Terrain constrained path planning for long-haul cables

Zengfu Wang; Qing Wang; Bill Moran; Moshe Zukerman

doi:10.1364/OE.27.008221

Optics Express
Vol. 27,
Issue 6,
pp. 8221-8235
(2019)
•https://doi.org/10.1364/OE.27.008221

Terrain constrained path planning for long-haul cables

Zengfu Wang, Qing Wang, Bill Moran, and Moshe Zukerman

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Zengfu Wang, Qing Wang, Bill Moran, and Moshe Zukerman, "Terrain constrained path planning for long-haul cables," Opt. Express 27, 8221-8235 (2019)

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Abstract

A method for path planning for a long-haul submarine optical fiber cable connecting two locations on the surface of the Earth is presented. Previous work on path planning takes account of the laying cost of the cable including material, labor, and its survivability, with consideration of risk of future cable break arising from laying of the cable in sensitive and risky areas, such as, in particular, earthquake prone areas. Previous work has also taken account of variation in the cost per unit length to optimize shielding (and associated increased costs) in higher risk areas. The key novelty here is to take account of the important requirement to reduce the likelihood of capsize of a remotely operated cable laying vehicle as it buries the cable in an uneven terrain. This instability risk depends on the direction of the path and slope of the terrain and is included here in the laying cost. Minimization of the cable laying cost and the expected number of potential cable repairs are the two objectives used to formulate the multi-objective optimal control problem. Using a Pareto approach, we solve the problem via dynamic programming and a computationally efficient algorithm based on the Ordered Upwind Method. Numerical results are consistent with an intuitive assessment of path quality, e.g., we can observe that the algorithm avoids high slope areas when better solutions are clearly available. Pareto optimal solutions and an approximate Pareto front are obtained to provide insight and guidance for cable path design that considers trade-offs between cost effectiveness (that includes consideration for stability of the remotely operated cable laying vehicle) and seismic resilience.

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References

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Year
Author
Publication

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[Crossref]
M. Zhao, T. W. S. Chow, P. Tang, Z. Wang, J. Guo, and M. Zukerman, “Route selection for cabling considering cost minimization and earthquake survivability via a semi-supervised probabilistic model,” IEEE Trans. Ind. Informat. 13(2), 502–511 (2017).
[Crossref]
Z. Wang, Q. Wang, M. Zukerman, J. Guo, Y. Wang, G. Wang, J. Yang, and B. Moran, “Multiobjective path optimization for critical infrastructure links with consideration to seismic resilience,” Comput-Aided. Civ. Inf. 32(10), 836–855 (2017).
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Z. Wang, Q. Wang, B. Moran, and M. Zukerman, “Application of the fast marching method for path planning of long-haul optical fiber cables with shielding,” IEEE Access 6(1), 41367–41378 (2018).
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J. Zhang, E. Modiano, and D. Hay, “Enhancing network robustness via shielding,” IEEE/ACM Trans. Netw. 25(4), 2209–2222 (2017).
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J. A. Sethian and A. Vladimirsky, “Ordered upwind methods for static Hamilton–Jacobi equations: Theory and algorithms,” SIAM J. Numer. Anal. 41(1), 325–363 (2003).
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W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing (Cambridge University, 1992).
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[Crossref]

2018 (2)

S. Huang, E. Chen, and J. Guo, “Efficient seafloor classification and submarine cable route design using an autonomous underwater vehicle,” IEEE J. Oceanic Eng. 43(1), 7–18 (2018).
[Crossref]

Z. Wang, Q. Wang, B. Moran, and M. Zukerman, “Application of the fast marching method for path planning of long-haul optical fiber cables with shielding,” IEEE Access 6(1), 41367–41378 (2018).
[Crossref]

2017 (4)

J. Zhang, E. Modiano, and D. Hay, “Enhancing network robustness via shielding,” IEEE/ACM Trans. Netw. 25(4), 2209–2222 (2017).
[Crossref]

M. Zhao, T. W. S. Chow, P. Tang, Z. Wang, J. Guo, and M. Zukerman, “Route selection for cabling considering cost minimization and earthquake survivability via a semi-supervised probabilistic model,” IEEE Trans. Ind. Informat. 13(2), 502–511 (2017).
[Crossref]

Z. Wang, Q. Wang, M. Zukerman, J. Guo, Y. Wang, G. Wang, J. Yang, and B. Moran, “Multiobjective path optimization for critical infrastructure links with consideration to seismic resilience,” Comput-Aided. Civ. Inf. 32(10), 836–855 (2017).
[Crossref]

Z. Wang, Q. Wang, M. Zukerman, and B. Moran, “A seismic resistant design algorithm for laying and shielding of optical fiber cables,” J. Lightwave Technol. 35(14), 3060–3074 (2017).
[Crossref]

2016 (5)

J. Hecht, “The bandwidth bottleneck,” Nature 536(7615), 139–142 (2016).
[Crossref] [PubMed]

C. Cao, Z. Wang, M. Zukerman, J. H. Manton, A. Bensoussan, and Y. Wang, “Optimal cable laying across an earthquake fault line considering elliptical failures,” IEEE Trans. Rel. 65(3), 1536–1550 (2016).
[Crossref]

P. N. Tran and H. Saito, “Geographical route design of physical networks using earthquake risk information,” IEEE Commun. Mag. 54(7), 131–137 (2016).
[Crossref]

P. N. Tran and H. Saito, “Enhancing physical network robustness against earthquake disasters with additional links,” J. Lightwave Technol. 34(22), 5226–5238 (2016).
[Crossref]

D. L. Msongaleli, F. Dikbiyik, M. Zukerman, and B. Mukherjee, “Disaster-aware submarine fiber-optic cable deployment for mesh networks,” J. Lightwave Technol. 34(18), 4293–4303 (2016).
[Crossref]

2014 (1)

H. Saito, “Analysis of geometric disaster evaluation model for physical networks,” IEEE/ACM Trans. Netw. 23(6), 1777–1789 (2014).
[Crossref]

2013 (1)

C. Cao, M. Zukerman, W. Wu, J. H. Manton, and B. Moran, “Survivable topology design of submarine networks,” J. Lightwave Technol. 31(5), 715–730 (2013).
[Crossref]

2003 (1)

J. A. Sethian and A. Vladimirsky, “Ordered upwind methods for static Hamilton–Jacobi equations: Theory and algorithms,” SIAM J. Numer. Anal. 41(1), 325–363 (2003).
[Crossref]

2001 (1)

J. A. Sethian and A. Vladimirsky, “Ordered upwind methods for static Hamilton–Jacobi equations,” PNAS. 98(20), 11069–11074 (2001).
[Crossref] [PubMed]

2000 (1)

S. Smale, “Global analysis and economics I: Pareto optimum and a generalization of Morse theory,” The Collected Papers of Stephen Smale 1, 259–270 (2000).
[Crossref]

1999 (1)

D. J. Wald, “Relationships between peak ground acceleration, peak ground velocity, and modified Mercalli intensity in California,” Earthq. Spectra. 15(3), 557–564 (1999).
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1983 (1)

M. G. Crandall and P-L. Lions, “Viscosity solutions of Hamilton-Jacobi equations,” Trans. Am. Math. Soc. 277(1), 1–42 (1983).
[Crossref]

Ash, S.

A. McCurdy, A. Shelton, B. Burns, C. Bayly, E. Handa, G. Gerstell, J.-M. Fortain, J.-P. Joensuu, L. Duvernay, R. Lingle, and S. Ash, “Submarine telecoms industry report,” Submarine Telecoms Forum, Inc., Tech. Rep., (2018).

Bardi, M.

M. Bardi and I. Capuzzo-Dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations (Springer Science & Business Media, 2008).

Bartlett-McNeil, D.

L. Carter, D. Burnett, S. Drew, G. Marle, L. Hagadorn, D. Bartlett-McNeil, and N. Irvine, Submarine cables and the oceans: connecting the world (UNEP-WCMC Biodiversity Series No.31. ICPC/UNEP/UNEP-WCMC, 2009).

Bayly, C.

Beckman, R.

D. R. Burnett, R. Beckman, and T. M. Davenport, Submarine Cables: The Handbook of Law and Policy (Martinus Nijhoff Publishers, 2013).
[Crossref]

Bensoussan, A.

Borwick, D.

R. Rapp, M. Lawrence, D. Borwick, and T. Kuwabara, “Marine survey & cable routing,” in Proc. SubOptic, (2004).

Burago, D.

D. Burago, Y. Burago, and S. Ivanov, A Course in Metric Geometry (American Mathematical Society Providence, 2001).

Burago, Y.

D. Burago, Y. Burago, and S. Ivanov, A Course in Metric Geometry (American Mathematical Society Providence, 2001).

Burnett, D.

Burnett, D. R.

D. R. Burnett, R. Beckman, and T. M. Davenport, Submarine Cables: The Handbook of Law and Policy (Martinus Nijhoff Publishers, 2013).
[Crossref]

Burns, B.

Cao, C.

C. Cao, M. Zukerman, W. Wu, J. H. Manton, and B. Moran, “Survivable topology design of submarine networks,” J. Lightwave Technol. 31(5), 715–730 (2013).
[Crossref]

C. Cao, “Cost effective and survivable cabling design under major disasters,” Ph.D dissertation, City University of Hong Kong, (2015).

Capuzzo-Dolcetta, I.

M. Bardi and I. Capuzzo-Dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations (Springer Science & Business Media, 2008).

Carter, L.

Chen, E.

S. Huang, E. Chen, and J. Guo, “Efficient seafloor classification and submarine cable route design using an autonomous underwater vehicle,” IEEE J. Oceanic Eng. 43(1), 7–18 (2018).
[Crossref]

Chow, T. W. S.

Crandall, M. G.

M. G. Crandall and P-L. Lions, “Viscosity solutions of Hamilton-Jacobi equations,” Trans. Am. Math. Soc. 277(1), 1–42 (1983).
[Crossref]

Davenport, T. M.

D. R. Burnett, R. Beckman, and T. M. Davenport, Submarine Cables: The Handbook of Law and Policy (Martinus Nijhoff Publishers, 2013).
[Crossref]

Dikbiyik, F.

Drew, S.

Dübendorfer, T. P.

T. P. Dübendorfer, “Impact analysis, early detection and mitigation of large-scale Internet attacks,” Ph.D dissertation, Dipl. Informatik-Ing., ETH Zürich, Zürich, Switzerland (2005).

Duvernay, L.

Eriksson, K.

K. Eriksson, D. Estep, and C. Johnson, Applied Mathematics: Body and Soul: Derivatives and Geometry in IR3 (Springer, 2004).

Estep, D.

K. Eriksson, D. Estep, and C. Johnson, Applied Mathematics: Body and Soul: Derivatives and Geometry in IR3 (Springer, 2004).

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing (Cambridge University, 1992).

Fortain, J.-M.

Gerstell, G.

Guo, J.

S. Huang, E. Chen, and J. Guo, “Efficient seafloor classification and submarine cable route design using an autonomous underwater vehicle,” IEEE J. Oceanic Eng. 43(1), 7–18 (2018).
[Crossref]

Hagadorn, L.

Handa, E.

Hay, D.

J. Zhang, E. Modiano, and D. Hay, “Enhancing network robustness via shielding,” IEEE/ACM Trans. Netw. 25(4), 2209–2222 (2017).
[Crossref]

Hecht, J.

J. Hecht, “The bandwidth bottleneck,” Nature 536(7615), 139–142 (2016).
[Crossref] [PubMed]

Huang, S.

S. Huang, E. Chen, and J. Guo, “Efficient seafloor classification and submarine cable route design using an autonomous underwater vehicle,” IEEE J. Oceanic Eng. 43(1), 7–18 (2018).
[Crossref]

Irvine, N.

Ivanov, S.

D. Burago, Y. Burago, and S. Ivanov, A Course in Metric Geometry (American Mathematical Society Providence, 2001).

Joensuu, J.-P.

Johnson, C.

K. Eriksson, D. Estep, and C. Johnson, Applied Mathematics: Body and Soul: Derivatives and Geometry in IR3 (Springer, 2004).

Kuwabara, T.

R. Rapp, M. Lawrence, D. Borwick, and T. Kuwabara, “Marine survey & cable routing,” in Proc. SubOptic, (2004).

Lawrence, M.

R. Rapp, M. Lawrence, D. Borwick, and T. Kuwabara, “Marine survey & cable routing,” in Proc. SubOptic, (2004).

Lingle, R.

Lions, P-L.

M. G. Crandall and P-L. Lions, “Viscosity solutions of Hamilton-Jacobi equations,” Trans. Am. Math. Soc. 277(1), 1–42 (1983).
[Crossref]

Manton, J. H.

C. Cao, M. Zukerman, W. Wu, J. H. Manton, and B. Moran, “Survivable topology design of submarine networks,” J. Lightwave Technol. 31(5), 715–730 (2013).
[Crossref]

W. Wu, B. Moran, J. H. Manton, and M. Zukerman, “Topology design of undersea cables considering survivability under major disasters,” in Proc. WAINA, 1154–1159 (2009).

Marle, G.

McCurdy, A.

Modiano, E.

J. Zhang, E. Modiano, and D. Hay, “Enhancing network robustness via shielding,” IEEE/ACM Trans. Netw. 25(4), 2209–2222 (2017).
[Crossref]

Moran, B.

Z. Wang, Q. Wang, M. Zukerman, and B. Moran, “A seismic resistant design algorithm for laying and shielding of optical fiber cables,” J. Lightwave Technol. 35(14), 3060–3074 (2017).
[Crossref]

C. Cao, M. Zukerman, W. Wu, J. H. Manton, and B. Moran, “Survivable topology design of submarine networks,” J. Lightwave Technol. 31(5), 715–730 (2013).
[Crossref]

W. Wu, B. Moran, J. H. Manton, and M. Zukerman, “Topology design of undersea cables considering survivability under major disasters,” in Proc. WAINA, 1154–1159 (2009).

Msongaleli, D. L.

Mukherjee, B.

Nielsen, K.

K. Nielsen, “Submarine telecoms industry report,” Submarine Telecoms Forum, Inc., Tech. Rep., (2015).

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing (Cambridge University, 1992).

Rapp, R.

R. Rapp, M. Lawrence, D. Borwick, and T. Kuwabara, “Marine survey & cable routing,” in Proc. SubOptic, (2004).

Saito, H.

P. N. Tran and H. Saito, “Geographical route design of physical networks using earthquake risk information,” IEEE Commun. Mag. 54(7), 131–137 (2016).
[Crossref]

P. N. Tran and H. Saito, “Enhancing physical network robustness against earthquake disasters with additional links,” J. Lightwave Technol. 34(22), 5226–5238 (2016).
[Crossref]

H. Saito, “Analysis of geometric disaster evaluation model for physical networks,” IEEE/ACM Trans. Netw. 23(6), 1777–1789 (2014).
[Crossref]

Sethian, J. A.

J. A. Sethian and A. Vladimirsky, “Ordered upwind methods for static Hamilton–Jacobi equations: Theory and algorithms,” SIAM J. Numer. Anal. 41(1), 325–363 (2003).
[Crossref]

J. A. Sethian and A. Vladimirsky, “Ordered upwind methods for static Hamilton–Jacobi equations,” PNAS. 98(20), 11069–11074 (2001).
[Crossref] [PubMed]

Shelton, A.

Smale, S.

S. Smale, “Global analysis and economics I: Pareto optimum and a generalization of Morse theory,” The Collected Papers of Stephen Smale 1, 259–270 (2000).
[Crossref]

Tahchi, E.

E. Tahchi and EGS (Asia) Limited, Quarry Bay, Hong Kong SAR, China (personal communication, 2018).

Tang, P.

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing (Cambridge University, 1992).

Tran, P. N.

P. N. Tran and H. Saito, “Geographical route design of physical networks using earthquake risk information,” IEEE Commun. Mag. 54(7), 131–137 (2016).
[Crossref]

P. N. Tran and H. Saito, “Enhancing physical network robustness against earthquake disasters with additional links,” J. Lightwave Technol. 34(22), 5226–5238 (2016).
[Crossref]

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing (Cambridge University, 1992).

Vladimirsky, A.

J. A. Sethian and A. Vladimirsky, “Ordered upwind methods for static Hamilton–Jacobi equations: Theory and algorithms,” SIAM J. Numer. Anal. 41(1), 325–363 (2003).
[Crossref]

J. A. Sethian and A. Vladimirsky, “Ordered upwind methods for static Hamilton–Jacobi equations,” PNAS. 98(20), 11069–11074 (2001).
[Crossref] [PubMed]

Wald, D. J.

D. J. Wald, “Relationships between peak ground acceleration, peak ground velocity, and modified Mercalli intensity in California,” Earthq. Spectra. 15(3), 557–564 (1999).
[Crossref]

Wang, G.

Wang, Q.

Z. Wang, Q. Wang, M. Zukerman, and B. Moran, “A seismic resistant design algorithm for laying and shielding of optical fiber cables,” J. Lightwave Technol. 35(14), 3060–3074 (2017).
[Crossref]

Wang, Y.

Wang, Z.

Z. Wang, Q. Wang, M. Zukerman, and B. Moran, “A seismic resistant design algorithm for laying and shielding of optical fiber cables,” J. Lightwave Technol. 35(14), 3060–3074 (2017).
[Crossref]

Wu, W.

C. Cao, M. Zukerman, W. Wu, J. H. Manton, and B. Moran, “Survivable topology design of submarine networks,” J. Lightwave Technol. 31(5), 715–730 (2013).
[Crossref]

W. Wu, B. Moran, J. H. Manton, and M. Zukerman, “Topology design of undersea cables considering survivability under major disasters,” in Proc. WAINA, 1154–1159 (2009).

Yang, J.

Zhang, J.

J. Zhang, E. Modiano, and D. Hay, “Enhancing network robustness via shielding,” IEEE/ACM Trans. Netw. 25(4), 2209–2222 (2017).
[Crossref]

Zhao, M.

Zukerman, M.

Z. Wang, Q. Wang, M. Zukerman, and B. Moran, “A seismic resistant design algorithm for laying and shielding of optical fiber cables,” J. Lightwave Technol. 35(14), 3060–3074 (2017).
[Crossref]

C. Cao, M. Zukerman, W. Wu, J. H. Manton, and B. Moran, “Survivable topology design of submarine networks,” J. Lightwave Technol. 31(5), 715–730 (2013).
[Crossref]

W. Wu, B. Moran, J. H. Manton, and M. Zukerman, “Topology design of undersea cables considering survivability under major disasters,” in Proc. WAINA, 1154–1159 (2009).

Comput-Aided. Civ. Inf. (1)

Earthq. Spectra. (1)

D. J. Wald, “Relationships between peak ground acceleration, peak ground velocity, and modified Mercalli intensity in California,” Earthq. Spectra. 15(3), 557–564 (1999).
[Crossref]

IEEE Access (1)

IEEE Commun. Mag. (1)

P. N. Tran and H. Saito, “Geographical route design of physical networks using earthquake risk information,” IEEE Commun. Mag. 54(7), 131–137 (2016).
[Crossref]

IEEE J. Oceanic Eng. (1)

S. Huang, E. Chen, and J. Guo, “Efficient seafloor classification and submarine cable route design using an autonomous underwater vehicle,” IEEE J. Oceanic Eng. 43(1), 7–18 (2018).
[Crossref]

IEEE Trans. Ind. Informat. (1)

IEEE Trans. Rel. (1)

IEEE/ACM Trans. Netw. (2)

J. Zhang, E. Modiano, and D. Hay, “Enhancing network robustness via shielding,” IEEE/ACM Trans. Netw. 25(4), 2209–2222 (2017).
[Crossref]

H. Saito, “Analysis of geometric disaster evaluation model for physical networks,” IEEE/ACM Trans. Netw. 23(6), 1777–1789 (2014).
[Crossref]

J. Lightwave Technol. (4)

P. N. Tran and H. Saito, “Enhancing physical network robustness against earthquake disasters with additional links,” J. Lightwave Technol. 34(22), 5226–5238 (2016).
[Crossref]

C. Cao, M. Zukerman, W. Wu, J. H. Manton, and B. Moran, “Survivable topology design of submarine networks,” J. Lightwave Technol. 31(5), 715–730 (2013).
[Crossref]

Z. Wang, Q. Wang, M. Zukerman, and B. Moran, “A seismic resistant design algorithm for laying and shielding of optical fiber cables,” J. Lightwave Technol. 35(14), 3060–3074 (2017).
[Crossref]

Nature (1)

J. Hecht, “The bandwidth bottleneck,” Nature 536(7615), 139–142 (2016).
[Crossref] [PubMed]

PNAS. (1)

J. A. Sethian and A. Vladimirsky, “Ordered upwind methods for static Hamilton–Jacobi equations,” PNAS. 98(20), 11069–11074 (2001).
[Crossref] [PubMed]

SIAM J. Numer. Anal. (1)

J. A. Sethian and A. Vladimirsky, “Ordered upwind methods for static Hamilton–Jacobi equations: Theory and algorithms,” SIAM J. Numer. Anal. 41(1), 325–363 (2003).
[Crossref]

The Collected Papers of Stephen Smale (1)

S. Smale, “Global analysis and economics I: Pareto optimum and a generalization of Morse theory,” The Collected Papers of Stephen Smale 1, 259–270 (2000).
[Crossref]

Trans. Am. Math. Soc. (1)

M. G. Crandall and P-L. Lions, “Viscosity solutions of Hamilton-Jacobi equations,” Trans. Am. Math. Soc. 277(1), 1–42 (1983).
[Crossref]

Other (16)

M. Bardi and I. Capuzzo-Dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations (Springer Science & Business Media, 2008).

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing (Cambridge University, 1992).

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Fig. 1 (a) Region 𝔻 in Google Earth. (b) Contour map of the region 𝔻. (c) Slope map of the region 𝔻. The curve marked by pluses represents the path obtained by the OUM-based algorithm while the curve marked by circles represents the path obtained by the FMM-based method.

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Fig. 2 (a) Region 𝔻′ in Google Earth. (b) Slope map of the region 𝔻′. (c) Logarithmic PGV map of the region 𝔻′.

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Fig. 3 Six Pareto optimal paths based on Scenario B obtained by the OUM-based algorithm. The black and magenta lines represent the path or path segments that adopt design Level 2 and design Level 1, respectively.

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Fig. 4 Approximate Pareto front.

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Tables (2)

Algorithm 1 Path planning algorithm for the region of interest 𝔻.

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Table 1 The laying cost (ℍ(γ^*, a(·), u^*(·))) and the expected number of potential cable repairs (𝔾(γ^*, u^*(·))) of the six Pareto optimal paths.

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Equations (26)

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q_{1} (x, a) = \frac{‖ \dot{z} ‖}{\sqrt{{\dot{x}}^{2} + {\dot{y}}^{2}}} .

q_{2} (x, a) = \frac{‖ {(n \times a)}_{3} ‖}{‖ {(n \times a)}_{1, 2} ‖} = \frac{| \frac{\partial z}{\partial x} \dot{y} - \frac{\partial z}{\partial y} \dot{x} |}{\sqrt{{(\frac{\partial z}{\partial y} \dot{z} + \dot{y})}^{2} + {(\frac{\partial z}{\partial x} \dot{z} + \dot{x})}^{2}}},

h_{1} (x, a) = e^{q_{1} (x, a) - θ_{1}} + e^{q_{2} (x, a) - θ_{2}},

h (x, a, u) = h_{1} (x, a) + h_{2} (x, u) .

ℍ (γ, a (\cdot), u (\cdot)) = \int_{0}^{l (γ)} h (γ (s), a (s), u (s)) d s,

𝔾 (γ, u (\cdot)) = \int_{0}^{l (γ)} g (γ (s), u (s)) d s .

\begin{array}{l} min_{a (\cdot), u (\cdot)} Φ (γ, a (\cdot), u (\cdot)), \\ s . t . γ (0) = A, γ (l (γ)) = B, \end{array}

\begin{array}{l} min_{a (\cdot), u (\cdot)} Φ^{'} (γ, a (\cdot), u (\cdot)) = \int_{0}^{l (γ)} f (γ (s), a (s), u (s)) d s, \\ s . t . γ (0) = A, γ (l (γ)) = B, \end{array}

F_{min} < f_{min} (x) \leq f (x, a, u) \leq f_{max} (x) < F_{max},

f_{min} (x) = min_{a \in 𝔸, u \in 𝕌} f (x, a, u), f_{max} (x) = max_{a \in 𝔸, u \in 𝕌} f (x, a, u) .

φ (β, a (\cdot), u (\cdot)) = \int_{0}^{l (β)} f (β (s), a (s), u (s)) d s,

φ (x) = min_{a (\cdot), u (\cdot)} φ (β, a (\cdot), u (\cdot)) = φ (β^{*}, a^{*} (\cdot), u^{*} (\cdot)),

φ (β (s)) = min_{a (\cdot), u (\cdot)} {\int_{s}^{s + t} f (β (τ), a (τ), u (τ)) d τ + φ (β (s + t))} .

φ (γ^{*} (s)) = \int_{s}^{s + t} f (γ^{*} (τ), a^{*} (τ), u^{*} (τ)) d τ + φ (γ^{*} (s + t)) .

f (γ^{*} (s), a^{*} (s), u^{*} (s)) + \frac{φ (γ^{*} (s + t)) - φ (γ^{*} (s))}{t} \approx 0 .

\begin{matrix} min_{a \in 𝔸; u \in 𝕌} {(\nabla ϕ (x) \cdot a) f (x, a, u)} = 0, x \in 𝕄, \\ φ (B) = 0, \end{matrix}

\begin{matrix} (a^{*}, u^{*}) = {arg min}_{a \in 𝔸, u \in 𝕄} {\nabla φ (γ (s)) \cdot a + f (γ (s), a, u)}, \\ γ (0) = A . \end{matrix}

a^{*} (x) = - \frac{\nabla φ (x)}{‖ \nabla φ (x) ‖} .

‖ \nabla φ (x) ‖ = f (x), φ (B) = 0 .

‖ \nabla φ (x) ‖ = min_{u} f (x, u), φ (B) = 0 .

NF (x) = {x_{j} x_{k} \in AF | \exists \tilde{x} on x_{j} x_{k} s . t . ‖ \tilde{x} - x ‖ \leq ν ϒ},

φ_{x_{j}, x_{k}}^{'} (x) = min_{ζ \in [0, 1]; u \in 𝕌} {τ (ζ) f (x, a_{ζ}, u) + ζ \bar{ϕ} (x_{j}) + (1 - ζ) \bar{ϕ} (x_{k})},

\bar{ϕ} (x) = min_{x_{j} x_{k} \in NF (x)} ϕ_{x_{j} x_{k}}^{'} (x) .

\bar{ϕ} (x) ≜ min {\bar{ϕ} (x), min_{\bar{x} x_{i} \in NF (x)} {\bar{ϕ}}_{\bar{x}, x_{i}} (x)} .

g (x, u) = 0, h (x, a, u) = e^{q_{1} (x, a) - θ_{1}} + e^{q_{2} (x, a) - θ_{2}} + h_{2} (x, u),

{log}_{10} (ν) = 1.0548 \cdot {log}_{10} (PGA) - 1.5566,

	c	𝔿(γ^, a(·), u^(·))	𝔾(γ^, u^(·))
a	0	772.4504	46.6757
b	3.2	806.3512	31.3214
c	11	900.1517	20.9815
d	22	1082.1358	9.3630
e	45	1146.4565	6.1245
f	570	1200.7766	5.0165

	c	𝔿(γ^, a(·), u^(·))	𝔾(γ^, u^(·))
a	0	772.4504	46.6757
b	3.2	806.3512	31.3214
c	11	900.1517	20.9815
d	22	1082.1358	9.3630
e	45	1146.4565	6.1245
f	570	1200.7766	5.0165