Well-posed systems—The LTI case and beyond

Abstract

International audienceThis survey is an introduction to well-posed linear time-invartiant (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

The State Feedback Regulator Problem for Regular Linear Systems

“…Regular systems model many physical systems involving waves, beams, plates, shells, elastic media, heat propagation, etc, see [3], [8]- [10], [20]- [22], [44], [47], and they usually have unbounded control and observation operators. However, in the literature on the regulator problem, in order to avoid technical difficulties, it is usually assumed that these operators are bounded.…”

Stability Properties of Coupled Impedance Passive LTI Systems

“…holds when z is a classical solution of (2.1) and y is the corresponding output function, see for instance [32], [33] or [41,Section 6]. If P = I, then we say that the system is impedance passive instead of impedance I-passive.…”

“…These operators have to satisfy certain natural functional equations, for the formal definition we refer to Salamon [31] or Staffans [34], [35] or Weiss [47] or Tucsnak and Weiss [41]. We recall some facts about well-posed linear systems following [47].…”

“…It is well known that any impedance passive system has a positive transfer function P, which means that P is analytic on C 0 and Re P(s) ≥ 0 for all s ∈ C 0 , see for instance [32]- [34], [41]. Positive transfer functions were introduced in Brune [9] (based on his PhD thesis).…”

“…A more detailed introduction and relevant references can be found in Staffans [34] and in Tucsnak and Weiss [40], [41].…”