Border Avoidance: Necessary Regularity for Coefficients and Viscosity Approach

Dan Goreac (1)

1 PS

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.

Mots clés: Invariance; Near-Viability; Viscosity Solutions; Brownian Diffusion; PDMP; Gene Networks

Domaines: Optimisation et contrôle [math.OC]; Probabilités [math.PR]

Dan Goreac. Border Avoidance: Necessary Regularity for Coefficients and Viscosity Approach. SIAM Journal on Control and Optimization, 2019, 57 (6), pp.4175-4204. ⟨10.1137/18M1220108⟩. ⟨hal-01798133v3⟩