Identification of boundary conditions at one of the ends of a segment

We consider a boundary value problem on an interval for a fourth-order differential equation. The boundary conditions at one end of the segment are known, but at the other end of the segment they are unknown. The eigenvalues of the boundary value problem are known as well. The problem is to reconstruct the unknown boundary conditions at one of the ends of the segment.
Four theorems are proved in the paper. The first two theorems are algebraic. They show that the matrix can be reconstructed accurate within linear transformations of rows with respect to its minors of maximal order. In this case, the matching conditions (so called Plucker relations) must be satisfied for the minors. In two other theorems, on the basis of the first two theorems, we prove the duality of the reconstruction of boundary conditions. The third theorem is devoted to the identification of boundary conditions by the entire spectrum of eigenvalues, and the fourth is to identify boundary conditions with respect to a finite number of eigenvalues. It is shown that it is sufficient to use four eigenvalues to identify the boundary conditions. Examples of the identification problem’s solution are given.