163 research outputs found
Generalized solutions of Riccati equalities and inequalities
The Riccati inequality and equality are studied for infinite dimensional
linear discrete time stationary systems with respect to the scattering supply
rate. The results obtained are an addition to and based on our earlier work on
the Kalman-Yakubovich-Popov inequality in [6]. The main theorems are closely
related to the results of Yu. M. Arlinski\u{\i} in [3]. The main difference is
that we do not assume the original system to be a passive scattering system,
and we allow the solutions of the Riccati inequality and equality to satisfy
weaker conditions.Comment: Published in Methods of Functional Analysis and Topology (MFAT),
available at http://mfat.imath.kiev.ua/article/?id=84
Szegö-Kac-Achiezer formulas in terms of realizations of the symbol
AbstractFor rational and analytic matrix functions new formulas are obtained for the limits in the Szegö-Kac-Achiezer limit theorems. In the rational case the new expressions are given in terms of finite matrices which come from special representations of the matrix functions. These representations are known as realizations in mathematical systems theory
A time-varying generalization of the canonical factorization theorem for Toeplitz operators
AbstractThe canonical factorization theorem for the symbol of a Toeplitz operator is generalized to a class of non-Toeplitz operators. The operators in this class may be described as input-output operators of time-varying linear systems. Dichotomy of difference equations plays an important role
Column reduced rational matrix functions with given null-pole data in the complex plane
AbstractExplicit formulas are given for rational matrix functions which have a prescribed null-pole structure in the complex plane and are column reduced at infinity. A full parametrization of such functions is obtained. The results are specified and developed further for matrix polynomials
Minimal representations of semiseparable kernels and systems with separable boundary conditions
AbstractThe simplest representations of triangular parts of finite rank kernels are analysed. Criteria for uniqueness up to similarity are given. The results are applied to the problem of minimal realization of systems with separable boundary conditions
Discrete skew selfadjoint canonical systems and the isotropic Heisenberg magnet model
A discrete analog of a skew selfadjoint canonical (Zakharov-Shabat or AKNS)
system with a pseudo-exponential potential is introduced. For the corresponding
Weyl function the direct and inverse problem are solved explicitly in terms of
three parameter matrices. As an application explicit solutions are obtained for
the discrete integrable nonlinear equation corresponding to the isotropic
Heisenberg magnet model. State space techniques from mathematical system theory
play an important role in the proofs
Semiseparable integral operators and explicit solution of an inverse problem for the skew-self-adjoint Dirac-type system
Inverse problem to recover the skew-self-adjoint Dirac-type system from the
generalized Weyl matrix function is treated in the paper. Sufficient conditions
under which the unique solution of the inverse problem exists, are formulated
in terms of the Weyl function and a procedure to solve the inverse problem is
given. The case of the generalized Weyl functions of the form
, where is a strictly proper rational
matrix function and is a diagonal matrix, is treated in greater
detail. Explicit formulas for the inversion of the corresponding semiseparable
integral operators and recovery of the Dirac-type system are obtained for this
case
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