On the controllability of entropy solutions of scalar conservation laws at a junction via Lyapunov methods

created by denitti on 16 Nov 2021
modified on 21 Nov 2021

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Submitted Paper

Inserted: 16 nov 2021
Last Updated: 21 nov 2021

Year: 2021
Notes:

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 (in the exterior nodes of the star-shaped graph), then $u \equiv v$ 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.