Transfer functions of regular linear systems Part II: The system operator and the Lax–Phillips semigroup

Abstract

Abstract. This paper is a sequel to a paper by the second author on regular linear systems (1994), referred to here as "Part I". We introduce the system operator of a well-posed linear system, which for a finite-dimensional system described byẋ = Ax + Bu, y = Cx + Du would be the s-dependent matrixIn the general case, S Σ (s) is an unbounded operator, and we show that it can be split into four blocks, as in the finite-dimensional case, but the splitting is not unique (the upper row consists of the uniquely determined blocks A−sI and B, as in the finite-dimensional case, but the lower row is more problematic). For weakly regular systems (which are introduced and studied here), there exists a special splitting of S Σ (s) where the right lower block is the feedthrough operator of the system. Using S Σ (0), we give representation theorems which generalize those from Part I to well-posed linear systems and also to the situation when the "initial time" is −∞. We also introduce the Lax-Phillips semigroup T induced by a well-posed linear system, which is in fact an alternative representation of a system, used in scattering theory. Our concept of a Lax-Phillips semigroup differs in several respects from the classical one, for example, by allowing an index ω ∈ R which determines an exponential weight in the input and output spaces. This index allows us to characterize the spectrum of A and also the points where S Σ (s) is not invertible, in terms of the spectrum of the generator of T (for various values of ω). The system Σ is dissipative if and only if T (with index zero) is a contraction semigroup.

Spectral theory for structured perturbations of linear operators

“…(v) We note that the operator T in (2.8) shows similarities to the system operator S Σ (λ) studied in some detail in [17].…”

Infinite-dimensional Lur’e systems with almost periodic forcing

“…Below we will provide a brief review of some material from the theory of well-posed systems; for more details, we refer the reader to [39,42,43,[45][46][47]. Throughout, we shall be considering a well-posed linear system Σ = (T, Φ, Ψ , G) with state space X , input space U and output space Y .…”

“…Here X , U and Y are separable complex Hilbert spaces, T = (T t ) t≥0 is a strongly continuous semigroup on X , Φ = (Φ t ) t≥0 is a family of bounded linear operators from L 2 (R + , U ) to X (input-to-state maps), Ψ = (Ψ t ) t≥0 is a family of bounded linear operators from X to L 2 (R + , Y ) (state-to-output maps) and G = (G t ) t≥0 is a family of bounded linear operators from L 2 (R + , U ) to L 2 (R + , Y ) (input-to-output maps). In order for Σ to qualify as a well-posed system, these families of operators need to satisfy certain natural conditions, see [39,43,45,46]. Particular consequences of these conditions are the following properties:…”

“…We note that well-posed linear systems allow for considerable unboundedness of the control and observation operators and they encompass many of the most commonly studied partial differential equations with boundary control and observation, and a large class of functional differential equations of retarded and neutral type with delays in the inputs and outputs. There exists a highly developed state-space and frequencydomain theory for well-posed infinite-dimensional systems; see, for example, [29,30,39,40,42,43,[45][46][47].…”

Strong Stabilization of a Non-uniform SCOLE Model

“…Through some extensions of operators, wellposed systems can get the similar realization as the finitedimensional systems. For more detailed background about well-posed systems we refer to Salamon [1987], Staffans [2002], Staffans and Weiss [2002], Weiss [1994], Weiss et al [2001] and Weiss and Tucsnak [2003]. Necessary and sufficient conditions for well-posedness have been given in Curtain and Weiss [1989].…”