Abstract
We show under mild assumptions that a composition of internally well-posed, impedance passive (or conservative) boundary control systems through Kirchhoff type connections is also an internally well-posed, impedance passive (resp., conservative) boundary control system. The proof is based on results of Malinen and Staffans [21]. We also present an example of such composition involving Webster's equation on a Y-shaped graph.
Highlights
We treat the solvability of dynamical boundary control systems that are composed by interconnecting a finite number of more simple boundary control subsystems that are already known to be solvable forward in time
In this work we treat linear boundary control systems described by operator differential equations of the form (5) involving linear mappings G, L, and K: Definition 2.1
(i) Ξ is a colligation on the Hilbert spaces (U, X, Y) if G, L, and K have the same domain Z = dom(Ξ) ⊂ X and values in U, X, and Y, respectively; (ii) A colligation Ξ is strong if is closed as an operator X →
Summary
We treat the solvability (forward in time) of dynamical boundary control systems that are composed by interconnecting a finite number of more simple boundary control subsystems that are already known to be solvable forward in time. Defining operator G by (4), that is, by Gz(t) = u(j)(t), and K in a similar manner, we obtain an internally well-posed boundary node Ξ(j) = (G, L, K) that is impedance conservative, see Definitions 2.2 and 2.3. We have boundary nodes Ξ(j), j = A, ..., D and coupling conditions of the form (2) for all vertices except the one that defines the external input and output through (3). (ii) If Ξ is impedance passive, it is an internally well-posed boundary node if and only if its input operator G is surjective.
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