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The maximum-likelihood approach to the lumped-parameter filtering estimation theory is extended to a general class of nonlinear distributed-parameter systems with additive Gaussian disturbances and measurement noise. The concept of conventional finite-dimensional-likelihood function is replaced by the likelihood functional which determines the statistical characteristics of an infinite-dimensional Gaussian random variable. First, the nonlinear filtering problem is considered. Using the differential dynamic-programming technique, an approximate nonlinear filter is derived which is shown to be the most natural nonlinear analogue of Kalman's linear distributed-parameter filter, presented in two recent papers. Secondly, the nonlinear prediction is treated by simple extrapolation. Thirdly, the smoothing problem is solved by the well known technique of decomposing the likelihood function(a1) in two parts. Finally, computational results are provided which show the effectiveness of the theory.