Exponential convergence towards consensus for non-symmetric linear first-order systems in finite and infinite dimensions

Laurent Boudin (1) , Francesco Salvarani (2, 3) , Emmanuel Trélat (1, 4)

1 LJLL (UMR_7598) - Laboratoire Jacques-Louis Lions
2 PULV - Pôle Universitaire Léonard de Vinci
3 Dipartimento di matematica F. Casorati
4 CaGE - Control And GEometry

We consider finite and infinite-dimensional first-order consensus systems with timeconstant interaction coefficients. For symmetric coefficients, convergence to consensus is classically established by proving, for instance, that the usual variance is an exponentially decreasing Lyapunov function. We investigate here the convergence to consensus in the non-symmetric case: we identify a positive weight which allows to define a weighted mean corresponding to the consensus, and obtain exponential convergence towards consensus. Moreover, we compute the sharp exponential decay rate.

Domaines: Equations aux dérivées partielles [math.AP]

Laurent Boudin, Francesco Salvarani, Emmanuel Trélat. Exponential convergence towards consensus for non-symmetric linear first-order systems in finite and infinite dimensions. SIAM Journal on Mathematical Analysis, 2022, 54 (3), pp.2727--2752. ⟨10.1137/21m1416102⟩. ⟨hal-03210969⟩