Volume 48, Issue 4
Cauchy Matrices in the Observation of Diffusion Equations

Faker Ben Belgacem & Sidi-Mahmoud Kaber

J. Math. Study, 48 (2015), pp. 330-344.

Published online: 2015-12

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  • Abstract
Observability Gramians of diffusion equations have been recently connected to infinite Pick and Cauchy matrices. In fact, inverse or observability inequalities can be obtained after estimating the extreme eigenvalues of these structured matrices,with respect to the diffusion semi-group matrix. The purpose is hence to conduct a spectral study of a subclass of symmetric Cauchy matrices and present an algebraic way to show the desired observability results. We revisit observability inequalities for three different observation problems of the diffusion equation and show how they can be (re)stated through simple proofs.
  • Keywords

Pick matrices Cauchy matrices Hadamard product diffusion equation observability inequalities

  • AMS Subject Headings

15B05, 15B34, 93B07, 35K05

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

faker.ben-belgacem@utc.fr (Faker Ben Belgacem)

kaber@ljll.math.upmc.fr (Sidi-Mahmoud Kaber)

  • BibTex
  • RIS
  • TXT
@Article{JMS-48-330, author = {Ben Belgacem , Faker and Kaber , Sidi-Mahmoud}, title = {Cauchy Matrices in the Observation of Diffusion Equations}, journal = {Journal of Mathematical Study}, year = {2015}, volume = {48}, number = {4}, pages = {330--344}, abstract = {Observability Gramians of diffusion equations have been recently connected to infinite Pick and Cauchy matrices. In fact, inverse or observability inequalities can be obtained after estimating the extreme eigenvalues of these structured matrices,with respect to the diffusion semi-group matrix. The purpose is hence to conduct a spectral study of a subclass of symmetric Cauchy matrices and present an algebraic way to show the desired observability results. We revisit observability inequalities for three different observation problems of the diffusion equation and show how they can be (re)stated through simple proofs.}, issn = {2617-8702}, doi = {https://doi.org/10.4208/jms.v48n4.15.02}, url = {http://global-sci.org/intro/article_detail/jms/9937.html} }
TY - JOUR T1 - Cauchy Matrices in the Observation of Diffusion Equations AU - Ben Belgacem , Faker AU - Kaber , Sidi-Mahmoud JO - Journal of Mathematical Study VL - 4 SP - 330 EP - 344 PY - 2015 DA - 2015/12 SN - 48 DO - http://doi.org/10.4208/jms.v48n4.15.02 UR - https://global-sci.org/intro/article_detail/jms/9937.html KW - Pick matrices KW - Cauchy matrices KW - Hadamard product KW - diffusion equation KW - observability inequalities AB - Observability Gramians of diffusion equations have been recently connected to infinite Pick and Cauchy matrices. In fact, inverse or observability inequalities can be obtained after estimating the extreme eigenvalues of these structured matrices,with respect to the diffusion semi-group matrix. The purpose is hence to conduct a spectral study of a subclass of symmetric Cauchy matrices and present an algebraic way to show the desired observability results. We revisit observability inequalities for three different observation problems of the diffusion equation and show how they can be (re)stated through simple proofs.
Faker Ben Belgacem & Sidi-Mahmoud Kaber. (2019). Cauchy Matrices in the Observation of Diffusion Equations. Journal of Mathematical Study. 48 (4). 330-344. doi:10.4208/jms.v48n4.15.02
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