Identification of a convolution kernel in a control problem for the heat equation with a boundary memory term

We consider the evolution of the temperature u in a material with thermal memory characterized by a time-dependent convolution kernel h. The material occupies a bounded region Ω with a feedback device controlling the external temperature located on the boundary Γ. Assuming both u and h unknown, we formulate an inverse control problem for an integrodifferential equation with a nonlinear and nonlocal boundary condition. Existence and uniqueness results of a solution to the inverse problem are proved.