Cauchy Matrices in the Observation of Diffusion Equations (Journal of Mathematical Study, 2021)
Observability Gramians of diffusion equations have been recently connectedto infinite Pick and Cauchy matrices. In fact, inverse or observability inequalities canbe obtained after estimating the extreme eigenvalues of these structured matrices,withrespect to the diffusion semi-group matrix. The purpose is hence to conduct a spectralstudy of a subclass of symmetric Cauchy matrices and present an algebraic way toshow the desired observability results. We revisit observability inequalities for threedifferent observation problems of the diffusion equation and show how they can be(re)stated through simple proofs.