Conservative linear systems, unitary colligations and Lax–Phillips scattering: Multidimensional generalizations

Joseph A. Ball, Cora Sadosky, Victor Vinnikov

Research output: Contribution to journalArticlepeer-review

5 Scopus citations

Abstract

Connections between conservative linear systems, Lax–Phillips scattering, and operator model theory are well known. A common thread in all the theories is a contractive, analytic, operator-valued function on the unit disc T(z) having a representation of the form T(z) = D + zC (I − zA)−1 B, known as the transfer or frequency-response function in the system-theory community, the scattering function in the mathematical physics community, and the characteristic operator function in the operator theory community. Here we consider analogues of this circle of ideas in the more general setting of multidimensional systems/multi-evolution scattering systems/multivariable function-theoretic operator theory. Three particular extensions are discussed; from the point of view of system theory, these involve (1) a multidimensional linear system with transfer function a contractive analytic operator function on the unit polydisc in complex Euclidean space, (2) a non-commutative multidimensional linear system with evolution along a free semigroup and with transfer function equal to a formal power series in non-commuting indeterminants, and (3) an overdetermined multidimensional linear system with transfer function identified as a bundle mapping between two Hermitian vector bundles over an algebraic curve embedded in complex projective space. This survey updates an earlier survey by the first author appearing in 2000.

Original languageEnglish
Pages (from-to)802-811
Number of pages10
JournalInternational Journal of Control
Volume77
Issue number9
DOIs
StatePublished - 1 Jan 2004

ASJC Scopus subject areas

  • Control and Systems Engineering
  • Computer Science Applications

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