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In this paper, the coupled dynamics and control of the vessel, crane, flexible cable and payload under environmental disturbances with attached thrusters for subsea installation operations is investigated. For the practical system with physical constraints, we employ Barrier Lyapunov Function in the design of positioning control for the flexible crane-cable-payload subsystem to ensure that the constraints are not violated. Uniform stability of the flexible subsystem is shown and asymptotic positioning of the boundaries is achieved. Next, we tackle the scenario where nonuniformity of the cable, uncertainties and environmental disturbances are considered. Boundary controls are formulated using the nonlinear partial differential equation of the cable. Numerical simulations are provided to illustrate the performance of the proposed controls.