On the observability inequalities of time discrete 2‐D integro‐differential systems in square domains

Abstract

In this article, we consider an observability inequality of time discrete approximation schemes for a class of integro‐differential equations in square domains. The equation is discretized in time by the back‐ward Euler method in combination with Lubich's convolution quadrature. We derive uniform observability inequality for time discretization schemes in which the high frequency components have been filtered. In this way, the well‐known exact observability inequality of the integro‐differential systems (see Loreti and Sforza, Evolution Equations and Control Theory, 2018, Vol. 7, No. 1, pp. 61–77, Thm. 1.2) can be reproduced as the limit, as the time step Δt → 0. The time discrete observability inequality is established by means of a time‐discrete version of the classical harmonic analysis approach. Numerical experiments are presented to show the theoretical analysis.