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

The Euler scheme for state constrained ordinary differential inclusions

Janosch Rieger 1,

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.
Keywords: Differential inclusions, temporal and spatial discretization, state constraints, Euler scheme, proof of convergence..
Mathematics Subject Classification: Primary: 34A60, 65L20; Secondary: 49J1.
Citation: Janosch Rieger. The Euler scheme for state constrained ordinary differential inclusions. Discrete & 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.  Google Scholar

[2]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

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

[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.  Google Scholar

[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.  Google Scholar

[13]

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

[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.  Google Scholar

[15]

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

[16]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

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

show all references

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.  Google Scholar

[2]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

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

[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.  Google Scholar

[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.  Google Scholar

[13]

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

[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.  Google Scholar

[15]

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

[16]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

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

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