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This study addresses the problem of trajectory control of a flexible two-link manipulator on the basis of the partial differential equation (PDE) dynamic model. One of the key contributions of this study is that a novel non-linear PDE observer is proposed to estimate distributed positions and velocities along flexible links, which cannot be achieved by the typical ordinary differential equation observer. In addition, the rigidity-flexibility coupling dynamics is decomposed using the singular perturbation approach, thus providing convenience for control design. Based on the proposed observer and the decoupled PDE model, a boundary control scheme is designed to regulate the end effector along reference trajectory in task space and suppress vibration simultaneously. The asymptotic stability of both the proposed observer and the control algorithm is validated by theoretical analysis and demonstrated by simulation results, respectively.