On the Observability of Time Discrete Integro-differential Systems



In this paper we study the observability properties of time discrete approximation schemes for some integro-differential equations. The equation is discretized in time by the back-ward Euler method in combination with convolution quadrature. We prove uniform observability results for time discretization schemes in which the high frequency components have been filtered. In this way, the well-known exact observability estimates of the integro-differential systems can be reproduced as the limit, as the time step \( {\varDelta }t\rightarrow 0 \). The discrete observability estimates are established by means of a time-discrete version of the classical harmonic analysis approach.


Integro-differential equation Observability Time discretization Filtering Harmonic analysis Ingham inequalities 

Mathematics Subject Classification

45K05 65D30 65N06 93B07 42A38 



This work was supported in part by the National Natural Science Foundation of China, contract Grant Numbers 11271123 and 11671131.


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Authors and Affiliations

  1. 1.Department of MathematicsHunan Normal UniversityChangshaPeople’s Republic of China

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