Control and stabilization of 2 × 2 hyperbolic systems on graphs

Abstract

We consider 2 × 2 (first order) hyperbolic systems on networks subject to general transmission conditions and to some dissipative boundary conditions on some external vertices. We find sufficient but natural conditions on these transmission conditions that guarantee the exponential decay of the full system on graphs with dissipative conditions at all except one external vertices. This result is obtained with the help of a perturbation argument and an observability estimate for an associated wave type equation. An exact controllability result is also deduced.

Linear hyperbolic systems on networks: well-posedness and qualitative properties

“…on a real interval (0, ) generalizes the first order reduction of the wave equation and offers a general framework to treat models that appear in several applications. The analysis of this system on networks with different boundary conditions has been performed in [56].…”

“…One may e.g. consider the vertex transmission conditions (see [56], p. 56) -p is continuous across the vertices and e∈Ev q e (v)ι ve = 0 for all v ∈ V; or -q is continuous across the vertices and e∈Ev p e (v)ι ve = 0 for all v ∈ V.…”

“…In this paper, we opt for yet another parametrization of the boundary conditions, inspired by a classical Sturm-Liouville formalism borrowed by Kuchment to discuss self-adjoint extensions of Laplacians on networks in [43] (see also [56] for the "telegrapher's equation" on networks with similar boundary conditions). More precisely, we impose boundary conditions of the form…”

Global solutions for a chemotaxis hyperbolic-parabolic system on networks with nonhomogeneous boundary conditions

“…Results about hyperbolic models on networks can be found in [6,7,23,19,22], with different kinds of transmission conditions; moreover parabolic chemotaxis models on networks were studied in [1,5,17], with continuity conditions at the nodes.…”

Boundary controllability of the Korteweg-de Vries equation on a tree-shaped network

“…For this particular kind of coupled systems we already find some boundary controllability results. In fact, we can mention [7] for parabolic systems and [4,14,15,18,21,25] for hyperbolic systems.…”