Optimization Formulation of the Coefficient Inverse Parabolic Interface Problem with an Additional Integral Condition

The optimization formulation of the coefficient inverse problem for semi-linear parabolic equations with imperfect contact matching conditions and an additional integral condition is considered. Imperfect-contact matching condition is given at the inner boundary between media. Well-posedness of the initial boundary value problem and the optimization formulation of the inverse problem is investigated. The differentiability of the objective functional is proved and the formula for its gradient is found. A necessary optimality condition is established in the form of a variational inequality.