# American Institute of Mathematical Sciences

October  2016, 21(8): 2729-2744. doi: 10.3934/dcdsb.2016070

## The Euler scheme for state constrained ordinary differential inclusions

 1 Department of Mathematics, Imperial College London, 180 Queen's Gate, London SW7 2AZ, United Kingdom

Received  July 2015 Revised  July 2016 Published  September 2016

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.
Citation: Janosch Rieger. The Euler scheme for state constrained ordinary differential inclusions. Discrete and Continuous Dynamical Systems - B, 2016, 21 (8) : 2729-2744. doi: 10.3934/dcdsb.2016070
##### References:
 [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. [2] J.-P. Aubin and H. Frankowska, Set-valued Analysis, Systems & Control: Foundations & Applications 2, Birkhäuser Boston, Inc., Boston, MA, 1990. [3] 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. [4] 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. [5] 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. [6] 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. [7] 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. [8] 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. [9] 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. [10] A. Dontchev and F. Lempio, Difference methods for differential inclusions: A survey, SIAM Rev., 34 (1992), 263-294. doi: 10.1137/1034050. [11] 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. [12] 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. [13] G. Grammel, Towards fully discretized differential inclusions, Set-Valued Anal., 11 (2003), 1-8. doi: 10.1023/A:1021981217050. [14] 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. [15] F. Lempio and V. Veliov, Discrete approximations of differential inclusions, Bayreuth. Math. Schr., 54 (1998), 149-232. [16] J. Rieger, Semi-implicit Euler schemes for ordinary differential inclusions, SIAM J. Numer. Anal., 52 (2014), 895-914. doi: 10.1137/110842727. [17] 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. [18] M. Sandberg, Convergence of the forward Euler method for nonconvex differential inclusions, SIAM J. Numer. Anal., 47 (2008), 308-320. doi: 10.1137/070686093. [19] 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. [20] 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. [21] J. Warga, Optimal Control of Differential and Functional Equations, Academic Press, New York, 1972.

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##### References:
 [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. [2] J.-P. Aubin and H. Frankowska, Set-valued Analysis, Systems & Control: Foundations & Applications 2, Birkhäuser Boston, Inc., Boston, MA, 1990. [3] 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. [4] 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. [5] 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. [6] 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. [7] 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. [8] 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. [9] 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. [10] A. Dontchev and F. Lempio, Difference methods for differential inclusions: A survey, SIAM Rev., 34 (1992), 263-294. doi: 10.1137/1034050. [11] 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. [12] 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. [13] G. Grammel, Towards fully discretized differential inclusions, Set-Valued Anal., 11 (2003), 1-8. doi: 10.1023/A:1021981217050. [14] 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. [15] F. Lempio and V. Veliov, Discrete approximations of differential inclusions, Bayreuth. Math. Schr., 54 (1998), 149-232. [16] J. Rieger, Semi-implicit Euler schemes for ordinary differential inclusions, SIAM J. Numer. Anal., 52 (2014), 895-914. doi: 10.1137/110842727. [17] 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. [18] M. Sandberg, Convergence of the forward Euler method for nonconvex differential inclusions, SIAM J. Numer. Anal., 47 (2008), 308-320. doi: 10.1137/070686093. [19] 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. [20] 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. [21] J. Warga, Optimal Control of Differential and Functional Equations, Academic Press, New York, 1972.
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