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Border Avoidance: Necessary Regularity for Coefficients and Viscosity Approach

Dan Goreac 1 
1 PS
LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées
Abstract : Motivated by the result of invariance of regular-boundary open sets in \cite{CannarsaDaPratoFrankowska2009} and multi-stability issues in gene networks, our paper focuses on three closely related aims. First, we give a necessary local Lipschitz-like condition in order to expect invariance of open sets (for deterministic systems). Comments on optimality are provided via examples. Second, we provide a border avoidance (near-viability) counterpart of \cite{CannarsaDaPratoFrankowska2009} for controlled Brownian diffusions and piecewise deterministic switched Markov processes (PDsMP). We equally discuss to which extent Lipschitz-continuity of the driving coefficients is needed. Finally, by applying the theoretical result on PDsMP to Hasty's model of bacteriophage (\cite{hasty_pradines_dolnik_collins_00}, \cite{crudu_debussche_radulescu_09}), we show the necessity of explicit modeling for the environmental cue triggering lysis.
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Submitted on : Tuesday, June 11, 2019 - 11:33:34 AM
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Dan Goreac. Border Avoidance: Necessary Regularity for Coefficients and Viscosity Approach. SIAM Journal on Control and Optimization, Society for Industrial and Applied Mathematics, 2019, 57 (6), pp.4175-4204. ⟨10.1137/18M1220108⟩. ⟨hal-01798133v3⟩



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