Additive inverse eigenvalue problems and pole placement of linear systems

This dissertation studies the additive inverse eigenvalue problem and its applications in linear systems. Lie perturbations and linear perturbations are studied in detail.; For Lie perturbations, we have proved that the inverse eigenvalue problem can be solved for any matrix {dollar}A{dollar} and any eigenvalues by using perturbations belonging to a Lie algebra {dollar}{lcub}cal L{rcub}{dollar} iff rank {dollar}{lcub}cal L{rcub}{dollar} {dollar}geq{dollar} {dollar}n{dollar} and {dollar}{lcub}cal L{rcub}{dollar} has at least one element which has distinct eigenvalues.; For linear perturbations, we have proved that the inverse eigenvalue problem can be solved for almost all matrices and almost all eigenvalues by using perturbations belonging to a linear subspace {dollar}{lcub}cal L{rcub}{dollar} of gl({dollar}n{dollar}) iff dim {dollar}{lcub}cal L{rcub}{dollar} {dollar}geq{dollar} {dollar}n{dollar} and {dollar}{lcub}cal L{rcub}{dollar} {dollar}notsubset{dollar} sl({dollar}n{dollar}). We also give several criteria to determine whether the problem can be solved for almost all eigenvalues for specific {dollar}A{dollar} and {dollar}{lcub}cal L{rcub}{dollar}.; The output feedback Lie algebra {dollar}{lcub}BKC{rcub}{dollar} and its Cartan subalgebra are then investigated and several applications in linear systems are given.; Using methods from algebraic geometry the central projection model of the pole placement map for output feedback is studied. When the system is nondegenerate, the pole placement map is a finite morphism, so it is finite to one and its image is a closed irreducible variety of dimension mp. It is noted that a system is degenerate iff some minors of its transfer function become zero after changing coordinates and applying output feedback. A simple proof of the branch point theorem of multivariable root locus is also given.; When mp = {dollar}n{dollar}, a necessary condition for the pole placement map {dollar}chi{dollar} to be almost onto is that {dollar}G(s){dollar} is a nondegenerate rational normal curve of degree {dollar}n{dollar} in some projective {dollar}n{dollar}-space, or equivalently, that dim {dollar}E = {lcub}m+pchoose p{rcub} - n - 2,{dollar} where {dollar}E{dollar} is the centre of {dollar}chi{dollar}. This condition is not sufficient; however, for 2 {dollar}times{dollar} 2 and 2 {dollar}times{dollar} 3 systems, {dollar}chi{dollar} is almost onto iff dim {dollar}E = {lcub}m+pchoose p{rcub} - n - 2.{dollar} Generally, {dollar}chi{dollar} is almost onto iff dim {dollar}E = {lcub}m+pchoose p{rcub} - n - 2{dollar} and Grass{dollar}(p, m + p){dollar} is not contained in the hyperplane {dollar}sigma{dollar}({dollar}E{dollar}) of Grass{dollar}(n + 1,N + 1){dollar} under the imbedding {dollar}i{dollar} induced by the mapping which sends each point of Grass{dollar}(p, m + p){dollar} to the tangent space of Grass{dollar}(p, m + p){dollar} at that point.