Abstract
Abstract In a square domain with respect of spatial variables, a boundary value problem is considered for a loaded heat equation, where the loading line is formed by fixing one of the spatial variables. It is required to provide stabilization for a finite period of time with the help of boundary control. Such problems on a semi-infinite time interval for parabolic equations were previously considered in the works of A.V. Fursikov. The stabilization problem posed in our work is transformed into some optimal control problem. The justification of this transformation is given. Theorems on the solvability of initial boundary value problems are formulated. The optimal control problem is solved by the Pontryagin maximum principle method. In accordance with the latter, a conjugate problem is introduced, for which another equivalent formulation is given. An algorithm for solving the optimal control problem using the Ricatti transformation method is presented.