On the Controllability of Entropy Solutions of Scalar Conservation Laws at a Junction via Lyapunov Methods

Abstract

In this note, we prove a controllability result for entropy solutions of scalar conservation laws on a star-shaped graph. Using a Lyapunov-type approach, we show that, under a monotonicity assumption on the flux, if u and v are two entropy solutions corresponding to different initial data and same in-flux boundary data (at the exterior nodes of the star-shaped graph), then u ≡ v for a sufficiently large time. In order words, we can drive u to the target profile v in a sufficiently large control time by inputting the trace of v at the exterior nodes as in-flux boundary data for u. This result can also be shown to hold on tree-shaped networks by an inductive argument. We illustrate the result with some numerical simulations.

Control of hyperbolic and parabolic equations on networks and singular limits

“…For the well-posedness, controllability, and stabilization of several hyperbolic problems, we also point to [50,49,48,51,6] and references therein (for models of thermoelastic beams, linked plates, plate-beam systems, etc.). Moreover, we refer to [9,Chapter 1.15], [43,45,71,31,42,44] for models arising in water flow, gas transport, etc., and to [30] for the controllability of scalar conservation laws on networks in the context of entropy solutions.…”