Realization Theorems for Operator-Valued R-Functions

Belyi, Sergey; Tsekanovskiı̆, Eduard

doi:10.1007/978-3-0348-8910-0_5

Cited by 17 publications

(76 citation statements)

References 9 publications

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“…[8], [16]. A classical result due to Brodskiȋ and Livšic [9] states that the compactly supported Herglotz-Nevanlinna functions of the form b a dΣ(t)/(t − z) correspond to minimal systems Θ of the form (7) via (4) with W (z) = W Θ (z) given by (5) and V (z) = V Θ (z) given by (6). Next consider a linear, stationary, conservative dynamical system Θ of the form…”

Section: Special Realization Problemsmentioning

confidence: 99%

Problem 1.2 The realization problem for Herglotz-Nevanlinna functions

Hassi¹,

Snoo²,

Tsekanovskĭ³

2009

Unsolved Problems in Mathematical Systems and Control Theory

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MOTIVATION AND HISTORY OF THE PROBLEMRoughly speaking, realization theory concerns itself with identifying a given holomorphic function as the transfer function of a system or as its linear fractional transformation. Linear, conservative, time-invariant systems whose main operator is bounded have been investigated thoroughly. However, many realizations in different areas of mathematics including system theory, electrical engineering, and scattering theory involve unbounded main operators, and a complete theory is still lacking. The aim of the present proposal is to outline the necessary steps needed to obtain a general realization theory along the lines of M. S. Brodskiȋ and M. S. Livšic [8], [9], [16], who have

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Section: Special Realization Problemsmentioning

confidence: 99%

Problem 1.2 The realization problem for Herglotz-Nevanlinna functions

Hassi¹,

Snoo²,

Tsekanovskĭ³

2009

Unsolved Problems in Mathematical Systems and Control Theory

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“…The proper definition of "unbounded node" or "unbounded colligation" originates in the work of Salamon [32] and continues with the work of Weiss and Staffans (see [35] for a full account). For the more structured case where the transfer function has values which are contractive or with positive real part on a half-plane, while there has been some work on "unbounded Livšic nodes" (see [7,8,9,19]), the version closest to what we use here originates in the work ofŠmuljan [33] as later codified and refined by Arov and Nudelman [4]; the latter obtained the analogue of (1.7) without the assumption that S is analytic with value I U at infinity. Further results along this line (including extensions of (1.8) to the case where ϕ need not be analytic in a neighborhood of infinity) were obtained in [34].…”

Section: If S(z) Is An L(u Y)-valued Function Analytic On the Unit Dmentioning

confidence: 99%

“…In the continuous-time setting, the approach through Lagrangian subspaces (see Section 4 below) appears to yield cleaner results than the approach through integral representations (see [7,8,9,19]). …”

Section: Conservative Realizations 175mentioning

confidence: 99%

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Conservative State-Space Realizations of Dissipative System Behaviors

Ball

Staffans

2005

Integr. equ. oper. theory

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It is well known that a Schur-class function S (contractive operatorvalued function on the unit disk) can be realized as the transfer functionC D ] unitary). One method of proof of this result (the "lurking isometry" method) identifies a solution U of the problem as a unitary extension of a partially defined isometry V determined by the problem data. Reformulated in terms of the graphs of V and U , solutions are identified with embeddings of an isotropic subspace of a certain Kreȋn space K constructed from the problem data into a Lagrangian subspace (maximal isotropic subspace of K). The contribution here is the observation that this reformulation applies to other types of realization problems as well, e.g., realization of positive-real or J-contractive operatorvalued functions over the unit disk (respectively over the right half plane) as the transfer function of a discrete-time (respectively, continuous-time) conservative system, i.e., an input-state-output system for which there is a quadratic storage function on the state space for which all system trajectories satisfy an energy-balance equation with respect to the appropriate supply rate on input-output pairs. The approach allows for unbounded state dynamics, unbounded input/output operators and descriptor-type state-space representations where needed in a systematic way. These results complement recent results of Arov-Nudelman, Hassi-de Snoo-Tsekanovskiȋ, Belyi-Tsekanovskiȋ and Staffans and fit into the behavioral frameworks of Trentelman-Willems and Georgiou-Smith. Mathematics Subject Classification (2000). Primary: 93B28; Secondary: 47A20, 47A48, 93C20.

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“…where A R is the real part of A (see, for example, [2]). The significance of rigged Hilbert spaces in system theory and natural appearing of systems (1.1), (1.3) in electrical engineering and the scattering theory can be seen in [23].…”

Section: Introductionmentioning

confidence: 99%

Regular Couplings of Dissipative and Anti-Dissipative Unbounded Operators, Asymptotics of the Corresponding Non-Dissipative Processes and the Scattering Theory

Kirchev

Borisova

2006

Integr. equ. oper. theory

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In this paper a triangular model of a class of unbounded nonselfadjoint K r -operators A presented as a coupling of dissipative and antidissipative operators in a Hilbert space with real absolutely continuous spectra and with different domains of A and A * is considered. The asymptotic behaviour of the corresponding non-dissipative processes Ttf = e itA f , generated from the semigroups Tt with generators iA, as t → ±∞ are obtained. The strong wave operators, the scattering operator for the couple (A * , A) and the similarity of A and the operator of multiplication by the independent variable are obtained explicitly. The considerations are based on the triangular models and characteristic functions of A. Kuzhel for unbounded operators and the limit values of the multiplicative integrals, describing the characteristic function of the considered model. Mathematics Subject Classification (2000). Primary 47A48; Secondary 60G12.

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Realization Theorems for Operator-Valued R-Functions

Cited by 17 publications

References 9 publications

Problem 1.2 The realization problem for Herglotz-Nevanlinna functions

Problem 1.2 The realization problem for Herglotz-Nevanlinna functions

Conservative State-Space Realizations of Dissipative System Behaviors

Regular Couplings of Dissipative and Anti-Dissipative Unbounded Operators, Asymptotics of the Corresponding Non-Dissipative Processes and the Scattering Theory

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