On a nonlinear parabolic problem with a boundary control and on its applications

We consider the optimal control in a system consisting of the boundary value problem of the first kind for a quasilinear parabolic equation with an unknown coefficient and an additional equation describing a time dependence of this coefficient. Two variational problems with a boundary control regime are substantiated for the given final observations. Some conditions of continuity and differentiability of the corresponding minimization functionals are formulated and proved. An exact representation for the differentials in terms of the solutions of the conjugate problems is obtained. The form of these conjugate problems and their unique solvability in a class of smooth functions are shown. This study is connected with modeling and control of physical-chemical processes in which the inner properties of materials are subjected to changes.