented based on a delay-dependent partial differential inequality. Furthermore, using the presented condition, a design method of state feedback controller is shown such that the closed-loop system satisfies the L2-gain constraint and asymptotical stability. Numerical examples are presented to demonstrate the presented theoretical results.