Passive state/signal systems and conservative boundary relations (Q2849107)

Given a Hilbert space \(\mathcal{X}\) (state space) and a Krein space \(\mathcal{W}\) (signal space), let the space \( \mathcal{K}= \mathcal{X} \times \mathcal{W}\times\mathcal{W}\) be equipped with the indefinite inner product induced by the quadratic form NEWLINE\[NEWLINE [f_{1}, f_{2}]_{\mathcal{K}} = -(z_{1}, x_{2})_{ \mathcal{X} } - (x_{1}, z_{2})_{ \mathcal{X} } + [w_{1}, w_{2}]_{\mathcal{W}} \qquad ( f_{i}=[z_{i} , x_{i}, w_{i}]\in \mathcal{K} ; i=1, 2 ) . NEWLINE\]NEWLINE A pair \([x, w]\) is called a classical trajectory generated by a closed subspace \(V \subseteq \mathcal{K}\) on \(\mathbb{R^{+}}\) if \(x\in C^{1}(\mathbb{R^{+}}, \mathcal{X})\), \(w \in C(\mathbb{R^{+}}, \mathcal{W})\) and \([\dot{x}(t), x(t), w(t)]\in V\), \(t \in \mathbb{R^{+}}\). If, for any \([z_{0}, x_{0}, w_{0}]\in V\), there exists such a classical trajectory \([x, w]\), generated by \(V\) on \(\mathbb{R^{+}}\) that \([\dot{x}(0), x(0), w(0)] = [z_{0}, x_{0}, w_{0}]\), and \(V\) is a graph of an (closed) operator with domain in \(\mathcal{X} \times \mathcal{W}\) and image in \(\mathcal{X}\), then such a triple \(\Sigma =(V, \mathcal{X}, \mathcal{W})\) is called a \textit{state/signal node} (s/s node). The corresponding \textit{s/s system} is the node \(\Sigma\) together with its set of classical trajectories and the closure of this set in \([C(\mathbb{R^{+}}, \mathcal{X}) ; L^{2}_{\mathrm{loc}}(\mathbb{R^{+}}, \mathcal{W}) ]\). Such an s/s system is said to be \textit{passive} if \(V\) is a maximal non-negative subspace of \(\mathcal{K}\); the s/s system is \textit{conservative} if \(V = V^{[\bot]}\). The authors consider different \textit{input/state/output} representations of s/s systems via different direct-sum decompositions of signal spaces \(W = W_{1} \dot{+} W_{2}\), in particular, fundamental and Lagrangian decompositions.NEWLINENEWLINEFrom the abstract: ``The main aim is to clarify the basic connections between the state/signal theory and that of (conservative) boundary relations. It is described how one can represent a state/signal system using input/state/output systems in different ways by making different choices of input signal and output signal. There is an `almost one-to-one' relationship between conservative state/signal systems and boundary relations, and this connection is used in order to introduce dynamics to a boundary relation. Consequently, a boundary relation is such a general object that it mathematically has rather little to do with boundary control. The Weyl family and \(\gamma\)-field of a boundary relation are connected to the frequency-domain characteristics of a state/signal system.''NEWLINENEWLINEFor the entire collection see [Zbl 1269.47001].