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 $$\Omega $$ with a feedback device controlling the external temperature located on the boundary $$\Gamma $$. 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.