Stability analysis and robustness results for a nonlinear system with distributed delays describing hematopoiesis

Walid Djema 1 Frederic Mazenc 1 Catherine Bonnet 1
1 DISCO - Dynamical Interconnected Systems in COmplex Environments
L2S - Laboratoire des signaux et systèmes, Inria Saclay - Ile de France, SUPELEC, CNRS - Centre National de la Recherche Scientifique : UMR8506
Abstract : A nonlinear system with distributed delays describing cell dynamics in hematopo-iesis is analyzed-in the time-domain-via a construction of suitable Lyapunov-Krasovskii functionals (LKFs). Two interesting biological situations lead us to re-investigate the stability properties of two meaningful steady states: the 0-equilibrium for unhealthy hematopoiesis and the positive equilibrium for the healthy case. Biologically, convergence to the 0-equilibrium means the extinction of all the generations of blood cells while the positive equilibrium reflects the normal process where blood cells survive. Their analyses are slightly different in the sense that we take advantage of positivity of the system to construct linear functionals to analyze the 0-equilibrium, while we use some quadratic functionals to investigate the stability properties of the positive equilibrium. For both equilibria, we establish the exponential stability of solutions and we provide an estimate of their rates of convergence. Moreover, a robustness analysis is performed when the model is subject to some nonvanishing perturbations. Numerical examples are provided.
Keywords : Biological model Delay Positive system Lyapunov Stability
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Walid Djema, Frederic Mazenc, Catherine Bonnet. Stability analysis and robustness results for a nonlinear system with distributed delays describing hematopoiesis. Systems and Control Letters, Elsevier, 2017, 102, pp.93 - 101. ⟨10.1016/j.sysconle.2017.01.007⟩. ⟨hal-01627125⟩

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