American Institute of Mathematical Sciences

Advanced
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

[1]

A. Marigo, Benedetto Piccoli. Cooperative controls for air traffic management. Communications on Pure & Applied Analysis, 2003, 2 (3) : 355-369. doi: 10.3934/cpaa.2003.2.355
[2]

Lino J. Alvarez-Vázquez, Néstor García-Chan, Aurea Martínez, Miguel E. Vázquez-Méndez. Optimal control of urban air pollution related to traffic flow in road networks. Mathematical Control & Related Fields, 2018, 8 (1) : 177-193. doi: 10.3934/mcrf.2018008
[3]

Mustapha Mokhtar-Kharroubi. On permanent regimes for non-autonomous linear evolution equations in Banach spaces with applications to transport theory. Kinetic & Related Models, 2010, 3 (3) : 473-499. doi: 10.3934/krm.2010.3.473
[4]

Mary Luz Mouronte, Rosa María Benito. Structural analysis and traffic flow in the transport networks of Madrid. Networks & Heterogeneous Media, 2015, 10 (1) : 127-148. doi: 10.3934/nhm.2015.10.127
[5]

Dengfeng Sun, Issam S. Strub, Alexandre M. Bayen. Comparison of the performance of four Eulerian network flow models for strategic air traffic management. Networks & Heterogeneous Media, 2007, 2 (4) : 569-595. doi: 10.3934/nhm.2007.2.569
[6]

Yacine Chitour, Guilherme Mazanti, Mario Sigalotti. Stability of non-autonomous difference equations with applications to transport and wave propagation on networks. Networks & Heterogeneous Media, 2016, 11 (4) : 563-601. doi: 10.3934/nhm.2016010
[7]

Abdelaziz Rhandi, Roland Schnaubelt. Asymptotic behaviour of a non-autonomous population equation with diffusion in $L^1$. Discrete & Continuous Dynamical Systems - A, 1999, 5 (3) : 663-683. doi: 10.3934/dcds.1999.5.663
[8]

Tomás Caraballo, Antonio M. Márquez-Durán, Rivero Felipe. Asymptotic behaviour of a non-classical and non-autonomous diffusion equation containing some hereditary characteristic. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 1817-1833. doi: 10.3934/dcdsb.2017108
[9]

Felipe Rivero. Time dependent perturbation in a non-autonomous non-classical parabolic equation. Discrete & Continuous Dynamical Systems - B, 2013, 18 (1) : 209-221. doi: 10.3934/dcdsb.2013.18.209
[10]

Tomás Caraballo, P.E. Kloeden. Non-autonomous attractors for integro-differential evolution equations. Discrete & Continuous Dynamical Systems - S, 2009, 2 (1) : 17-36. doi: 10.3934/dcdss.2009.2.17
[11]

Mahesh G. Nerurkar. Spectral and stability questions concerning evolution of non-autonomous linear systems. Conference Publications, 2001, 2001 (Special) : 270-275. doi: 10.3934/proc.2001.2001.270
[12]

Peter E. Kloeden, Jacson Simsen. Pullback attractors for non-autonomous evolution equations with spatially variable exponents. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2543-2557. doi: 10.3934/cpaa.2014.13.2543
[13]

Xinguang Yang, Baowei Feng, Thales Maier de Souza, Taige Wang. Long-time dynamics for a non-autonomous Navier-Stokes-Voigt equation in Lipschitz domains. Discrete & Continuous Dynamical Systems - B, 2019, 24 (1) : 363-386. doi: 10.3934/dcdsb.2018084
[14]

Suping Wang, Qiaozhen Ma. Existence of pullback attractors for the non-autonomous suspension bridge equation with time delay. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 0-0. doi: 10.3934/dcdsb.2019221
[15]

Sylvia Novo, Rafael Obaya, Ana M. Sanz. Exponential stability in non-autonomous delayed equations with applications to neural networks. Discrete & Continuous Dynamical Systems - A, 2007, 18 (2&3) : 517-536. doi: 10.3934/dcds.2007.18.517
[16]

Minzilia A. Sagadeeva, Sophiya A. Zagrebina, Natalia A. Manakova. Optimal control of solutions of a multipoint initial-final problem for non-autonomous evolutionary Sobolev type equation. Evolution Equations & Control Theory, 2019, 8 (3) : 473-488. doi: 10.3934/eect.2019023
[17]

Chunyou Sun, Daomin Cao, Jinqiao Duan. Non-autonomous wave dynamics with memory --- asymptotic regularity and uniform attractor. Discrete & Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 743-761. doi: 10.3934/dcdsb.2008.9.743
[18]

María Anguiano, Tomás Caraballo. Asymptotic behaviour of a non-autonomous Lorenz-84 system. Discrete & Continuous Dynamical Systems - A, 2014, 34 (10) : 3901-3920. doi: 10.3934/dcds.2014.34.3901
[19]

Shaoyong Lai, Yong Hong Wu. The asymptotic solution of the Cauchy problem for a generalized Boussinesq equation. Discrete & Continuous Dynamical Systems - B, 2003, 3 (3) : 401-408. doi: 10.3934/dcdsb.2003.3.401
[20]

Rodrigo Samprogna, Tomás Caraballo. Pullback attractor for a dynamic boundary non-autonomous problem with Infinite Delay. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 509-523. doi: 10.3934/dcdsb.2017195

2018 Impact Factor: 0.871

Tools

Metrics

  • PDF downloads (6)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences