Approximate state recovering of a distributed parameter system

In this paper the problem of recovering the initial state of time-invariant distributed parabolic systems on the basis of a localized noisy observation is considered, and a criterion for the optimal selection of the measurement point is proposed which takes care of the statistical assumptions about the components of the initial state and of the class of approximating functions describing the spatial content of the solution. An error estimate is derived and the optimal sensor location is shown to be dependent upon the set of approximating functions, the measurement noise, and the a priori statistics of the initial state. A numerical example is reported which allows a comparison of two different approximating systems of the same order in the light of the proposed criterion: the first one obtained by an eigenfunction expansion of the spatial differential operator; the second by a cubic spline interpolation.