Theory and computations of partial eigenvalue and eigenstructure assignment problems in matrix second-order and distributed-parameter systems

This dissertation is devoted to the study of the two feedback control problems in the matrix second-order and distributed-parameter systems: the partial eigenvalue assignment problem and the partial eigenstructure assignment problem. Contributions are made to both the theory and computations of these problems. The existence and uniqueness results for both the problems in the matrix second-order case and for the partial eigenvalue assignment problem in the distributed-parameter case are derived. New results on orthogonality relations between the eigenvectors (eigenfunctions) of the quadratic matrix (operator) pencil are proved. Computational contributions include development of a novel “direct and partial modal” approach for the solution of these problems. The approach is direct because each problem is solved in its own formulation. That is, the problem given in a matrix second-order setting is solved without reformulation to a first-order form. Similarly, the problem when formulated in its own natural distributed-parameter setting is solved without discretization to a reduced-order second-order model. The approach is partial modal in the sense that it requires only partial knowledge of eigenvalues and eigenvectors (eigenfunctions). The latter makes the approach completely viable for practical applications because the state-of-the-art techniques are capable of computing only a small part of the spectra of the associated quadratic pencil. This “partial modal” aspect of our solutions is especially remarkable for the distributed-parameter systems since in this case the solution of an infinite dimensional operator problem is obtained by solving a small finite dimensional linear algebraic system. The results on numerical experiments on some real-life examples are given.