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This paper addresses the robust H2 guaranteed cost control of continuous-time Markov jump linear systems with transition parameters taking values in a countably infinite set. In the finite case, an adjoint approach to the robust control of MJLS in face of linear structured uncertainty is developed. Regarding the scenario of uncertain transition rates of the Markov process, the design of robust controllers is characterized by uncertainty-dependent linear matrix inequality problems. The main results are applied to the robust control of an underactuated robotic manipulator system.