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 & 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, (1984). doi: 10.1007/978-3-642-69512-4.

[2]

J.-P. Aubin and H. Frankowska, Set-valued Analysis,, Systems & Control: Foundations & Applications 2, (1990).

[3]

R. Baier, M. Gerdts and I. Xausa, Approximation of reachable sets using optimal control algorithms,, Numer. Algebra Control Optim., 3 (2013), 519. 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. 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. 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. 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. 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. 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. doi: 10.1137/S0363012995293694.

[10]

A. Dontchev and F. Lempio, Difference methods for differential inclusions: A survey,, SIAM Rev., 34 (1992), 263. 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. 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, (2013), 491. doi: 10.1007/978-3-642-36062-6_49.

[13]

G. Grammel, Towards fully discretized differential inclusions,, Set-Valued Anal., 11 (2003), 1. 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, (2013), 102. 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.

[16]

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

[17]

J. Rieger, Robust boundary tracking for reachable sets of nonlinear differential inclusions,, Found. Comput. Math., 15 (2015), 1129. 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. 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. 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. doi: 10.1016/0167-6911(89)90073-X.

[21]

J. Warga, Optimal Control of Differential and Functional Equations,, Academic Press, (1972).

show all references

References:
[1]

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

[2]

J.-P. Aubin and H. Frankowska, Set-valued Analysis,, Systems & Control: Foundations & Applications 2, (1990).

[3]

R. Baier, M. Gerdts and I. Xausa, Approximation of reachable sets using optimal control algorithms,, Numer. Algebra Control Optim., 3 (2013), 519. 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. 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. 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. 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. 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. 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. doi: 10.1137/S0363012995293694.

[10]

A. Dontchev and F. Lempio, Difference methods for differential inclusions: A survey,, SIAM Rev., 34 (1992), 263. 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. 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, (2013), 491. doi: 10.1007/978-3-642-36062-6_49.

[13]

G. Grammel, Towards fully discretized differential inclusions,, Set-Valued Anal., 11 (2003), 1. 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, (2013), 102. 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.

[16]

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

[17]

J. Rieger, Robust boundary tracking for reachable sets of nonlinear differential inclusions,, Found. Comput. Math., 15 (2015), 1129. 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. 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. 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. doi: 10.1016/0167-6911(89)90073-X.

[21]

J. Warga, Optimal Control of Differential and Functional Equations,, Academic Press, (1972).

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