V. S. Deineka

Limiting Representations of Weighted Pseudoinverse Matrices with Positive Definite Weights. Problem Regularization

Limiting representations for weighted pseudoinverse matrices with positive definite weights are derived. It is shown that regularized problems can be constructed based on such limiting representations intended for evaluation of weighted pseudoinverse matrices and weighted normal pseudosolutions with positive definite weights. The results obtained, concerning regularization of problems on evaluation of weighted normal pseudosolutions, are employed for regularization of least-squares problems with constraints.

Expansions and Polynomial Limit Representations of Weighted Pseudoinverses

Expansions of weighted pseudoinverses with positive definite or singular weights in matrix power series or power products with negative exponents and arbitrary positive parameters are proposed and analyzed. Based on these expansions, polynomial limit representations of weighted pseudoinverses are obtained. Issues related to the construction of direct and iterative methods for calculating weighted pseudoinverses and weighted normal pseudosolutions, as well as solving constrained least squares problems, are examined.

Weighted pseudoinverses and weighted normal pseudosolutions with singular weights

Weighted pseudoinverses with singular weights can be defined by a system of matrix equations. For one of such definitions, necessary and sufficient conditions are given for the corresponding system to have a unique solution. Representations of the pseudoinverses in terms of the characteristic polynomials of symmetrizable and symmetric matrices, as well as their expansions in matrix power series or power products, are obtained. A relationship is found between the weighted pseudoinverses and the weighted normal pseudosolutions, and iterative methods for calculating both pseudoinverses and pseudosolutions are constructed. The properties of the weighted pseudoinverses with singular weights are shown to extend the corresponding properties of weighted pseudoinverses with positive definite weights.

Problems with conjugation conditions and their high-accuracy computational discretization algorithms

New classes of problems with discontinuous solutions are considered. The corresponding generalized problems are obtained. Numerical schemes, which are as asymptotically accurate as similar schemes for problems with smooth solutions, are proposed for the new classes of problems.

Optimal Control of an Elliptic-Parabolic System with Conjugation Conditions

New optimal control problems for distributed systems described by initial boundary-value problems are considered in the paper for an elliptic-parabolic equation with conjugation conditions and quadratic cost functions. Theorems of existence of a unique optimal control are proved for all the considered cases.

Optimal Control of a Conventionally Correct System with Conjugation Conditions

New optimal control problems are constructed and considered for distributed systems whose states are described by Neumann boundary-value problems with conjugation conditions and non-unique solutions. The paper proposes accurate higher-order computation schemes for discretization of optimization problems for the case where a control set coincides with a full Hilbert space.

Necessary and sufficient conditions for the existence of a weighted singular value decomposition of matrices with singular weights

UDC 512.61 A weighted singular-valued decomposition of matrices with singular weights is obtained by using orthogonal matrices. The necessary and sufficient conditions for the existence of the constructed weighted singular-valued decomposition are established. The indicated singular-valued decomposition of matrices is used to obtain a decomposition of their weighted pseudoinverse matrices and decompose them into matrix power series and products. The applications of these decompositions are discussed.

Optimal Control of a System Described by a Two-Dimensional Quartic Equation with Conjugation Conditions

New problems of optimal control of distributed systems described by partial differential two-dimensional quartic equations with conjugation conditions and quadratic cost functions are considered in the paper. The theorems of existence of unique optimal solutions are proved for every considered case.

Optimal Control of a System Described by Hyperbolic Equation with Conjugation Conditions

New problems of optimal control of distributed systems described by a hyperbolic equation with conjugation conditions and a quadratic cost function are considered in the paper. Unique optimal control existence theorems are proved for all the analyzed cases.

The Dirichlet and Neumann Problems for Elliptical Equations with Conjugation Conditions and High-Precision Algorithms of Their Discretization

New Dirichlet and Neumann boundary-value problems for elliptic equations with conjugation conditions are considered and existence and uniqueness of their solutions are studied. High-precision discretization algorithms are constucted based on the classes of discontinuous admissible functions.