Projection Methods For Problems In Control Theory And Computation

The present Ph.D. dissertation is concerned with projection methods for large-scale problems in control theory and computation. We propose Galerkin method and minimal residual method for iteratively solving generalized Sylvester equations. The algorithms use Krylov subspace for which orthogonal basis are generated by the Arnoldi procedure and reduce the storage space required by using the structure of the matrix. We propose a new projection method based on global Arnoldi procedure for solving large Sylvester equations and the large generalized Sylvester equations. For solving large-scale quadratic eigenvalue problem (QEP), we first introduce a block second-order Krylov subspace based on a pair of square matrices A₁ and A₂ and an orthonormal matrix Q₁, then we present a block second-order Arnoldi procedure for generating an orthonormal basis of the space and a block second-order biorthogonalization procedure for generating biorthonormal basis. By applying the projection techniques, we derive two block second-order Krylov subspace methods. These methods are applied to the QEP directly. Hence they preserve essential structures and properties of the QEP. We present a structure-preserving model-order reduction method for solving large-scale second-order multi-input multi-output dynamical systems. It is a projection method based on a block second-order Krylov subspace. We use the block second-order Arnoldi procedure to generate an orthonormal basis of the projection subspace. The reduced system preserves the second-order structure of the original system. Finally, we propose a modified simple iterative method and a modified Newton method for nonsymmetric algebraic Riccati equations arising in transport theory.