Discrete Observability of Parabolic Initial Boundary Value Problems

Abstract

The general problem involves the observabiLity and approximate reconstruction of initial data for parabolic initial boundary value problems from incomplete measurements of the solution at discrete sets of spatial and temporal values. Typically this type of problem is ill-posed in the sense of Hadamard, and as is well known, this difficulty is often reflected in the instabiLity of the inversion scheme. We consider a particular method for carrying out such an approximate inversion based on results estabLished in [3], [4]. This method, based on eigenfunction expansions, is presented for quite general paraboLic systems with selfadjoint spatial part but the general technique can easily be extended to discrete spectral operators with at most finitely many unstable eigenvalues assuming certain decay rates on the spectrum of the spatial operator. Part of our future effort will be directed toward the analysis of more general equations including hyperbolic problems and various problems involving dissipative spatial operators.KeywordsRank ConditionDirichlet SeriesInfinite SystemEigenfunction ExpansionSpatial OperatorThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.