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This paper studies mobility strategies to control positions of mobile robots in a mobile sensor network in order to maximize lifetime of the network. With communication and mobility energy costs modeled, the problem is formulated into a nonlinear non-convex optimization problem, then reformulated into a convex optimization problem. The separable property of the system is then exploited by Lagrangian duality, and the solution is obtained by distributed saddle-point computations. Computer simulations showed that the proposed distributed algorithm can quickly converge to the optimal solution, and it also justifies the use of mobility for energy efficiency by showing its significant improvement to the network lifetime and relatively low cost in mobility. Furthermore, the proposed energy optimization framework can accommodate different mobile sensor network models.