Construction of the minimum time function for linear systems via higher-order set-valued methods

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June 2019, 9(2): 223-255. doi: 10.3934/mcrf.2019012

Construction of the minimum time function for linear systems via higher-order set-valued methods

1.	Universität Bayreuth, Mathematisches Institut, 95440 Bayreuth, Germany
2.	Otto von Guericke University Magdeburg, Department of Mathematics, Universitätsplatz 2, 39106 Magdeburg, Germany

* Corresponding author: Thuy T. T. Le

Received December 2016 Revised March 2018 Published November 2018

Fund Project: The second author is supported by a PhD fellowship for foreign students at the Università di Padova funded by Fondazione CARIPARO. This paper was developed while the second author was visiting the Department of Mathematics of the University of Bayreuth

The paper is devoted to introducing an approach to compute the approximate minimum time function of control problems which is based on reachable set approximation and uses arithmetic operations for convex compact sets. In particular, in this paper the theoretical justification of the proposed approach is restricted to a class of linear control systems. The error estimate of the fully discrete reachable set is provided by employing the Hausdorff distance to the continuous-time reachable set. The detailed procedure solving the corresponding discrete set-valued problem is described. Under standard assumptions, by means of convex analysis and knowledge of the regularity of the true minimum time function, we estimate the error of its approximation. Higher-order discretization of the reachable set of the linear control problem can balance missing regularity (e.g., if only Hölder continuity holds) of the minimum time function for smoother problems. To illustrate the error estimates and to demonstrate differences to other numerical approaches we provide a collection of numerical examples which either allow higher order of convergence with respect to time discretization or where the continuity of the minimum time function cannot be sufficiently granted, i.e., we study cases in which the minimum time function is Hölder continuous or even discontinuous.

Keywords: Minimum time function, reachable sets, linear control problems, higher-order set-valued methods, direct discretization methods.

Mathematics Subject Classification: 49N60, 93B03 (49N05, 49M25, 52A27).

Citation: Robert Baier, Thuy T. T. Le. Construction of the minimum time function for linear systems via higher-order set-valued methods. Mathematical Control & Related Fields, 2019, 9 (2) : 223-255. doi: 10.3934/mcrf.2019012

References:

[1]	W. Alt, R. Baier, M. Gerdts and F. Lempio, Approximations of linear control problems with bang-bang solutions, Optimization, 62 (2013), 9-32. doi: 10.1080/02331934.2011.568619.
[2]	M. Althoff, Reachability Analysis and its Application to the Safety Assessment of Autonomous Cars, PhD thesis, Fakultät für Elektrotechnik und Informationstechnik, Technische Universität München, Munich, Germany, 2010, 221 S.
[3]	J.-P. Aubin, A. M. Bayen and P. Saint-Pierre, Viability Theory. New Directions, 2nd edition, Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-16684-6.
[4]	J.-P. Aubin and A. Cellina, Differential Inclusions, vol. 264 of Grundlehren der mathematischen Wissenschaften, Springer-Verlag, Berlin-Heidelberg-New York-Tokyo, 1984.
[5]	R. J. Aumann, Integrals of set-valued functions, J. Math. Anal. Appl., 12 (1965), 1-12. doi: 10.1016/0022-247X(65)90049-1.
[6]	R. Baier, Mengenwertige Integration und die diskrete Approximation erreichbarer Mengen, [Set-valued Integration and the Discrete Approximation of Reachable Sets], PhD thesis, Universität Bayreuth, 1994.
[7]	R. Baier, Selection strategies for set-valued Runge-Kutta methods, in Numerical Analysis and Its Applications, Third International Conference, NAA 2004, Rousse, Bulgaria, June 29 - July 3, 2004, Revised Selected Papers (eds. Z. Li, L. G. Vulkov and J. Wasniewski), vol. 3401 of Lecture Notes in Comput. Sci., Springer, Berlin-Heidelberg, 2005, 149-157.
[8]	R. Baier, C. Büskens, I. A. Chahma and M. Gerdts, Approximation of reachable sets by direct solution methods of optimal control problems, Optim. Methods Softw., 22 (2007), 433-452. doi: 10.1080/10556780600604999.
[9]	R. Baier and F. Lempio, Approximating reachable sets by extrapolation methods, in Curves and Surfaces in Geometric Design. Papers from the Second International Conference on Curves and Surfaces, held in Chamonix-Mont-Blanc, France, July 10-16, 1993 (eds. P. J. Laurent, A. L. Méhauteé and L. L. Schumaker), A K Peters, Wellesley, 1994, 9-18.
[10]	R. Baier and T. T. T. Le, Construction of the minimum time function via reachable sets of linear control systems. Part 1: error estimates, part 2: numerical computations, preprint, Dec 2015. arXiv: 1512.08630 and arXiv: 1512.08617.
[11]	R. Baier and F. Lempio, Computing Aumann's integral, in Modeling Techniques for Uncertain Systems, Proceedings of a Conference held in Sopron, Hungary, July 6-10, 1992 (eds. A. B. Kurzhanski and V. M. Veliov), vol. 18 of Progress in Systems and Control Theory, Birkhäuser, Basel, 1994, 71-92. doi: 10.1007/978-3-642-78787-4_7.
[12]	M. Bardi and I. Capuzzo-Dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations, Systems & Control: Foundations & Applications, Birkhäuser Boston Inc., Boston, MA, 1997, with appendices by Maurizio Falcone and Pierpaolo Soravia. doi: 10.1007/978-0-8176-4755-1.
[13]	M. Bardi and M. Falcone, An approximation scheme for the minimum time function, SIAM J. Control Optim., 28 (1990), 950-965. doi: 10.1137/0328053.
[14]	M. Bardi and M. Falcone, Discrete approximation of the minimal time function for systems with regular optimal trajectories, in Analysis and Optimization of Systems. Proceedings of the 9th International Conference Antibes, June 12-15, 1990 (eds. A. Bensoussan and J. L. Lions), vol. 144 of Lecture Notes in Control and Inform. Sci., Springer, Berlin-Heidelberg, 1990, 103-112. doi: 10.1007/BFb0120033.
[15]	M. Bardi, M. Falcone and P. Soravia, Numerical methods for pursuit-evasion games via viscosity solutions, in Stochastic and Differential Games, vol. 4 of Ann. Internat. Soc. Dynam. Games, Birkhäuser Boston, Boston, MA, 1999, 105-175.
[16]	O. Bokanowski, A. Briani and H. Zidani, Minimum time control problems for non-autonomous differential equations, Systems Control Lett., 58 (2009), 742-746. doi: 10.1016/j.sysconle.2009.08.003.
[17]	O. Bokanowski, N. Forcadel and H. Zidani, Reachability and minimal times for state constrained nonlinear problems without any controllability assumption, SIAM J. Control Optim., 48 (2010), 4292-4316. doi: 10.1137/090762075.
[18]	F. H. Clarke, Y. S. Ledyaev, R. J. Stern and P. R. Wolenski, Nonsmooth Analysis and Control Theory, Springer-Verlag, New York, 1998.
[19]	G. Colombo and T. T. T. Le, Higher order discrete controllability and the approximation of the minimum time function, Discrete Contin. Dyn. Syst., 35 (2015), 4293-4322. doi: 10.3934/dcds.2015.35.4293.
[20]	G. Colombo, A. Marigonda and P. R. Wolenski, Some new regularity properties for the minimal time function, SIAM J. Control Optim., 44 (2006), 2285-2299. doi: 10.1137/050630076.
[21]	G. Colombo, K. T. Nguyen and L. V. Nguyen, Non-Lipschitz points and the SBV regularity of the minimum time function, Calc. Var. Partial Differential Equations, 51 (2014), 439-463. doi: 10.1007/s00526-013-0682-9.
[22]	B. D. Doitchinov and V. M. Veliov, Parametrizations of integrals of set-valued mappings and applications, J. Math. Anal. Appl., 179 (1993), 483-499. doi: 10.1006/jmaa.1993.1363.
[23]	T. D. Donchev and E. M. Farkhi, Moduli of smoothness of vector valued functions of a real variable and applications, Numer. Funct. Anal. Optim., 11 (1990), 497-509. doi: 10.1080/01630569008816385.
[24]	A. L. Dontchev and E. M. Farkhi, Error estimates for discretized differential inclusions, Computing, 41 (1989), 349-358. doi: 10.1007/BF02241223.
[25]	M. Falcone, Numerical Solution of Dynamic Programming Equations. Appendix A, in Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations (eds. M. Bardi and I. Capuzzo-Dolcetta), Systems & Control: Foundations & Applications, Birkhäuser Boston Inc., Boston, MA, 1997, 471-504. doi: 10.1007/978-0-8176-4755-1.
[26]	M. Gerdts, Optimal control of ODEs and DAEs, de Gruyter Textbook, Walter de Gruyter & Co., Berlin, 2012. doi: 10.1515/9783110249996.
[27]	A. Girard, C. Le Guernic and O. Maler, Efficient computation of reachable sets of linear time-invariant systems with inputs, in Hybrid Systems: Computation and Control, vol. 3927 of Lecture Notes in Comput. Sci., Springer, Berlin, 2006, 257-271. doi: 10.1007/11730637_21.
[28]	L. Grüne and T. T. T. Le, A double-sided dynamic programming approach to the minimum time problem and its numerical approximation, Applied Numerical Mathematics, 121 (2017), 68-81. doi: 10.1016/j.apnum.2017.06.008.
[29]	H. Hermes and J. LaSalle, Functional Analysis and Time Optimal Control, vol. 56 of Mathematics in science and engineering, Academic Press, New York, 1969.
[30]	A. B. Kurzhanski and P. Varaiya, Dynamics and Control of Trajectory Tubes. Theory and Computation, Systems & Control: Foundations & Applications, Springer, Cham-Heidelberg-New York-Dordrecht-London, 2014. doi: 10.1007/978-3-319-10277-1.
[31]	T. T. T. Le, Results on Controllability and Numerical Approximation of the Minimum Time Function, PhD thesis, Dipartimento di Matematica, Padova, Italy, 2016.
[32]	C. Le Guernic, Reachability Analysis of Hybrid Systems with Linear Continuous Dynamics, PhD thesis, École Doctorale Mathématiques, Sciences et Technologies de l'Information, Informatique, Grenoble, France, 2009, 169 pages.
[33]	E. Lee and L. Markus, Foundations of Optimal Control Theory, SIAM Series in Applied Mathematics, John Wiley & Sons, Inc., New York-London-Sydney, 1967.
[34]	A. Marigonda, Second order conditions for the controllability of nonlinear systems with drift, Commun. Pure Appl. Anal., 5 (2006), 861-885. doi: 10.3934/cpaa.2006.5.861.
[35]	D. Pallaschke and R. Urbański, Pairs of Compact Convex Sets, vol. 548 of Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, 2002. doi: 10.1007/978-94-015-9920-7.
[36]	N. N. Petrov, On the Bellman function for the time-optimality process problem, Prikl. Mat. Meh., 34 (1970), 820-826.
[37]	R. T. Rockafellar and R. J.-B. Wets, Variational Analysis, vol. 317 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer Science & Business Media, Berlin, 2009. doi: 10.1007/978-3-642-02431-3.
[38]	P. Saint-Pierre, Approximation of the viability kernel, Appl. Math. Optim., 29 (1994), 187-209. doi: 10.1007/BF01204182.
[39]	A. Tolstonogov, Differential Inclusions in a Banach Space, vol. 524 of Mathematics and its Applications, Kluwer Academic Publishers, Dordrecht, 2000, translated from the 1986 Russian original and revised by the author. doi: 10.1007/978-94-015-9490-5.
[40]	V. M. Veliov, Discrete approximations of integrals of multivalued mappings, C. R. Acad. Bulgare Sci., 42 (1989), 51-54.
[41]	V. M. Veliov, Second order discrete approximation to linear differential inclusions, SIAM J. Numer. Anal., 29 (1992), 439-451. doi: 10.1137/0729026.
[42]	M. D. Wills, Hausdorff distance and convex sets, J. Convex Anal., 14 (2007), 109-117.
[43]	P. R. Wolenski, The exponential formula for the reachable set of a Lipschitz differential inclusion, SIAM J. Control Optim., 28 (1990), 1148-1161. doi: 10.1137/0328062.

show all references

References:

[1]	W. Alt, R. Baier, M. Gerdts and F. Lempio, Approximations of linear control problems with bang-bang solutions, Optimization, 62 (2013), 9-32. doi: 10.1080/02331934.2011.568619.
[2]	M. Althoff, Reachability Analysis and its Application to the Safety Assessment of Autonomous Cars, PhD thesis, Fakultät für Elektrotechnik und Informationstechnik, Technische Universität München, Munich, Germany, 2010, 221 S.
[3]	J.-P. Aubin, A. M. Bayen and P. Saint-Pierre, Viability Theory. New Directions, 2nd edition, Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-16684-6.
[4]	J.-P. Aubin and A. Cellina, Differential Inclusions, vol. 264 of Grundlehren der mathematischen Wissenschaften, Springer-Verlag, Berlin-Heidelberg-New York-Tokyo, 1984.
[5]	R. J. Aumann, Integrals of set-valued functions, J. Math. Anal. Appl., 12 (1965), 1-12. doi: 10.1016/0022-247X(65)90049-1.
[6]	R. Baier, Mengenwertige Integration und die diskrete Approximation erreichbarer Mengen, [Set-valued Integration and the Discrete Approximation of Reachable Sets], PhD thesis, Universität Bayreuth, 1994.
[7]	R. Baier, Selection strategies for set-valued Runge-Kutta methods, in Numerical Analysis and Its Applications, Third International Conference, NAA 2004, Rousse, Bulgaria, June 29 - July 3, 2004, Revised Selected Papers (eds. Z. Li, L. G. Vulkov and J. Wasniewski), vol. 3401 of Lecture Notes in Comput. Sci., Springer, Berlin-Heidelberg, 2005, 149-157.
[8]	R. Baier, C. Büskens, I. A. Chahma and M. Gerdts, Approximation of reachable sets by direct solution methods of optimal control problems, Optim. Methods Softw., 22 (2007), 433-452. doi: 10.1080/10556780600604999.
[9]	R. Baier and F. Lempio, Approximating reachable sets by extrapolation methods, in Curves and Surfaces in Geometric Design. Papers from the Second International Conference on Curves and Surfaces, held in Chamonix-Mont-Blanc, France, July 10-16, 1993 (eds. P. J. Laurent, A. L. Méhauteé and L. L. Schumaker), A K Peters, Wellesley, 1994, 9-18.
[10]	R. Baier and T. T. T. Le, Construction of the minimum time function via reachable sets of linear control systems. Part 1: error estimates, part 2: numerical computations, preprint, Dec 2015. arXiv: 1512.08630 and arXiv: 1512.08617.
[11]	R. Baier and F. Lempio, Computing Aumann's integral, in Modeling Techniques for Uncertain Systems, Proceedings of a Conference held in Sopron, Hungary, July 6-10, 1992 (eds. A. B. Kurzhanski and V. M. Veliov), vol. 18 of Progress in Systems and Control Theory, Birkhäuser, Basel, 1994, 71-92. doi: 10.1007/978-3-642-78787-4_7.
[12]	M. Bardi and I. Capuzzo-Dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations, Systems & Control: Foundations & Applications, Birkhäuser Boston Inc., Boston, MA, 1997, with appendices by Maurizio Falcone and Pierpaolo Soravia. doi: 10.1007/978-0-8176-4755-1.
[13]	M. Bardi and M. Falcone, An approximation scheme for the minimum time function, SIAM J. Control Optim., 28 (1990), 950-965. doi: 10.1137/0328053.
[14]	M. Bardi and M. Falcone, Discrete approximation of the minimal time function for systems with regular optimal trajectories, in Analysis and Optimization of Systems. Proceedings of the 9th International Conference Antibes, June 12-15, 1990 (eds. A. Bensoussan and J. L. Lions), vol. 144 of Lecture Notes in Control and Inform. Sci., Springer, Berlin-Heidelberg, 1990, 103-112. doi: 10.1007/BFb0120033.
[15]	M. Bardi, M. Falcone and P. Soravia, Numerical methods for pursuit-evasion games via viscosity solutions, in Stochastic and Differential Games, vol. 4 of Ann. Internat. Soc. Dynam. Games, Birkhäuser Boston, Boston, MA, 1999, 105-175.
[16]	O. Bokanowski, A. Briani and H. Zidani, Minimum time control problems for non-autonomous differential equations, Systems Control Lett., 58 (2009), 742-746. doi: 10.1016/j.sysconle.2009.08.003.
[17]	O. Bokanowski, N. Forcadel and H. Zidani, Reachability and minimal times for state constrained nonlinear problems without any controllability assumption, SIAM J. Control Optim., 48 (2010), 4292-4316. doi: 10.1137/090762075.
[18]	F. H. Clarke, Y. S. Ledyaev, R. J. Stern and P. R. Wolenski, Nonsmooth Analysis and Control Theory, Springer-Verlag, New York, 1998.
[19]	G. Colombo and T. T. T. Le, Higher order discrete controllability and the approximation of the minimum time function, Discrete Contin. Dyn. Syst., 35 (2015), 4293-4322. doi: 10.3934/dcds.2015.35.4293.
[20]	G. Colombo, A. Marigonda and P. R. Wolenski, Some new regularity properties for the minimal time function, SIAM J. Control Optim., 44 (2006), 2285-2299. doi: 10.1137/050630076.
[21]	G. Colombo, K. T. Nguyen and L. V. Nguyen, Non-Lipschitz points and the SBV regularity of the minimum time function, Calc. Var. Partial Differential Equations, 51 (2014), 439-463. doi: 10.1007/s00526-013-0682-9.
[22]	B. D. Doitchinov and V. M. Veliov, Parametrizations of integrals of set-valued mappings and applications, J. Math. Anal. Appl., 179 (1993), 483-499. doi: 10.1006/jmaa.1993.1363.
[23]	T. D. Donchev and E. M. Farkhi, Moduli of smoothness of vector valued functions of a real variable and applications, Numer. Funct. Anal. Optim., 11 (1990), 497-509. doi: 10.1080/01630569008816385.
[24]	A. L. Dontchev and E. M. Farkhi, Error estimates for discretized differential inclusions, Computing, 41 (1989), 349-358. doi: 10.1007/BF02241223.
[25]	M. Falcone, Numerical Solution of Dynamic Programming Equations. Appendix A, in Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations (eds. M. Bardi and I. Capuzzo-Dolcetta), Systems & Control: Foundations & Applications, Birkhäuser Boston Inc., Boston, MA, 1997, 471-504. doi: 10.1007/978-0-8176-4755-1.
[26]	M. Gerdts, Optimal control of ODEs and DAEs, de Gruyter Textbook, Walter de Gruyter & Co., Berlin, 2012. doi: 10.1515/9783110249996.
[27]	A. Girard, C. Le Guernic and O. Maler, Efficient computation of reachable sets of linear time-invariant systems with inputs, in Hybrid Systems: Computation and Control, vol. 3927 of Lecture Notes in Comput. Sci., Springer, Berlin, 2006, 257-271. doi: 10.1007/11730637_21.
[28]	L. Grüne and T. T. T. Le, A double-sided dynamic programming approach to the minimum time problem and its numerical approximation, Applied Numerical Mathematics, 121 (2017), 68-81. doi: 10.1016/j.apnum.2017.06.008.
[29]	H. Hermes and J. LaSalle, Functional Analysis and Time Optimal Control, vol. 56 of Mathematics in science and engineering, Academic Press, New York, 1969.
[30]	A. B. Kurzhanski and P. Varaiya, Dynamics and Control of Trajectory Tubes. Theory and Computation, Systems & Control: Foundations & Applications, Springer, Cham-Heidelberg-New York-Dordrecht-London, 2014. doi: 10.1007/978-3-319-10277-1.
[31]	T. T. T. Le, Results on Controllability and Numerical Approximation of the Minimum Time Function, PhD thesis, Dipartimento di Matematica, Padova, Italy, 2016.
[32]	C. Le Guernic, Reachability Analysis of Hybrid Systems with Linear Continuous Dynamics, PhD thesis, École Doctorale Mathématiques, Sciences et Technologies de l'Information, Informatique, Grenoble, France, 2009, 169 pages.
[33]	E. Lee and L. Markus, Foundations of Optimal Control Theory, SIAM Series in Applied Mathematics, John Wiley & Sons, Inc., New York-London-Sydney, 1967.
[34]	A. Marigonda, Second order conditions for the controllability of nonlinear systems with drift, Commun. Pure Appl. Anal., 5 (2006), 861-885. doi: 10.3934/cpaa.2006.5.861.
[35]	D. Pallaschke and R. Urbański, Pairs of Compact Convex Sets, vol. 548 of Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, 2002. doi: 10.1007/978-94-015-9920-7.
[36]	N. N. Petrov, On the Bellman function for the time-optimality process problem, Prikl. Mat. Meh., 34 (1970), 820-826.
[37]	R. T. Rockafellar and R. J.-B. Wets, Variational Analysis, vol. 317 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer Science & Business Media, Berlin, 2009. doi: 10.1007/978-3-642-02431-3.
[38]	P. Saint-Pierre, Approximation of the viability kernel, Appl. Math. Optim., 29 (1994), 187-209. doi: 10.1007/BF01204182.
[39]	A. Tolstonogov, Differential Inclusions in a Banach Space, vol. 524 of Mathematics and its Applications, Kluwer Academic Publishers, Dordrecht, 2000, translated from the 1986 Russian original and revised by the author. doi: 10.1007/978-94-015-9490-5.
[40]	V. M. Veliov, Discrete approximations of integrals of multivalued mappings, C. R. Acad. Bulgare Sci., 42 (1989), 51-54.
[41]	V. M. Veliov, Second order discrete approximation to linear differential inclusions, SIAM J. Numer. Anal., 29 (1992), 439-451. doi: 10.1137/0729026.
[42]	M. D. Wills, Hausdorff distance and convex sets, J. Convex Anal., 14 (2007), 109-117.
[43]	P. R. Wolenski, The exponential formula for the reachable set of a Lipschitz differential inclusion, SIAM J. Control Optim., 28 (1990), 1148-1161. doi: 10.1137/0328062.

Figure 1. Part of the triangulation

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Figure 2. Minimum time functions for Example 5.1 with different control sets

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Figure 3. Minimum time function for Example 5.1 with U = [−1, 1]², $\mathcal{S} = \left\{ 0 \right\}$

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Figure 4. Minimum time function for Example 5.2a) with target set {0} resp. B_0.05(0)

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Figure 5. Minimum time functions for Example 5.2b)

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Figure 6. Approximate optimal trajectories for Example 5.2a) resp. b)

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Figure 7. Minimum time functions for Examples 5.3 and 5.4

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Figure 8. Euler and Heun's iterates, minimum time function for Example 5.5 resp

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Figure 9. Reachable sets and minimum time functions for Example 5.6

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Figure 10. Reachable sets with various end times t_f for Examples 5.7 and 5.8

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Figure 11. Reachable sets with various end times and different target sets for Example 5.9

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Figure 12. Reachable sets with various end times and different control sets for Example 5.10

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Table 1. error estimates for Example 5.1 with different control and target sets

$ N_{\mathcal{R}} = N_U $	$U=B_1(0)$, $\mathcal{S}=B_{0.25}(0)$	$U=[-1, 1]^2$, $\mathcal{S}=B_{0.25}(0)$	$U=[-1, 1]^2$, $\mathcal{S}=\left\{ 0 \right\}$
0.04	50	0.2951	0.2265
0.02	100	0.1862	0.1180
0.01	200	0.1332	0.0122
0.005	400	0.1132	0.0062
0.0025	800	0.0683	0.0062

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$ N_{\mathcal{R}} = N_U $	$U=B_1(0)$, $\mathcal{S}=B_{0.25}(0)$	$U=[-1, 1]^2$, $\mathcal{S}=B_{0.25}(0)$	$U=[-1, 1]^2$, $\mathcal{S}=\left\{ 0 \right\}$
0.04	50	0.2951	0.2265
0.02	100	0.1862	0.1180
0.01	200	0.1332	0.0122
0.005	400	0.1132	0.0062
0.0025	800	0.0683	0.0062

Table 2. Error estimates for Ex. 5.2 a) for combination methods of order 1 and 2

h	$N_{\mathcal{R}}$	Euler scheme & Riemann sum	Heun's scheme & trapezoid rule
0.04	50	0.2951	0.2265
0.02	100	0.1862	0.1180
0.01	200	0.1332	0.0122
0.005	400	0.1132	0.0062
0.0025	800	0.0683	0.0062

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h	$N_{\mathcal{R}}$	Euler scheme & Riemann sum	Heun's scheme & trapezoid rule
0.04	50	0.2951	0.2265
0.02	100	0.1862	0.1180
0.01	200	0.1332	0.0122
0.005	400	0.1132	0.0062
0.0025	800	0.0683	0.0062

Table 3. Error estimates for Ex. 5.2 a) for Runge-Kutta meth. of order 1 and 2

h	$N_{\mathcal{R}}$	set-valued Euler method	set-valued Heun method
0.04	50	0.2330	0.2265
0.02	100	0.1681	0.1180
0.01	200	0.1149	0.0122
0.005	400	0.0753	0.0062
0.0025	800	0.0318	0.0062

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h	$N_{\mathcal{R}}$	set-valued Euler method	set-valued Heun method
0.04	50	0.2330	0.2265
0.02	100	0.1681	0.1180
0.01	200	0.1149	0.0122
0.005	400	0.0753	0.0062
0.0025	800	0.0318	0.0062

Table 4. Error estimates for Example 5.3 for methods of order 1 and 2

h	Euler scheme & Riemann sum	Heun's scheme & trapezoid rule
0.05	0.170	0.1153
0.025	0.095	0.0470
0.0125	0.0599	0.0133
0.00625	0.0285	0.0032

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h	Euler scheme & Riemann sum	Heun's scheme & trapezoid rule
0.05	0.170	0.1153
0.025	0.095	0.0470
0.0125	0.0599	0.0133
0.00625	0.0285	0.0032

Table 5. Error estimates for Example 5.5 with set-valued methods of order 1 and 2

h	$N_{\mathcal{R}}$	set-valued Euler scheme	set-valued Heun's scheme
0.5	50	0.0848	0.1461
0.1	100	0.0060	0.0076
0.05	200	0.0015	0.0020
0.025	400	0.00042	0.000502
0.0125	800	0.000108	0.000126

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h	$N_{\mathcal{R}}$	set-valued Euler scheme	set-valued Heun's scheme
0.5	50	0.0848	0.1461
0.1	100	0.0060	0.0076
0.05	200	0.0015	0.0020
0.025	400	0.00042	0.000502
0.0125	800	0.000108	0.000126

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American Institute of Mathematical Sciences