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In this paper we present a framework to address filtering and smoothing problems for distributed parameter systems when mobile (dynamic) sensors are used to provide system measurements. This framework can be used for systems governed by parabolic and hyperbolic partial differential equations and hence has application to a diverse set of problems such as estimating locations of biological and chemical sources, target tracking and estimation. We formulate the problems as hybrid systems on infinite dimensional spaces (coupled systems of partial, ordinary and delay differential equations) and use infinite dimensional theory to develop computational algorithms for the problems. A simple numerical example illustrates the approach.