Compositions of passive boundary control systems (Q1941663)

The subject of this paper is best described by an example from acoustics given by the authors. The basic entity is a graph of interconnected segments. Propagation on each edge (wave guide) is governed by the linear wave equation \[ {\partial^2 \psi \over \partial t^2} = c^2 {\partial^2 \psi \over \partial x^2} \, . \] At the vertices of the graph, there are two linear boundary conditions which involve the solution at the wave guides making up the vertex; one states the equality of \(\partial \psi / \partial t\) for all these solutions at the vertex, the other equates to zero a linear combination of \(\partial \psi / \partial x\) at the vertex. In the acoustic application in mind, \(\psi\) is a velocity potential, \(\partial \psi / \partial t\) is (a multiple of) the perturbation pressure and \(\partial \psi / \partial x\) is the perturbation velocity. The input or control \(u(t)\) is applied at a vertex where only two wave guides meet; \(u(t)\) is the common value of \(\partial \psi / \partial t.\) Finally, the output is a linear combination of \(\partial \psi / \partial x\) for the two waveguides. The objective is to show that, under adequate assumptions, the dynamical system \(u \to y\) (which is a boundary control system in a generalized sense) is well posed and conservative. The authors solve the problem by means of an operator setup that encompasses this and other examples, such as a model of the acoustics of the human vocal tract.