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Approximation of the viability kernel

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Abstract

We study recursive inclusionsx n+1 ε G(x n). For instance, such systems appear for discrete finite-difference inclusionsx n+1 εG ρ (x n) whereG ρ :=1+ρF. The discrete viability kernel ofG ρ , i.e., the largest discrete viability domain, can be an internal approximation of the viability kernel ofK underF. We study discrete and finite dynamical systems. In the Lipschitz case we get a generalization to differential inclusions of the Euler and Runge-Kutta methods. We prove first that the viability kernel ofK underF can be approached by a sequence of discrete viability kernels associated withx n+1 εГ ρ (xn) whereГ ρ (x) =x + ρF(x) + (ML/2) ρ 2ℬ. Secondly, we show that it can be approached by finite viability kernels associated withx n+1 h ε (Г ρ (x n+1h ) +α(hℬ) ∩X h .

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  1. CEREMADE, Université Paris-Dauphine, Place du Maréchal de Lattre de Tassigny, 75775, Paris cedex 16, France

    Patrick Saint-Pierre

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  1. Patrick Saint-Pierre
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Saint-Pierre, P. Approximation of the viability kernel. Appl Math Optim 29, 187–209 (1994). https://doi.org/10.1007/BF01204182

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  • DOI: https://doi.org/10.1007/BF01204182

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