American Institute of Mathematical Sciences

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December  2013, 8(4): 843-855. doi: 10.3934/nhm.2013.8.843

Asymptotic periodicity of flows in time-depending networks

Fatih Bayazit 1, , Britta Dorn 2, and  Marjeta Kramar Fijavž 3,

1. 

Universität Tübingen, Mathematisch-Naturwissenschaftliche Fakultät, Auf der Morgenstelle 10, D-72076 Tübingen
2. 

WSI für Informatik, Universität Tübingen, Sand 13, D-72076 Tübingen, Germany
3. 

University of Ljubljana, Faculty of Civil and Geodetic Engineering, Jamova 2, SI-1000 Ljubljana, Slovenia

Received  February 2013 Revised  July 2013 Published  November 2013

We consider a linear transport equation on the edges of a network with time-varying coefficients. Using methods for non-autonomous abstract Cauchy problems, we obtain well-posedness of the problem and describe the asymptotic profile of the solutions under certain natural conditions on the network. We further apply our theory to a model used for air traffic flow management.
Keywords: air traffic flow management., evolution family, asymptotic periodicity, non-autonomous Cauchy problem, transport equation, Time-depending networks.
Mathematics Subject Classification: Primary: 35R02; Secondary: 47N20, 37B5.
Citation: Fatih Bayazit, Britta Dorn, Marjeta Kramar Fijavž. Asymptotic periodicity of flows in time-depending networks. Networks & Heterogeneous Media, 2013, 8 (4) : 843-855. doi: 10.3934/nhm.2013.8.843
References:
[1]

F. Bayazit, Positive evolution families solving nonautonomous difference equations,, Positivity, 16 (2012), 653.  doi: 10.1007/s11117-011-0139-3.  Google Scholar

[2]

F. Bayazit, On the Asymptotic Behavior of Periodic Evolution Families on Banach Spaces,, Ph.D thesis, (2012).   Google Scholar

[3]

A. M. Bayen, R. L. Raffard and C. L. Tomlin, Eulerian network model of air traffic flow in congested areas,, Proc. of the American Control Conference, (2004), 5520.   Google Scholar

[4]

A. M. Bayen, R. L. Raffard and C. L. Tomlin, Adjoint-based control of Eulerian transportation networks: application to air traffic control,, Proc. of the American Control Conference, (2004), 5539.   Google Scholar

[5]

A. M. Bayen, R. L. Raffard and C. L. Tomlin, Adjoint-based control of a new Eulerian network model of air traffic flow,, IEEE Transactions on Control Systems Technology, 14 (2006), 804.  doi: 10.1109/TCST.2006.876904.  Google Scholar

[6]

B. Dorn, Semigroups for flows in infinite networks,, Semigroup Forum, 76 (2008), 341.  doi: 10.1007/s00233-007-9036-2.  Google Scholar

[7]

B. Dorn, Flows in Infinite Networks - A Semigroup Approach,, Ph.D thesis, (2008).   Google Scholar

[8]

B. Dorn, V. Keicher and E. Sikolya, Asymptotic periodicity of recurrent flows in infinite networks,, Math. Z., 263 (2009), 69.  doi: 10.1007/s00209-008-0410-x.  Google Scholar

[9]

B. Dorn, M. Kramar Fijavž, R. Nagel and A. Radl, The semigroup approach to flows in networks,, Physica D, 239 (2010), 1416.  doi: 10.1016/j.physd.2009.06.012.  Google Scholar

[10]

K.-J. Engel, M. Kramar Fijavž, B. Klöss, R. Nagel and E. Sikolya, Maximal controllability for boundary control problems,, Appl. Math. Optim., 62 (2010), 205.  doi: 10.1007/s00245-010-9101-1.  Google Scholar

[11]

K.-J. Engel, M. Kramar Fijavž, R. Nagel and E. Sikolya, Vertex control of flows in networks,, J. Networks Heterogeneous Media, 3 (2008), 709.  doi: 10.3934/nhm.2008.3.709.  Google Scholar

[12]

K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations,, Graduate Texts in Math. 194, (2000).   Google Scholar

[13]

M. Garavello and B. Piccoli, Traffic Flow on Networks,, American Institute of Mathematical Sciences, (2006).   Google Scholar

[14]

M. Kramar and E. Sikolya, Spectral properties and asymptotic periodicity of flows in networks,, Math. Z., 249 (2005), 139.  doi: 10.1007/s00209-004-0695-3.  Google Scholar

[15]

M. J. Lighthill and G. B. Whitham, On kinematic waves. II. A theory of traffic flow on long crowded roads,, Proc. of the Royal Society of London, 229 (1956), 317.  doi: 10.1098/rspa.1955.0089.  Google Scholar

[16]

T. Mátrai and E. Sikolya, Asymptotic behavior of flows in networks,, Forum Math., 19 (2007), 429.  doi: 10.1515/FORUM.2007.018.  Google Scholar

[17]

P. K. Menon, G. D. Sweriduk and K. Bilimoria, A new approach for modeling, analysis and control of air traffic flow,, AIAA Journal of Guidance, 27 (2004), 737.   Google Scholar

[18]

H. Minc, Nonnegative Matrices,, John Wiley & Sons, (1988).   Google Scholar

[19]

R. Nagel, Semigroup methods for nonautonomous Cauchy problems,, Evolution Equations, 168 (1995), 301.   Google Scholar

[20]

R. Nagel and G. Nickel, Well-posedness for nonautonomous abstract Cauchy problems,, Prog. Nonlinear Differential Equations Appl., 50 (2002), 279.   Google Scholar

[21]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Springer-Verlag, (1983).  doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[22]

A. Radl, Transport processes in networks with scattering ramification nodes,, J. Appl. Funct. Anal., 3 (2008), 461.   Google Scholar

[23]

P. I. Richards, Shock waves on the highway,, Oper. Res., 4 (1956), 42.  doi: 10.1287/opre.4.1.42.  Google Scholar

[24]

C.-A. Robelin, D. Sun, G. Wu and A. M. Bayen, MILP control of aggregate Eulerian network airspace models,, Proc. of the American Control Conference, (2006), 5257.  doi: 10.1109/ACC.2006.1657558.  Google Scholar

[25]

H. H. Schaefer, Banach Lattices and Positive Operators,, Grundlehren Math. Wiss., (1974).   Google Scholar

[26]

E. Sikolya, Flows in networks with dynamic ramification nodes,, J. Evol. Equ., 5 (2005), 441.  doi: 10.1007/s00028-005-0221-z.  Google Scholar

[27]

B. Sridhar and P. K. Menon, Comparison of linear dynamic models for air traffic flow management,, Proc. 16th IFAC World Congress, (2005).   Google Scholar

[28]

D. Sun, I. S. Strub and A. M. Bayen, Comparison of the performance of four Eulerian network flow models for strategic air traffic management,, Netw. Heterog. Media, 2 (2007), 569.  doi: 10.3934/nhm.2007.2.569.  Google Scholar

show all references

References:
[1]

F. Bayazit, Positive evolution families solving nonautonomous difference equations,, Positivity, 16 (2012), 653.  doi: 10.1007/s11117-011-0139-3.  Google Scholar

[2]

F. Bayazit, On the Asymptotic Behavior of Periodic Evolution Families on Banach Spaces,, Ph.D thesis, (2012).   Google Scholar

[3]

A. M. Bayen, R. L. Raffard and C. L. Tomlin, Eulerian network model of air traffic flow in congested areas,, Proc. of the American Control Conference, (2004), 5520.   Google Scholar

[4]

A. M. Bayen, R. L. Raffard and C. L. Tomlin, Adjoint-based control of Eulerian transportation networks: application to air traffic control,, Proc. of the American Control Conference, (2004), 5539.   Google Scholar

[5]

A. M. Bayen, R. L. Raffard and C. L. Tomlin, Adjoint-based control of a new Eulerian network model of air traffic flow,, IEEE Transactions on Control Systems Technology, 14 (2006), 804.  doi: 10.1109/TCST.2006.876904.  Google Scholar

[6]

B. Dorn, Semigroups for flows in infinite networks,, Semigroup Forum, 76 (2008), 341.  doi: 10.1007/s00233-007-9036-2.  Google Scholar

[7]

B. Dorn, Flows in Infinite Networks - A Semigroup Approach,, Ph.D thesis, (2008).   Google Scholar

[8]

B. Dorn, V. Keicher and E. Sikolya, Asymptotic periodicity of recurrent flows in infinite networks,, Math. Z., 263 (2009), 69.  doi: 10.1007/s00209-008-0410-x.  Google Scholar

[9]

B. Dorn, M. Kramar Fijavž, R. Nagel and A. Radl, The semigroup approach to flows in networks,, Physica D, 239 (2010), 1416.  doi: 10.1016/j.physd.2009.06.012.  Google Scholar

[10]

K.-J. Engel, M. Kramar Fijavž, B. Klöss, R. Nagel and E. Sikolya, Maximal controllability for boundary control problems,, Appl. Math. Optim., 62 (2010), 205.  doi: 10.1007/s00245-010-9101-1.  Google Scholar

[11]

K.-J. Engel, M. Kramar Fijavž, R. Nagel and E. Sikolya, Vertex control of flows in networks,, J. Networks Heterogeneous Media, 3 (2008), 709.  doi: 10.3934/nhm.2008.3.709.  Google Scholar

[12]

K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations,, Graduate Texts in Math. 194, (2000).   Google Scholar

[13]

M. Garavello and B. Piccoli, Traffic Flow on Networks,, American Institute of Mathematical Sciences, (2006).   Google Scholar

[14]

M. Kramar and E. Sikolya, Spectral properties and asymptotic periodicity of flows in networks,, Math. Z., 249 (2005), 139.  doi: 10.1007/s00209-004-0695-3.  Google Scholar

[15]

M. J. Lighthill and G. B. Whitham, On kinematic waves. II. A theory of traffic flow on long crowded roads,, Proc. of the Royal Society of London, 229 (1956), 317.  doi: 10.1098/rspa.1955.0089.  Google Scholar

[16]

T. Mátrai and E. Sikolya, Asymptotic behavior of flows in networks,, Forum Math., 19 (2007), 429.  doi: 10.1515/FORUM.2007.018.  Google Scholar

[17]

P. K. Menon, G. D. Sweriduk and K. Bilimoria, A new approach for modeling, analysis and control of air traffic flow,, AIAA Journal of Guidance, 27 (2004), 737.   Google Scholar

[18]

H. Minc, Nonnegative Matrices,, John Wiley & Sons, (1988).   Google Scholar

[19]

R. Nagel, Semigroup methods for nonautonomous Cauchy problems,, Evolution Equations, 168 (1995), 301.   Google Scholar

[20]

R. Nagel and G. Nickel, Well-posedness for nonautonomous abstract Cauchy problems,, Prog. Nonlinear Differential Equations Appl., 50 (2002), 279.   Google Scholar

[21]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Springer-Verlag, (1983).  doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[22]

A. Radl, Transport processes in networks with scattering ramification nodes,, J. Appl. Funct. Anal., 3 (2008), 461.   Google Scholar

[23]

P. I. Richards, Shock waves on the highway,, Oper. Res., 4 (1956), 42.  doi: 10.1287/opre.4.1.42.  Google Scholar

[24]

C.-A. Robelin, D. Sun, G. Wu and A. M. Bayen, MILP control of aggregate Eulerian network airspace models,, Proc. of the American Control Conference, (2006), 5257.  doi: 10.1109/ACC.2006.1657558.  Google Scholar

[25]

H. H. Schaefer, Banach Lattices and Positive Operators,, Grundlehren Math. Wiss., (1974).   Google Scholar

[26]

E. Sikolya, Flows in networks with dynamic ramification nodes,, J. Evol. Equ., 5 (2005), 441.  doi: 10.1007/s00028-005-0221-z.  Google Scholar

[27]

B. Sridhar and P. K. Menon, Comparison of linear dynamic models for air traffic flow management,, Proc. 16th IFAC World Congress, (2005).   Google Scholar

[28]

D. Sun, I. S. Strub and A. M. Bayen, Comparison of the performance of four Eulerian network flow models for strategic air traffic management,, Netw. Heterog. Media, 2 (2007), 569.  doi: 10.3934/nhm.2007.2.569.  Google Scholar

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