Some identification problems for integro-differential operator equations

Abstract

We consider, in a Hilbert space $H$, the convolution integro-differential equation $u^{\prime \prime }(t)-h\ast Au(t)=f(t)$, $0\le t\le T$, $h\ast v(t)={\int }_0^{t}h(t-s)v(s)ds$, where $A$ is a linear closed densely defined (possibly selfadjoint and/or positive definite) operator in $H$. Under suitable assumptions on the data we solve the inverse problem consisting of finding the kernel $h$ from the extra data (measured data) of the type $g(t):=(u(t),\varphi )$, where $\varphi$ is some eigenvector of $A^{\ast }$. An inverse problem for the first-order equation $u^{\prime }(t)-l\ast Au(t)=f(t)$, $0\le t\le T$, is also studied when $A$ enjoys the same properties as in the previous case.

15pp