N. Nitti, De Nitti N., Zuazua E.. On the Controllability of Entropy Solutions of Scalar Conservation Laws at a Junction via Lyapunov methods (2023) Vietnam J. Math. https://doi.org/10.1007/s10013-022-00598-9
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.