J-energy preserving well-posed linear systems

Olof Staffans

International Journal of Applied Mathematics and Computer Science (2001)

  • Volume: 11, Issue: 6, page 1361-1378
  • ISSN: 1641-876X

Abstract

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The following is a short survey of the notion of a well-posed linear system. We start by describing the most basic concepts, proceed to discuss dissipative and conservative systems, and finally introduce J-energy-preserving systems, i.e., systems that preserve energy with respect to some generalized inner products (possibly semi-definite or indefinite) in the input, state and output spaces. The class of well-posed linear systems contains most linear time-independent distributed parameter systems: internal or boundary control of PDE’s, integral equations, delay equations, etc. These systems have existed in an implicit form in the mathematics literature for a long time, and they are closely connected to the scattering theory by Lax and Phillips and to the model theory by Sz.-Nagy and Foiaş. The theory has been developed independently by many different schools, and it is only recently that these different approaches have begun to converge. One of the most interesting objects of the present study is the Riccati equation theory for this class of infinite-dimensional systems (H^2 - and H^∞ -theories).

How to cite

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Staffans, Olof. "J-energy preserving well-posed linear systems." International Journal of Applied Mathematics and Computer Science 11.6 (2001): 1361-1378. <http://eudml.org/doc/207559>.

@article{Staffans2001,
abstract = {The following is a short survey of the notion of a well-posed linear system. We start by describing the most basic concepts, proceed to discuss dissipative and conservative systems, and finally introduce J-energy-preserving systems, i.e., systems that preserve energy with respect to some generalized inner products (possibly semi-definite or indefinite) in the input, state and output spaces. The class of well-posed linear systems contains most linear time-independent distributed parameter systems: internal or boundary control of PDE’s, integral equations, delay equations, etc. These systems have existed in an implicit form in the mathematics literature for a long time, and they are closely connected to the scattering theory by Lax and Phillips and to the model theory by Sz.-Nagy and Foiaş. The theory has been developed independently by many different schools, and it is only recently that these different approaches have begun to converge. One of the most interesting objects of the present study is the Riccati equation theory for this class of infinite-dimensional systems (H^2 - and H^∞ -theories).},
author = {Staffans, Olof},
journal = {International Journal of Applied Mathematics and Computer Science},
keywords = {Lax-Phillips semigroup; dissipative system; system node; J-energy-preserving system; conservative system; well-posed linear system; Riccati equation; transfer function; model theory; Lyapunov equation; conservative realization; well posed linear systems; survey; conservative systems; indefinite metric},
language = {eng},
number = {6},
pages = {1361-1378},
title = {J-energy preserving well-posed linear systems},
url = {http://eudml.org/doc/207559},
volume = {11},
year = {2001},
}

TY - JOUR
AU - Staffans, Olof
TI - J-energy preserving well-posed linear systems
JO - International Journal of Applied Mathematics and Computer Science
PY - 2001
VL - 11
IS - 6
SP - 1361
EP - 1378
AB - The following is a short survey of the notion of a well-posed linear system. We start by describing the most basic concepts, proceed to discuss dissipative and conservative systems, and finally introduce J-energy-preserving systems, i.e., systems that preserve energy with respect to some generalized inner products (possibly semi-definite or indefinite) in the input, state and output spaces. The class of well-posed linear systems contains most linear time-independent distributed parameter systems: internal or boundary control of PDE’s, integral equations, delay equations, etc. These systems have existed in an implicit form in the mathematics literature for a long time, and they are closely connected to the scattering theory by Lax and Phillips and to the model theory by Sz.-Nagy and Foiaş. The theory has been developed independently by many different schools, and it is only recently that these different approaches have begun to converge. One of the most interesting objects of the present study is the Riccati equation theory for this class of infinite-dimensional systems (H^2 - and H^∞ -theories).
LA - eng
KW - Lax-Phillips semigroup; dissipative system; system node; J-energy-preserving system; conservative system; well-posed linear system; Riccati equation; transfer function; model theory; Lyapunov equation; conservative realization; well posed linear systems; survey; conservative systems; indefinite metric
UR - http://eudml.org/doc/207559
ER -

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