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The Euler scheme for state constrained ordinary differential inclusions

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  • We propose and analyze a variation of the Euler scheme for state constrained ordinary differential inclusions under weak assumptions on the right-hand side and the state constraints. Convergence results are given for the space-continuous and the space-discrete versions of this scheme, and a numerical example illustrates in which sense these limits have to be interpreted.
    Mathematics Subject Classification: Primary: 34A60, 65L20; Secondary: 49J15.


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  • [1]

    J.-P. Aubin and A. Cellina, Differential Inclusions, Grundlehren der Mathematischen Wissenschaften 264, Springer-Verlag, Berlin, 1984.doi: 10.1007/978-3-642-69512-4.


    J.-P. Aubin and H. Frankowska, Set-valued Analysis, Systems & Control: Foundations & Applications 2, Birkhäuser Boston, Inc., Boston, MA, 1990.


    R. Baier, M. Gerdts and I. Xausa, Approximation of reachable sets using optimal control algorithms, Numer. Algebra Control Optim., 3 (2013), 519-548.doi: 10.3934/naco.2013.3.519.


    R. Baier, I. A. Chahma and F. Lempio, Stability and convergence of Euler's method for state-constrained differential inclusions, SIAM J. Optim., 18 (2007), 1004-1026.doi: 10.1137/060661867.


    P. Bettiol, A. Bressan and R. Vinter, On trajectories satisfying a state constraint: $W^{1,1}$ estimates and counterexamples, SIAM J. Control Optim., 48 (2010), 4664-4679.doi: 10.1137/090769788.


    P. Bettiol, H. Frankowska and R. Vinter, $L^\infty$ estimates on trajectories confined to a closed subset, J. Differential Equations, 252 (2012), 1912-1933.doi: 10.1016/j.jde.2011.09.007.


    W.-J. Beyn and J. Rieger, Numerical fixed grid methods for differential inclusions, Computing, 81 (2007), 91-106.doi: 10.1007/s00607-007-0240-4.


    W.-J. Beyn and J. Rieger, The implicit Euler scheme for one-sided Lipschitz differential inclusions, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 409-428.doi: 10.3934/dcdsb.2010.14.409.


    T. Donchev and E. Farkhi, Stability and Euler approximation of one-sided Lipschitz differential inclusions, SIAM J. Control Optim., 36 (1998), 780-796.doi: 10.1137/S0363012995293694.


    A. Dontchev and F. Lempio, Difference methods for differential inclusions: A survey, SIAM Rev., 34 (1992), 263-294.doi: 10.1137/1034050.


    M. Gerdts, R. Henrion, D. Hömberg and C. Landry, Path planning and collision avoidance for robots, Numer. Algebra Control Optim., 2 (2012), 437-463.doi: 10.3934/naco.2012.2.437.


    M. Gerdts and I. Xausa, Avoidance trajectories using reachable sets and parametric sensitivity analysis. System modeling and optimization, 491-500, IFIP Adv. Inf. Commun. Technol. 391, Springer, Heidelberg, 2013.doi: 10.1007/978-3-642-36062-6_49.


    G. Grammel, Towards fully discretized differential inclusions, Set-Valued Anal., 11 (2003), 1-8.doi: 10.1023/A:1021981217050.


    C. Landry, M. Gerdts, R. Henrion and D. Hömberg, Path-planning with collision avoidance in automotive industry. System modeling and optimization, 102-111, IFIP Adv. Inf. Commun. Technol. 391, Springer, Heidelberg, 2013.doi: 10.1007/978-3-642-36062-6_11.


    F. Lempio and V. Veliov, Discrete approximations of differential inclusions, Bayreuth. Math. Schr., 54 (1998), 149-232.


    J. Rieger, Semi-implicit Euler schemes for ordinary differential inclusions, SIAM J. Numer. Anal., 52 (2014), 895-914.doi: 10.1137/110842727.


    J. Rieger, Robust boundary tracking for reachable sets of nonlinear differential inclusions, Found. Comput. Math., 15 (2015), 1129-1150.doi: 10.1007/s10208-014-9218-8.


    M. Sandberg, Convergence of the forward Euler method for nonconvex differential inclusions, SIAM J. Numer. Anal., 47 (2008), 308-320.doi: 10.1137/070686093.


    D. Szolnoki, Set oriented methods for computing reachable sets and control sets, Discrete Contin. Dyn. Syst. Ser. B, 3 (2003), 361-382.doi: 10.3934/dcdsb.2003.3.361.


    V. Veliov, Second order discrete approximations to strongly convex differential inclusions, Systems Control Lett., 13 (1989), 263-269.doi: 10.1016/0167-6911(89)90073-X.


    J. Warga, Optimal Control of Differential and Functional Equations, Academic Press, New York, 1972.

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