-
Previous Article
On the compressible Navier-Stokes-Korteweg equations
- DCDS-B Home
- This Issue
-
Next Article
Complex dynamics of a nutrient-plankton system with nonlinear phytoplankton mortality and allelopathy
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 |
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. |
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. |
[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. |
[1] |
Thomas Lorenz. Partial differential inclusions of transport type with state constraints. Discrete and Continuous Dynamical Systems - B, 2019, 24 (3) : 1309-1340. doi: 10.3934/dcdsb.2019018 |
[2] |
Wolf-Jüergen Beyn, Janosch Rieger. The implicit Euler scheme for one-sided Lipschitz differential inclusions. Discrete and Continuous Dynamical Systems - B, 2010, 14 (2) : 409-428. doi: 10.3934/dcdsb.2010.14.409 |
[3] |
Wojciech Kryszewski, Dorota Gabor, Jakub Siemianowski. The Krasnosel'skii formula for parabolic differential inclusions with state constraints. Discrete and Continuous Dynamical Systems - B, 2018, 23 (1) : 295-329. doi: 10.3934/dcdsb.2018021 |
[4] |
Bahareh Akhtari, Esmail Babolian, Andreas Neuenkirch. An Euler scheme for stochastic delay differential equations on unbounded domains: Pathwise convergence. Discrete and Continuous Dynamical Systems - B, 2015, 20 (1) : 23-38. doi: 10.3934/dcdsb.2015.20.23 |
[5] |
Matthias Gerdts, Martin Kunkel. Convergence analysis of Euler discretization of control-state constrained optimal control problems with controls of bounded variation. Journal of Industrial and Management Optimization, 2014, 10 (1) : 311-336. doi: 10.3934/jimo.2014.10.311 |
[6] |
Elimhan N. Mahmudov. Optimal control of second order delay-discrete and delay-differential inclusions with state constraints. Evolution Equations and Control Theory, 2018, 7 (3) : 501-529. doi: 10.3934/eect.2018024 |
[7] |
Monika Eisenmann, Etienne Emmrich, Volker Mehrmann. Convergence of the backward Euler scheme for the operator-valued Riccati differential equation with semi-definite data. Evolution Equations and Control Theory, 2019, 8 (2) : 315-342. doi: 10.3934/eect.2019017 |
[8] |
Benoît Merlet, Morgan Pierre. Convergence to equilibrium for the backward Euler scheme and applications. Communications on Pure and Applied Analysis, 2010, 9 (3) : 685-702. doi: 10.3934/cpaa.2010.9.685 |
[9] |
Jiao-Yan Li, Xiao Hu, Zhong Wan. An integrated bi-objective optimization model and improved genetic algorithm for vehicle routing problems with temporal and spatial constraints. Journal of Industrial and Management Optimization, 2020, 16 (3) : 1203-1220. doi: 10.3934/jimo.2018200 |
[10] |
Nidhal Gammoudi, Hasnaa Zidani. A differential game control problem with state constraints. Mathematical Control and Related Fields, 2022 doi: 10.3934/mcrf.2022008 |
[11] |
Jacson Simsen, Mariza Stefanello Simsen, José Valero. Convergence of nonautonomous multivalued problems with large diffusion to ordinary differential inclusions. Communications on Pure and Applied Analysis, 2020, 19 (4) : 2347-2368. doi: 10.3934/cpaa.2020102 |
[12] |
Andrei V. Dmitruk, Nikolai P. Osmolovskii. Proof of the maximum principle for a problem with state constraints by the v-change of time variable. Discrete and Continuous Dynamical Systems - B, 2019, 24 (5) : 2189-2204. doi: 10.3934/dcdsb.2019090 |
[13] |
Elimhan N. Mahmudov. Optimization of fourth order Sturm-Liouville type differential inclusions with initial point constraints. Journal of Industrial and Management Optimization, 2020, 16 (1) : 169-187. doi: 10.3934/jimo.2018145 |
[14] |
Elimhan N. Mahmudov. Optimal control of Sturm-Liouville type evolution differential inclusions with endpoint constraints. Journal of Industrial and Management Optimization, 2020, 16 (5) : 2503-2520. doi: 10.3934/jimo.2019066 |
[15] |
Maurizio Grasselli, Morgan Pierre. Convergence to equilibrium of solutions of the backward Euler scheme for asymptotically autonomous second-order gradient-like systems. Communications on Pure and Applied Analysis, 2012, 11 (6) : 2393-2416. doi: 10.3934/cpaa.2012.11.2393 |
[16] |
Jian Su, Yinnian He. The almost unconditional convergence of the Euler implicit/explicit scheme for the three dimensional nonstationary Navier-Stokes equations. Discrete and Continuous Dynamical Systems - B, 2017, 22 (9) : 3421-3438. doi: 10.3934/dcdsb.2017173 |
[17] |
Michel H. Geoffroy, Alain Piétrus. A fast iterative scheme for variational inclusions. Conference Publications, 2009, 2009 (Special) : 250-258. doi: 10.3934/proc.2009.2009.250 |
[18] |
Umberto Mosco. Impulsive motion on synchronized spatial temporal grids. Discrete and Continuous Dynamical Systems, 2017, 37 (12) : 6069-6098. doi: 10.3934/dcds.2017261 |
[19] |
Aniello Raffaele Patrone, Otmar Scherzer. On a spatial-temporal decomposition of optical flow. Inverse Problems and Imaging, 2017, 11 (4) : 761-781. doi: 10.3934/ipi.2017036 |
[20] |
Mariusz Michta. On solutions to stochastic differential inclusions. Conference Publications, 2003, 2003 (Special) : 618-622. doi: 10.3934/proc.2003.2003.618 |
2020 Impact Factor: 1.327
Tools
Metrics
Other articles
by authors
[Back to Top]