When is a linear system conservative?

Abstract

Abstract. We consider infinite-dimensional linear systems without a-priori wellposedness assumptions, in a framework based on the works of M. Livšic, M. S. Brodskiȋ, Y. L. Smuljan, and others. We define the energy in the system as the norm of the state squared (other, possibly indefinite quadratic forms will also be considered). We derive a number of equivalent conditions for a linear system to be energy preserving and hence, in particular, well posed. Similarly, we derive equivalent conditions for a system to be conservative, which means that both the system and its dual are energy preserving. For systems whose control operator is one-to-one and whose observation operator has dense range, the equivalent conditions for being conservative become simpler, and reduce to three algebraic equations. Introduction.In this paper we give a number of algebraic characterizations of energy preserving and of conservative linear systems in terms of the operators appearing in a differential/algebraic state space description of the system. Such algebraic characterizations may be useful for recognizing that certain linear partial differential or delay equations describe energy preserving or conservative systems. Our conditions do not include an a-priori well-posedness assumption, but well-posedness follows if the system is energy preserving.

The Cayley Transform as a Time Discretization Scheme

“…System nodes are a functional analytic framework for presenting linear dynamical systems with possibly infinite-dimensional state spacesincluding boundary control systems defined by PDEs. System nodes are discussed in, e.g., Malinen, Staffans, and Weiss [25] but we review the construction below. 2 Let X be a Hilbert space and let A: dom(A) ⊂ X → X be a closed, densely defined linear operator with a nonempty resolvent set (A).…”

“…Much classic literature exists for them, see, e.g., Arov and Nudelman [2], Ball and Staffans [3], Brodskiȋ [5][6][7], Livšic [23], Livšic and Yantsevich [22], Sz.-Nagy and Foiaş [36], Smuljan [30], and Staffans [31][32][33]. Operator theory techniques for proving conservativity in applications are given in Malinen, Staffans, and Weiss [25] and Tucsnak and Weiss [37,39]. The special case of boundary control systems is further studied in Malinen [24] and Malinen and Staffans [26,27]; see also Gorbachuk and Gorbachuk [19] and the references therein.…”

Boundary Control Systems and the System Node

“…In order to give a proper definition of the system node we introduce the following proposition, see (Staffans, 2005, §3). Malinen et al, 2003)). Let X be a Hilbert space and let A : D(A) ⊂ X → X be a closed, densely defined linear operator with a nonempty resolvent set ρ(A).…”

“…Now it is possible to define the system node as follows Definition 2. ( (Malinen et al, 2003)). Let U , X, and Y be Hilbert spaces.…”

“…It is known (see (Malinen et al, 2003)) that if A is the generator of a C 0 -semigroup then S defines a linear dynamical system as follows…”