Discrete Filippov-type stability for one-sided Lipschitzian difference inclusions

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Baier, Robert ; Farkhi, Elza:
Discrete Filippov-type stability for one-sided Lipschitzian difference inclusions.
In: Feichtinger, Gustav ; Kovacevic, Raimund M. ; Tragler, Gernot (Hrsg.): Control Systems and Mathematical Methods in Economics : Essays in Honor of Vladimir M. Veliov. - Cham : Springer , 2018 . - S. 27-55 . - (Lecture Notes in Economics and Mathematical Systems ; 687 )
ISBN 978-3-319-75168-9
DOI: https://doi.org/10.1007/978-3-319-75169-6_3

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Abstract

We state and prove Filippov-type stability theorems for discrete difference inclusions obtained by the Euler discretization of a differential inclusion with perturbations in the set of initial points, in the right-hand side and in the state variable. We study the cases in which the right-hand side of the inclusion is not necessarily Lipschitz, but satisfies a weaker one-sided Lipschitz (OSL) or strengthened one-sided Lipschitz (SOSL) condition. The obtained estimates imply stability of the discrete solutions for infinite number of fixed time steps if the OSL constant is negative and the perturbations are bounded in certain norms. We show a better order of stability for SOSL right-hand sides and apply our theorems to estimate the distance from the solutions of other difference methods, as for the implicit Euler scheme to the set of solutions of the Euler scheme. We also prove a discrete relaxation stability theorem for the considered difference inclusion, which also extends a theorem of G. Grammel (2003) from the class of Lipschitz maps to the wider class of OSL ones.