Sampled-data H∞ filtering of a 2D heat equation under pointlike measurements

The existing sampled-data observers for 2D heat equations use spatially averaged measurements, i.e., the state values averaged over subdomains covering the entire space domain. In this paper, we introduce an observer for a 2D heat equation that uses pointlike measurements, which are modeled as the state values averaged over small subsets that do not cover the space domain. The key result, allowing for an efficient analysis of such an observer, is a new inequality that bounds the L₂-norm of the difference between the state and its point value by the reciprocally convex combination of the L₂-norms of the first and second order space derivatives of the state. The convergence conditions are formulated in terms of linear matrix inequalities feasible for large enough observer gain and number of pointlike sensors. The results are extended to solve the H∞ filtering problem under continuous and sampled in time pointlike measurements.