article

Approximation of the viability kernel

Author:

Patrick Saint-PierreAuthors Info & Claims

Applied Mathematics and Optimization, Volume 29, Issue 2

Pages 187 - 209

https://doi.org/10.1007/BF01204182

Published: 01 March 1994 Publication History

Abstract

We study recursive inclusionsxn+1ź G(xn). For instance, such systems appear for discrete finite-difference inclusionsxn+1 źGź(xn) whereGź:=1+źF. The discrete viability kernel ofGź, 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 withxn+1 źźź(xn) whereźź(x) =x + źF(x) + (ML/2) ź2ź. Secondly, we show that it can be approached by finite viability kernels associated withxhn+1 ź (źź(xhn+1) +ź(hź) źXh.

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Zhao NGao YLv JTang J(2023)On the computation of the robust viability kernel for switched systemsAsian Journal of Control10.1002/asjc.310425:5(3598-3615)Online publication date: 4-Sep-2023
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cover image Applied Mathematics and Optimization

Applied Mathematics and Optimization Volume 29, Issue 2

March 1994

111 pages

ISSN:0095-4616

Issue’s Table of Contents

Copyright © Copyright © 1994 Springer-Verlag New York Inc.

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Springer-Verlag

Berlin, Heidelberg

Publication History

Published: 01 March 1994

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Cited By

Tonda AAlvarez IMartin SSquillero GLutton ESilva SPaquete L(2023)Towards Evolutionary Control Laws for Viability ProblemsProceedings of the Genetic and Evolutionary Computation Conference10.1145/3583131.3590415(1464-1472)Online publication date: 15-Jul-2023
https://dl.acm.org/doi/10.1145/3583131.3590415
Zhao NGao YLv JTang J(2023)On the computation of the robust viability kernel for switched systemsAsian Journal of Control10.1002/asjc.310425:5(3598-3615)Online publication date: 4-Sep-2023
https://dl.acm.org/doi/10.1002/asjc.3104
Mehmood USheikhi SBak SSmolka SStoller S(2022)The Black-Box Simplex Architecture for Runtime Assurance of Autonomous CPSNASA Formal Methods10.1007/978-3-031-06773-0_12(231-250)Online publication date: 24-May-2022
https://dl.acm.org/doi/10.1007/978-3-031-06773-0_12
Mukhopadhyay SZhang F(2021)An algorithm for computing robust forward invariant sets of two dimensional nonlinear systemsAsian Journal of Control10.1002/asjc.236023:5(2403-2419)Online publication date: 1-Oct-2021
https://dl.acm.org/doi/10.1002/asjc.2360
Bouguerra MFraichard TFezari M(2019)Viability-Based Guaranteed Safe Robot NavigationJournal of Intelligent and Robotic Systems10.1007/s10846-018-0955-995:2(459-471)Online publication date: 1-Aug-2019
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Martin BMullier O(2018)Improving validated computation of Viability KernelsProceedings of the 21st International Conference on Hybrid Systems: Computation and Control (part of CPS Week)10.1145/3178126.3178141(227-236)Online publication date: 11-Apr-2018
https://dl.acm.org/doi/10.1145/3178126.3178141
Lv JGao YZhao N(2018)Viability Criteria for a Switched System on Bounded PolyhedronAsian Journal of Control10.1002/asjc.171920:6(2380-2387)Online publication date: 22-Nov-2018
https://dl.acm.org/doi/10.1002/asjc.1719
Gleason JVinod AOishi M(2017)Underapproximation of reach-avoid sets for discrete-time stochastic systems via Lagrangian methods2017 IEEE 56th Annual Conference on Decision and Control (CDC)10.1109/CDC.2017.8264291(4283-4290)Online publication date: 12-Dec-2017
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Rasmussen MRieger JWebster K(2017)Approximation of reachable sets using optimal control and support vector machinesJournal of Computational and Applied Mathematics10.1016/j.cam.2016.06.015311:C(68-83)Online publication date: 1-Feb-2017
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Liniger ALygeros JGirard ASankaranarayanan S(2015)A viability approach for fast recursive feasible finite horizon path planning of autonomous RC carsProceedings of the 18th International Conference on Hybrid Systems: Computation and Control10.1145/2728606.2728620(1-10)Online publication date: 14-Apr-2015
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