Finding the jump rate for fastest decay in the Goldstein-Taylor model

Helge Dietert (1, 2) , Josephine Evans (3)

1 IMJ-PRG (UMR_7586) - Institut de Mathématiques de Jussieu - Paris Rive Gauche
2 Leipzig University / Universität Leipzig
3 WMI - Warwick Mathematics Institute

For hypocoercive linear kinetic equations we first formulate an optimisation problem on a spatially dependent jump rate in order to find the fastest decay rate of perturbations. In the Goldstein-Taylor model we show (i) that for a locally optimal jump rate the spectral gap is determined by multiple, possible degenerate, eigenvectors and (ii) that globally the fastest decay is obtained with a spatially homogeneous jump rate. Our proofs rely on a connection to damped wave equations and a relationship to the spectral theory of Schrödinger operators.

Mots clés: wave equation; optimal control; Hypocoercivity; spatial weight; optimal design of damping set; Goldstein-Taylor model

Domaines: Equations aux dérivées partielles [math.AP]; Optimisation et contrôle [math.OC]

Helge Dietert, Josephine Evans. Finding the jump rate for fastest decay in the Goldstein-Taylor model. 2021. ⟨hal-03171498v2⟩