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doi: 10.3934/naco.2021002

Direct method to solve linear-quadratic optimal control problems

Mohamed Aliane 1,, , Mohand Bentobache 1, , Nacima Moussouni 2, and  Philippe Marthon 3,

1. 

Laboratory of Pure and Applied Mathematics, University of Laghouat, Bp 37G, Ghardaia road, 03000, Laghouat, Algeria
2. 

Laboratory L2CSP, University of Tizi-Ouzou, 15000, Tizi-Ouzou, Algeria
3. 

IRIT-ENSEEIHT, Université Fédérale Toulouse-Midi-Pyrénées, France

* Corresponding author: m.aliane@lagh-univ.dz

Received  April 2020 Revised  December 2020 Published  January 2021

In this work, we have proposed a new approach for solving the linear-quadratic optimal control problem, where the quality criterion is a quadratic function, which can be convex or non-convex. In this approach, we transform the continuous optimal control problem into a quadratic optimization problem using the Cauchy discretization technique, then we solve it with the active-set method. In order to study the efficiency and the accuracy of the proposed approach, we developed an implementation with MATLAB, and we performed numerical experiments on several convex and non-convex linear-quadratic optimal control problems. The obtained simulation results show that our method is more accurate and more efficient than the method using the classical Euler discretization technique. Furthermore, it was shown that our method fastly converges to the optimal control of the continuous problem found analytically using the Pontryagin's maximum principle.

Keywords: Linear-quadratic optimal control, Cauchy discretization technique, quadratic optimization problem, active-set method, numerical simulation.
Mathematics Subject Classification: Primary: 93C05; Secondary: 53C35.
Citation: Mohamed Aliane, Mohand Bentobache, Nacima Moussouni, Philippe Marthon. Direct method to solve linear-quadratic optimal control problems. Numerical Algebra, Control & Optimization, doi: 10.3934/naco.2021002
References:
[1]

J. Awerjcewicz, Modeling, Simulation and Control of Non-Linear Engineering Dynamical Systems, State-of-the-Art, Presperctives and Applications, Springer, Heidelberg, Germany, 2008. doi: 10.1007/978-3-319-08266-0.  Google Scholar

[2]

M. Aliane, N. Moussouni and M. Bentobache, Optimal control of a rectilinear motion of a rocket, Statistics, Optimization & Information Computing, 8 (2020), 281-295.  doi: 10.19139/soic-2310-5070-741.  Google Scholar

[3]

M. Aliane, N. Moussouni and M. Bentobache, Non-linear optimal control of the heel angle of a rocket, The 6th International Conference on Control, Decision and Information Technologies, CODIT'19, Paris, France, (2019), 756–760. doi: 10.19139/soic-2310-5070-741.  Google Scholar

[4]

M. Bentobache, M. Telli and A. Mokhtari, A sequential linear programming algorithm for continuous and mixed-integer non-convex quadratic programming, Optimization of Complex Systems: Theory, Models, Algorithms and Applications, WCGO 2019, Advances in Intelligent Systems and Computing, Springer, Cham, 991 (2020), 26–36. Google Scholar

[5]

M. O. Bibi and M. Bentobache, A hybrid direction algorithm for solving linear programs, International Journal of Computer Mathematics, 92 (2015), 201-216.  doi: 10.1080/00207160.2014.890188.  Google Scholar

[6]

L. D. Duncan, Basic Considerations in the Development of an Unguided Rocket Trajectory Simulation Model, Technical report $N^0$ 5076, Atmospheric Sciences Laboratory, United States Army Electronics Command, 1966. Google Scholar

[7]

A. Chinchuluun, R. Enkhbat and P. M. Pardalos, A novel approach for non-convex optimal control problems, Optimization, 58 (2009), 781-789.  doi: 10.1080/02331930902943962.  Google Scholar

[8]

R. Gabasov, F. M. Kirillova and S. V. Prischepova, Optimal Feedback Control, Springer-Verlag, London, 1995.  Google Scholar

[9]

G. R. Rose, Numerical Methods for Solving Optimal Control Problems, Master's Thesis, University of Tennessee, Knoxville, 2015. Google Scholar

[10]

P. Howlett, The optimal control of a train, Annals of Operations Research, 98 (2000), 65-87.  doi: 10.1023/A:1019235819716.  Google Scholar

[11]

N. Khimoum and M. O. Bibi, Primal-dual method for solving a linear-quadratic multi-input optimal control problem, Optimization Letters, 14 (2019), 653-669.  doi: 10.1007/s11590-018-1375-2.  Google Scholar

[12]

K. Louadj, P. Spiteri, F. Demim, M. Aidene, A. Nemra and F. Messine, Application optimal control for a problem aircraft flight, Engineering Science and Technology Review, 11 (2018), 156-164.   Google Scholar

[13]

N. Moussouni and M. Aidene, An algorithm for optimization of cereal output, Acta Applicandae Mathematicae, 11 (2011), 113-127.  doi: 10.1007/s10440-011-9664-0.  Google Scholar

[14]

N. Moussouni and M. Aidene, Optimization of cereal output in presence of locusts, An International Journal of Optimization and Control: Theories & Applications, 6 (2016), 1-10.  doi: 10.11121/ijocta.01.2016.00254.  Google Scholar

[15]

O. Oukacha, Direct Method for the Optimization of Control Problems, PhD Dissertation, University of Tizi-Ouzou, Algeria, 2016 (in French). Google Scholar

[16]

L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes, Intersciences Publisher, New York, 1962.  Google Scholar

[17]

E. Trélat, Optimal Control: Theory and Applications, Vuibert, Paris, France, 2005 (in French).  Google Scholar

[18]

J. Zhu and R. Zeng, A mathematical formulation for optimal control of air pollution, Science in China, 46 (2003), 994-1002.   Google Scholar

[19]

M. A. Zaitri, M. O. Bibi and M. Bentobache, A hybrid direction algorithm for solving optimal control problems, Cogent Mathematics & Statistics, 6 (2019), 1-12.  doi: 10.1080/25742558.2019.1612614.  Google Scholar

show all references

References:
[1]

J. Awerjcewicz, Modeling, Simulation and Control of Non-Linear Engineering Dynamical Systems, State-of-the-Art, Presperctives and Applications, Springer, Heidelberg, Germany, 2008. doi: 10.1007/978-3-319-08266-0.  Google Scholar

[2]

M. Aliane, N. Moussouni and M. Bentobache, Optimal control of a rectilinear motion of a rocket, Statistics, Optimization & Information Computing, 8 (2020), 281-295.  doi: 10.19139/soic-2310-5070-741.  Google Scholar

[3]

M. Aliane, N. Moussouni and M. Bentobache, Non-linear optimal control of the heel angle of a rocket, The 6th International Conference on Control, Decision and Information Technologies, CODIT'19, Paris, France, (2019), 756–760. doi: 10.19139/soic-2310-5070-741.  Google Scholar

[4]

M. Bentobache, M. Telli and A. Mokhtari, A sequential linear programming algorithm for continuous and mixed-integer non-convex quadratic programming, Optimization of Complex Systems: Theory, Models, Algorithms and Applications, WCGO 2019, Advances in Intelligent Systems and Computing, Springer, Cham, 991 (2020), 26–36. Google Scholar

[5]

M. O. Bibi and M. Bentobache, A hybrid direction algorithm for solving linear programs, International Journal of Computer Mathematics, 92 (2015), 201-216.  doi: 10.1080/00207160.2014.890188.  Google Scholar

[6]

L. D. Duncan, Basic Considerations in the Development of an Unguided Rocket Trajectory Simulation Model, Technical report $N^0$ 5076, Atmospheric Sciences Laboratory, United States Army Electronics Command, 1966. Google Scholar

[7]

A. Chinchuluun, R. Enkhbat and P. M. Pardalos, A novel approach for non-convex optimal control problems, Optimization, 58 (2009), 781-789.  doi: 10.1080/02331930902943962.  Google Scholar

[8]

R. Gabasov, F. M. Kirillova and S. V. Prischepova, Optimal Feedback Control, Springer-Verlag, London, 1995.  Google Scholar

[9]

G. R. Rose, Numerical Methods for Solving Optimal Control Problems, Master's Thesis, University of Tennessee, Knoxville, 2015. Google Scholar

[10]

P. Howlett, The optimal control of a train, Annals of Operations Research, 98 (2000), 65-87.  doi: 10.1023/A:1019235819716.  Google Scholar

[11]

N. Khimoum and M. O. Bibi, Primal-dual method for solving a linear-quadratic multi-input optimal control problem, Optimization Letters, 14 (2019), 653-669.  doi: 10.1007/s11590-018-1375-2.  Google Scholar

[12]

K. Louadj, P. Spiteri, F. Demim, M. Aidene, A. Nemra and F. Messine, Application optimal control for a problem aircraft flight, Engineering Science and Technology Review, 11 (2018), 156-164.   Google Scholar

[13]

N. Moussouni and M. Aidene, An algorithm for optimization of cereal output, Acta Applicandae Mathematicae, 11 (2011), 113-127.  doi: 10.1007/s10440-011-9664-0.  Google Scholar

[14]

N. Moussouni and M. Aidene, Optimization of cereal output in presence of locusts, An International Journal of Optimization and Control: Theories & Applications, 6 (2016), 1-10.  doi: 10.11121/ijocta.01.2016.00254.  Google Scholar

[15]

O. Oukacha, Direct Method for the Optimization of Control Problems, PhD Dissertation, University of Tizi-Ouzou, Algeria, 2016 (in French). Google Scholar

[16]

L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes, Intersciences Publisher, New York, 1962.  Google Scholar

[17]

E. Trélat, Optimal Control: Theory and Applications, Vuibert, Paris, France, 2005 (in French).  Google Scholar

[18]

J. Zhu and R. Zeng, A mathematical formulation for optimal control of air pollution, Science in China, 46 (2003), 994-1002.   Google Scholar

[19]

M. A. Zaitri, M. O. Bibi and M. Bentobache, A hybrid direction algorithm for solving optimal control problems, Cogent Mathematics & Statistics, 6 (2019), 1-12.  doi: 10.1080/25742558.2019.1612614.  Google Scholar

Figure 1.  Optimal control $ t\longmapsto u(t) $ for the problem of Example 1
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Figure 2.  CPU time and Error in terms of $ N $ for Example 1
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Figure 3.  CPU time and Error in terms of $ N $ for Example 2
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Figure 4.  CPU time and Error in terms of $ N $ for Example 3
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Figure 5.  CPU time and Error in terms of $ N $ for Example 4
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Table 1.  Numerical simulation results of EDM for Example 1
EDM(SQP) EDM(ASM)
N J(u) CPU Error J(u) CPU Error
4 0.953125 0.45 3.73E+00 0.953125 1.03 3.73E+00
10 2.588281 0.09 2.09E+00 2.588281 0.12 2.09E+00
30 3.865982 0.15 8.14E-01 3.865982 0.25 8.14E-01
50 4.176861 0.43 5.03E-01 4.176861 0.54 5.03E-01
70 4.316335 0.91 3.64E-01 4.316335 1.58 3.64E-01
100 4.422900 1.99 2.57E-01 4.422900 2.48 2.57E-01
200 4.549882 10.30 1.30E-01 4.549697 12.45 1.30E-01
400 4.614494 51.99 6.54E-02 4.614199 46.91 6.57E-02
600 4.636198 161.93 4.37E-02 4.635949 144.54 4.39E-02
800 4.647087 375.59 3.28E-02 4.646814 379.60 3.31E-02
1000 4.653632 696.03 2.63E-02 4.653362 865.84 2.65E-02
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EDM(SQP) EDM(ASM)
N J(u) CPU Error J(u) CPU Error
4 0.953125 0.45 3.73E+00 0.953125 1.03 3.73E+00
10 2.588281 0.09 2.09E+00 2.588281 0.12 2.09E+00
30 3.865982 0.15 8.14E-01 3.865982 0.25 8.14E-01
50 4.176861 0.43 5.03E-01 4.176861 0.54 5.03E-01
70 4.316335 0.91 3.64E-01 4.316335 1.58 3.64E-01
100 4.422900 1.99 2.57E-01 4.422900 2.48 2.57E-01
200 4.549882 10.30 1.30E-01 4.549697 12.45 1.30E-01
400 4.614494 51.99 6.54E-02 4.614199 46.91 6.57E-02
600 4.636198 161.93 4.37E-02 4.635949 144.54 4.39E-02
800 4.647087 375.59 3.28E-02 4.646814 379.60 3.31E-02
1000 4.653632 696.03 2.63E-02 4.653362 865.84 2.65E-02
Table 2.  Numerical simulation results of CDA for Example 1
CDA1 CDA2
N J(u) CPU Error J(u) CPU Error
4 4.494898 0.01 1.85E-01 4.494898 0.05 1.85E-01
10 4.658000 0.01 2.19E-02 4.658000 0.07 2.19E-02
30 4.674355 0.03 5.54E-03 4.674355 0.21 5.54E-03
50 4.677464 0.09 2.43E-03 4.677464 0.45 2.43E-03
70 4.678780 0.11 1.11E-03 4.678780 0.54 1.11E-03
100 4.679761 0.25 1.30E-04 4.679761 0.81 1.30E-04
200 4.679830 0.96 6.13E-05 4.679830 8.37 6.13E-05
400 4.679865 10.73 2.67E-05 4.679865 10.08 2.67E-05
600 4.679876 31.06 1.52E-05 4.679876 29.00 1.52E-05
800 4.679882 72.20 9.43E-06 4.679882 80.56 9.43E-06
1000 4.679886 160.93 5.97E-06 4.679886 178.74 5.97E-06
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CDA1 CDA2
N J(u) CPU Error J(u) CPU Error
4 4.494898 0.01 1.85E-01 4.494898 0.05 1.85E-01
10 4.658000 0.01 2.19E-02 4.658000 0.07 2.19E-02
30 4.674355 0.03 5.54E-03 4.674355 0.21 5.54E-03
50 4.677464 0.09 2.43E-03 4.677464 0.45 2.43E-03
70 4.678780 0.11 1.11E-03 4.678780 0.54 1.11E-03
100 4.679761 0.25 1.30E-04 4.679761 0.81 1.30E-04
200 4.679830 0.96 6.13E-05 4.679830 8.37 6.13E-05
400 4.679865 10.73 2.67E-05 4.679865 10.08 2.67E-05
600 4.679876 31.06 1.52E-05 4.679876 29.00 1.52E-05
800 4.679882 72.20 9.43E-06 4.679882 80.56 9.43E-06
1000 4.679886 160.93 5.97E-06 4.679886 178.74 5.97E-06
Table 3.  Numerical simulation results of EDM for Example 2
EDM(SQP) EDM(ASM)
N J(u) CPU Error J(u) CPU Error
4 -807.000006 0.42 6.33E+02 -807.000000 0.78 6.33E+02
10 -6.020544 0.04 1.68E+02 -6.021052 0.07 1.68E+02
30 -180.145998 0.17 6.15E+00 -180.145998 0.24 6.15E+00
50 -187.685200 0.74 1.37E+01 -187.685200 0.66 1.37E+01
70 -187.041677 1.80 1.30E+01 -187.041677 1.85 1.30E+01
100 -184.758275 5.04 1.08E+01 -184.758275 4.92 1.08E+01
200 -180.485036 37.17 6.49E+00 -180.485036 36.75 6.49E+00
400 -177.524433 293.36 3.52E+00 -177.523183 324.39 3.52E+00
600 -176.412810 1075.55 2.41E+00 -176.411448 1284.80 2.41E+00
800 -175.833379 2692.90 1.83E+00 -175.831972 3712.96 1.83E+00
1000 -175.478114 5611.01 1.48E+00 -175.477202 8598.05 1.48E+00
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EDM(SQP) EDM(ASM)
N J(u) CPU Error J(u) CPU Error
4 -807.000006 0.42 6.33E+02 -807.000000 0.78 6.33E+02
10 -6.020544 0.04 1.68E+02 -6.021052 0.07 1.68E+02
30 -180.145998 0.17 6.15E+00 -180.145998 0.24 6.15E+00
50 -187.685200 0.74 1.37E+01 -187.685200 0.66 1.37E+01
70 -187.041677 1.80 1.30E+01 -187.041677 1.85 1.30E+01
100 -184.758275 5.04 1.08E+01 -184.758275 4.92 1.08E+01
200 -180.485036 37.17 6.49E+00 -180.485036 36.75 6.49E+00
400 -177.524433 293.36 3.52E+00 -177.523183 324.39 3.52E+00
600 -176.412810 1075.55 2.41E+00 -176.411448 1284.80 2.41E+00
800 -175.833379 2692.90 1.83E+00 -175.831972 3712.96 1.83E+00
1000 -175.478114 5611.01 1.48E+00 -175.477202 8598.05 1.48E+00
Table 4.  Numerical simulation results of CDA for Example 2
CDA1 CDA2
N J(u) CPU Error J(u) CPU Error
4 -174.000000 0.29 9.95E-13 -174.000000 0.56 0.00E+00
10 -161.832000 0.02 1.22E+01 -161.832000 0.11 1.22E+01
30 -172.676444 0.07 1.32E+00 -172.676444 0.33 1.32E+00
50 -173.524339 0.19 4.76E-01 -173.524339 0.61 4.76E-01
70 -173.757431 0.44 2.43E-01 -173.757431 0.98 2.43E-01
100 -174.000000 0.98 3.01E-12 -174.000000 1.81 5.00E-12
200 -174.000000 8.02 4.01E-12 -174.000000 10.29 6.00E-12
400 -174.000000 83.60 3.00E-11 -174.000000 90.35 9.00E-11
600 -174.000000 354.27 2.10E-11 -174.000000 359.11 6.40E-11
800 -174.000000 996.67 6.20E-11 -174.000000 1045.79 1.56E-10
1000 -174.000000 2354.52 3.01E-12 -174.000000 2320.88 4.01E-12
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CDA1 CDA2
N J(u) CPU Error J(u) CPU Error
4 -174.000000 0.29 9.95E-13 -174.000000 0.56 0.00E+00
10 -161.832000 0.02 1.22E+01 -161.832000 0.11 1.22E+01
30 -172.676444 0.07 1.32E+00 -172.676444 0.33 1.32E+00
50 -173.524339 0.19 4.76E-01 -173.524339 0.61 4.76E-01
70 -173.757431 0.44 2.43E-01 -173.757431 0.98 2.43E-01
100 -174.000000 0.98 3.01E-12 -174.000000 1.81 5.00E-12
200 -174.000000 8.02 4.01E-12 -174.000000 10.29 6.00E-12
400 -174.000000 83.60 3.00E-11 -174.000000 90.35 9.00E-11
600 -174.000000 354.27 2.10E-11 -174.000000 359.11 6.40E-11
800 -174.000000 996.67 6.20E-11 -174.000000 1045.79 1.56E-10
1000 -174.000000 2354.52 3.01E-12 -174.000000 2320.88 4.01E-12
Table 5.  Numerical simulation results of EDM for Example 3
EDM(SQP) EDM(ASM)
N J(u) CPU Error J(u) CPU Error
4 32.000000 0.45 4.20E+01 32.000000 0.85 4.20E+01
10 53.422400 0.49 2.06E+01 53.422400 0.93 2.06E+01
30 66.485116 0.54 7.51E+00 66.485116 1.02 7.51E+00
50 69.407818 0.66 4.59E+00 69.407818 1.22 4.59E+00
70 70.694043 0.88 3.31E+00 70.694043 1.51 3.31E+00
100 71.672177 1.26 2.33E+00 71.672177 2.01 2.33E+00
200 72.828072 2.84 1.17E+00 72.828072 4.14 1.17E+00
400 73.412022 12.25 5.88E-01 73.412022 16.85 5.88E-01
600 73.607566 42.80 3.92E-01 73.607566 56.17 3.92E-01
800 73.705506 120.33 2.94E-01 73.705506 154.07 2.94E-01
1000 73.764324 284.73 2.36E-01 73.764324 360.15 2.36E-01
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EDM(SQP) EDM(ASM)
N J(u) CPU Error J(u) CPU Error
4 32.000000 0.45 4.20E+01 32.000000 0.85 4.20E+01
10 53.422400 0.49 2.06E+01 53.422400 0.93 2.06E+01
30 66.485116 0.54 7.51E+00 66.485116 1.02 7.51E+00
50 69.407818 0.66 4.59E+00 69.407818 1.22 4.59E+00
70 70.694043 0.88 3.31E+00 70.694043 1.51 3.31E+00
100 71.672177 1.26 2.33E+00 71.672177 2.01 2.33E+00
200 72.828072 2.84 1.17E+00 72.828072 4.14 1.17E+00
400 73.412022 12.25 5.88E-01 73.412022 16.85 5.88E-01
600 73.607566 42.80 3.92E-01 73.607566 56.17 3.92E-01
800 73.705506 120.33 2.94E-01 73.705506 154.07 2.94E-01
1000 73.764324 284.73 2.36E-01 73.764324 360.15 2.36E-01
Table 6.  Numerical simulation results of CDA for Example 3
CDA1 CDA2
N J(u) CPU Error J(u) CPU Error
4 74.000000 0.01 0.00E+00 74.000000 0.05 0.00E+00
10 74.000000 0.02 0.00E+00 74.000000 0.07 0.00E+00
30 74.000000 0.03 0.00E+00 74.000000 0.21 0.00E+00
50 74.000000 0.08 0.00E+00 74.000000 0.38 0.00E+00
70 74.000000 0.11 0.00E+00 74.000000 0.54 9.95E-14
100 74.000000 0.18 0.00E+00 74.000000 0.93 0.00E+00
200 74.000000 0.92 0.00E+00 74.000000 2.51 0.00E+00
400 74.000000 6.55 2.98E-13 74.000000 9.12 3.98E-13
600 74.000000 24.48 1.99E-13 74.000000 28.43 3.98E-13
800 74.000000 65.34 6.96E-13 74.000000 71.64 6.96E-13
1000 74.000000 144.16 0.00E+00 74.000000 152.09 0.00E+00
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CDA1 CDA2
N J(u) CPU Error J(u) CPU Error
4 74.000000 0.01 0.00E+00 74.000000 0.05 0.00E+00
10 74.000000 0.02 0.00E+00 74.000000 0.07 0.00E+00
30 74.000000 0.03 0.00E+00 74.000000 0.21 0.00E+00
50 74.000000 0.08 0.00E+00 74.000000 0.38 0.00E+00
70 74.000000 0.11 0.00E+00 74.000000 0.54 9.95E-14
100 74.000000 0.18 0.00E+00 74.000000 0.93 0.00E+00
200 74.000000 0.92 0.00E+00 74.000000 2.51 0.00E+00
400 74.000000 6.55 2.98E-13 74.000000 9.12 3.98E-13
600 74.000000 24.48 1.99E-13 74.000000 28.43 3.98E-13
800 74.000000 65.34 6.96E-13 74.000000 71.64 6.96E-13
1000 74.000000 144.16 0.00E+00 74.000000 152.09 0.00E+00
Table 7.  Numerical simulation results of EDM for Example 4
EDM(SQP) EDM(ASM)
N J(u) CPU Error J(u) CPU Error
4 0.171875 0.46 1.81E+00 0.171875 1.03 1.81E+00
10 0.922969 0.07 1.06E+00 0.922969 0.08 1.06E+00
30 1.560700 0.17 4.17E-01 1.560700 0.16 4.17E-01
50 1.719317 0.60 2.59E-01 1.719317 0.53 2.59E-01
70 1.790821 1.12 1.87E-01 1.790821 1.00 1.87E-01
100 1.845588 2.58 1.33E-01 1.844967 2.06 1.33E-01
200 1.910993 12.49 6.71E-02 1.910780 10.75 6.73E-02
400 1.944331 60.91 3.38E-02 1.944171 61.13 3.39E-02
600 1.955538 187.13 2.26E-02 1.955084 176.01 2.30E-02
800 1.961162 438.55 1.70E-02 1.960852 530.81 1.73E-02
1000 1.964543 820.38 1.36E-02 1.964187 1180.43 1.39E-02
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EDM(SQP) EDM(ASM)
N J(u) CPU Error J(u) CPU Error
4 0.171875 0.46 1.81E+00 0.171875 1.03 1.81E+00
10 0.922969 0.07 1.06E+00 0.922969 0.08 1.06E+00
30 1.560700 0.17 4.17E-01 1.560700 0.16 4.17E-01
50 1.719317 0.60 2.59E-01 1.719317 0.53 2.59E-01
70 1.790821 1.12 1.87E-01 1.790821 1.00 1.87E-01
100 1.845588 2.58 1.33E-01 1.844967 2.06 1.33E-01
200 1.910993 12.49 6.71E-02 1.910780 10.75 6.73E-02
400 1.944331 60.91 3.38E-02 1.944171 61.13 3.39E-02
600 1.955538 187.13 2.26E-02 1.955084 176.01 2.30E-02
800 1.961162 438.55 1.70E-02 1.960852 530.81 1.73E-02
1000 1.964543 820.38 1.36E-02 1.964187 1180.43 1.39E-02
Table 8.  Numerical simulation results of CDA for Example 4
CDA1 CDA2
N J(u) CPU Error J(u) CPU Error
4 1.882653 0.01 9.55E-02 1.882653 0.06 9.55E-02
10 1.966800 0.01 1.13E-02 1.966800 0.07 1.13E-02
30 1.975251 0.03 2.86E-03 1.975251 0.21 2.86E-03
50 1.976858 0.08 1.25E-03 1.976858 0.36 1.25E-03
70 1.977538 0.11 5.74E-04 1.977538 0.58 5.74E-04
100 1.978046 0.23 6.72E-05 1.978046 0.80 6.72E-05
200 1.978081 1.05 3.17E-05 1.978081 2.92 3.17E-05
400 1.978099 10.23 1.38E-05 1.978099 10.02 1.38E-05
600 1.978105 30.87 7.86E-06 1.978105 28.47 7.86E-06
800 1.978108 73.76 4.88E-06 1.978108 79.62 4.88E-06
1000 1.978110 157.80 3.09E-06 1.978110 177.02 3.09E-06
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CDA1 CDA2
N J(u) CPU Error J(u) CPU Error
4 1.882653 0.01 9.55E-02 1.882653 0.06 9.55E-02
10 1.966800 0.01 1.13E-02 1.966800 0.07 1.13E-02
30 1.975251 0.03 2.86E-03 1.975251 0.21 2.86E-03
50 1.976858 0.08 1.25E-03 1.976858 0.36 1.25E-03
70 1.977538 0.11 5.74E-04 1.977538 0.58 5.74E-04
100 1.978046 0.23 6.72E-05 1.978046 0.80 6.72E-05
200 1.978081 1.05 3.17E-05 1.978081 2.92 3.17E-05
400 1.978099 10.23 1.38E-05 1.978099 10.02 1.38E-05
600 1.978105 30.87 7.86E-06 1.978105 28.47 7.86E-06
800 1.978108 73.76 4.88E-06 1.978108 79.62 4.88E-06
1000 1.978110 157.80 3.09E-06 1.978110 177.02 3.09E-06
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