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In this paper, some necessary and sufficient solvability conditions are established for the generalized Sylvester operator equations AXB-CXD = E, AXB-CYD = E and AX + YB+CZ = E on Hilbert spaces, respectively. Moreover, we give a solvability condition for the *-Sylvester operator equation AX -X*B = C, which holds for finite matrices due to Wimmer (1994). [ABSTRACT FROM AUTHOR]

In this paper, we generalize an existing technique which is relevant to a pole placement problem and state feedback control design with constraints on control inputs. The existing technique concerns only a restrictive class of systems for which many assumptions should be satisfied. The generalization aims then at avoiding those assumptions. The advantages of generalizing this technique are to determine linear feedback controllers that asymptotically stabilize a wide class of systems which are subject to input constraints without violating these constraints, to derive a simple formula for the state feedback and to assign the whole spectrum when stable poles are not suitably located for performances. Moreover, there will be no need to assume a lack of overlap between open and closed-loop spectra. Illustrative examples are presented to point out these advantages. [ABSTRACT FROM AUTHOR]

The solutions of Lyapunov and generalized Lyapunov equations are a key player in many applications in systems and control theory. Their stable numerical computation, when the full solution is sought, is considered solved since the seminal work of Bartels and Stewart [1972] and its generalization by Penzl [1997]. Those variants do not go completely beyond BLAS level-2 style implementation. On modern computers, however, the formulation of level-3 BLAS type implementations is crucial to enable optimal usage of cache hierarchies and modern block scheduling methods based on directed acyclic graphs describing the interdependence of single block computations. Although there exists a recursive blocked solution scheme for (quasi-) triangular generalized Lyapunov equations [Jonsson and Kågström 2002b], we focus on standard blocking techniques. Using the standard blocking approach our contribution lifts the aforementioned level-2 algorithm by Penzl to BLAS level-3 for (quasi-) triangular equations. Especially, we consider the solution of the appearing Sylvester equations and provide a hybrid algorithm merging our strategy with the recursive blocking method above. [ABSTRACT FROM AUTHOR]

This work concerns the distance in the 2-norm from a given matrix polynomial to a nearest polynomial with a specified number of its eigenvalues at specified locations in the complex plane. Initially, we consider perturbations of the constant coefficient matrix only. A singular value optimization characterization is derived for the associated distance. We also consider the distance in the general setting, when all of the coefficient matrices are perturbed. In this general setting, we obtain a lower bound in terms of another singular value optimization problem. The singular value optimization problems derived facilitate the numerical computation of the distances. [ABSTRACT FROM AUTHOR]

Abstract: We investigate the factored alternating directions implicit (ADI) iteration for large and sparse Sylvester equations. A novel low-rank expression for the associated Sylvester residual is established which enables cheap computations of the residual norm along the iteration, and which yields a reformulated factored ADI iteration. The application to generalized Sylvester equations is considered as well. We also discuss the efficient handling of complex shift parameters and reveal interconnections between the ADI iterates w.r.t. those complex shifts. This yields a further modification of the factored ADI iteration which employs only an absolutely necessary amount of complex arithmetic operations and storage, and which produces low-rank solution factors consisting of entirely real data. Certain linear matrix equations, such as, e.g., cross Gramian Sylvester and Stein equations, are in fact special cases of generalized Sylvester equations and we show how specially tailored low-rank ADI iterations can be deduced from the generalized factored ADI iteration. [Copyright &y& Elsevier]

Abstract: In this paper, we give some necessary and sufficient solvability conditions for the mixed Sylvester matrix equations, and parameterize general solution when it is solvable. Moreover, we investigate the maximal and minimal ranks of the general solution, and maximal and minimal ranks and inertias of Hermitian part of solution, respectively. [Copyright &y& Elsevier]

This issue contains papers dealing with a variety of linear and nonlinear matrix equations arising in systems and control theory, model reduction, and in various other areas of applied mathematics, economics, engineering, and the sciences. Special emphasis is given to the numerical treatment of large-scale problems. Copyright © 2008 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

The distance to uncontrollability for a linear control system is the distance (in the 2-norm) to the nearest uncontrollable system. We present an algorithm based on methods of Gu and Burke-Lewis-Overton that estimates the distance to uncontrollability to any prescribed accuracy. The new method requires O(n4) operations on average, which is an improvement over previous methods which have complexity O(n6), where n is the order of the system. Numerical experiments indicate that the new method is reliable in practice. [ABSTRACT FROM AUTHOR]

Abstract: The solution of a generalized Sylvester equation associated to a linear descriptor system and subject to some rank and regional pole-placement constraints, is studied in this paper. Under the hypothesis of strong-detectability of the descriptor system, a sequence of coordinate transformations is proposed such that the considered problem can be solved through a Sylvester equation associated to a detectable reduced-order normal system. Thus, solutions to this equation can be found either by eigenstructure assignment or by regional pole-placement in LMI-type regions of the complex plane. The proposed results are motivated from its use for designing minimal-order observers and for computing output feedback control laws by a particular technique. Some numerical results are reported and possible extensions for the proposed approach are commented. [Copyright &y& Elsevier]

The design of a Luenberger observer for large control systems is an important problem in Control Theory. Recently, several computational methods have been proposed by Datta and collaborators. The present paper discusses numerical aspects of one of these methods, described by Datta and Saad (1991). Copyright © 2001 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]