On identification of memory kernels in linear theory of heat conduction

A linear integrodifferential equation describing the heat flow in a material with memory is considered. This equation contains a pair of time-dependent convolution kernels that are unknown. Such kernels are determined as solutions of an optimal control problem by using additional data obtained from measurements of average temperature around some fixed points of the domain over some finite time interval. We show the existence of an optimal solution of this problem and derive optimality conditions for it.