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In this paper, we consider a model case of stationary vibrations of a thin flat plate, one side of which is embedded, the opposite side is free, and the sides are freely leaned. In mathematical modeling, there is a local boundary value problem for the biharmonic equation in a rectangular domain. Boundary conditions are given on all boundary of the domain. We show that the considered problem is self-adjoint. Herewith, the problem is ill-posed. We show that the stability of solution to the problem is disturbed. Necessary and sufficient conditions of existence of the problem solution are found. Spaces of the ill-posedness of the considered problem are constructed. [ABSTRACT FROM AUTHOR]

In this paper, we consider the boundary value problem in two-dimensional rectangular domain for the biharmonic equation. The studied ill-posed boundary value problem is reduced to the optimal control problem. In terms of solutions of the adjoint boundary value problem, we set optimality conditions. It is found strong criterion for solvability of ill-posed boundary value problem. [ABSTRACT FROM AUTHOR]