ON THE COVER PROBLEMS OF GEOMETRIC-THEORY

For linear systems, a family of cover problems of the geometric theory are introduced as extensions of the standard cover problem and a matrix pencil formulation of such problems is given. It is shown that the solvability of such problems is reduced to a problem of Kronecker Invariant Transformation by Matrix Pencil Augmentation and a Matrix Pencil Realisation Problem. Necessary, as well as sufficient conditions for solvability of both problems are given, which lead to a number of conditions for solvability of the partial, as well as standard cover problem. The special cases of left regular, regular solutions of the cover problem are investigated and a parametrisation of such families of solutions is given.