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

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 by x = Ax + Bu, y = Cx + Du would be the s-dependent matrix S Σ (s) = [A-Si/C B D ]. In 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? (for various values of ω). The system Σ is dissipative if and only if? (with index zero) is a contraction semigroup.