TY - JOUR
T1 - Well-posed systems - The LTI case and beyond
AU - Tucsnak, Marius
AU - Weiss, George
N1 - Funding Information:
This work was supported mainly by the Lorraine Region via a grant “Chercheur d’excellence”. We also acknowledge the support of the French National Research Agency (ANR) via the grant 11-BS03-0002 HAMECMOPSYS . The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Editor John Baillieul.
PY - 2014/7
Y1 - 2014/7
N2 - This survey is an introduction to well-posed linear time-invariant (LTI) systems for non-specialists. We recall the more general concept of a system node, classical and generalized solutions of system equations, criteria for well-posedness, the subclass of regular linear systems, some of the available linear feedback theory. Motivated by physical examples, we recall the concepts of impedance passive and scattering passive systems, conservative systems and systems with a special structure that belong to these classes. We illustrate this theory by examples of systems governed by heat and wave equations. We develop local and global well-posedness results for LTI systems with nonlinear (in particular, bilinear) feedback, by extracting the abstract idea behind various proofs in the literature. We apply these abstract results to derive well-posedness results for the Burgers and Navier-Stokes equations.
AB - This survey is an introduction to well-posed linear time-invariant (LTI) systems for non-specialists. We recall the more general concept of a system node, classical and generalized solutions of system equations, criteria for well-posedness, the subclass of regular linear systems, some of the available linear feedback theory. Motivated by physical examples, we recall the concepts of impedance passive and scattering passive systems, conservative systems and systems with a special structure that belong to these classes. We illustrate this theory by examples of systems governed by heat and wave equations. We develop local and global well-posedness results for LTI systems with nonlinear (in particular, bilinear) feedback, by extracting the abstract idea behind various proofs in the literature. We apply these abstract results to derive well-posedness results for the Burgers and Navier-Stokes equations.
KW - Burgers equation
KW - Heat equation
KW - Impedance passive system
KW - Local well-posedness
KW - Navier-Stokes equations
KW - Non-linear feedback
KW - Operator semigroup
KW - Regular linear system
KW - Scattering conservative system
KW - Scattering passive system
KW - Wave equation
KW - Well-posed linear system
UR - http://www.scopus.com/inward/record.url?scp=84904720061&partnerID=8YFLogxK
U2 - 10.1016/j.automatica.2014.04.016
DO - 10.1016/j.automatica.2014.04.016
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AN - SCOPUS:84904720061
SN - 0005-1098
VL - 50
SP - 1757
EP - 1779
JO - Automatica
JF - Automatica
IS - 7
ER -