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This paper presents an abstract framework for the optimization of mobile sensing devices in distributed parameter systems. The optimization is performed on the repositioning of these devices throughout the spatial domain. It is assumed that a network of mobile sensing devices is available to obtain measurements from the spatially distributed process. Both a centralized state estimator with mobile sensors and a network of consensus distributed estimators are proposed, since both filters can be shown to result in the specific algebraic structure of a symmetric spatial operator with collocated filter and measurement operators. Extensive numerical simulations for a centralized filter for a diffusion equation are included to verify the effectiveness of a such a mobile sensor network in suppressing the effects of spatially varying disturbances and enhancing the estimator's performance.