Asymptotic periodicity of flows in time-depending networks

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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

Abstract
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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 and 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), 653-684. doi: 10.1007/s11117-011-0139-3.
[2]	F. Bayazit, On the Asymptotic Behavior of Periodic Evolution Families on Banach Spaces, Ph.D thesis, Eberhard Karls Universität Tübingen, 2012.
[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), 5520-5526.
[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), 5539-5545.
[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), 804-818. doi: 10.1109/TCST.2006.876904.
[6]	B. Dorn, Semigroups for flows in infinite networks, Semigroup Forum, 76 (2008), 341-356. doi: 10.1007/s00233-007-9036-2.
[7]	B. Dorn, Flows in Infinite Networks - A Semigroup Approach, Ph.D thesis, Eberhard Karls Universität Tübingen, 2008.
[8]	B. Dorn, V. Keicher and E. Sikolya, Asymptotic periodicity of recurrent flows in infinite networks, Math. Z., 263 (2009), 69-87. doi: 10.1007/s00209-008-0410-x.
[9]	B. Dorn, M. Kramar Fijavž, R. Nagel and A. Radl, The semigroup approach to flows in networks, Physica D, 239 (2010), 1416-1421. doi: 10.1016/j.physd.2009.06.012.
[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), 205-227. doi: 10.1007/s00245-010-9101-1.
[11]	K.-J. Engel, M. Kramar Fijavž, R. Nagel and E. Sikolya, Vertex control of flows in networks, J. Networks Heterogeneous Media, 3 (2008), 709-722. doi: 10.3934/nhm.2008.3.709.
[12]	K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Math. 194, Springer-Verlag, 2000.
[13]	M. Garavello and B. Piccoli, Traffic Flow on Networks, American Institute of Mathematical Sciences, 2006.
[14]	M. Kramar and E. Sikolya, Spectral properties and asymptotic periodicity of flows in networks, Math. Z., 249 (2005), 139-162. doi: 10.1007/s00209-004-0695-3.
[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), 317-345. doi: 10.1098/rspa.1955.0089.
[16]	T. Mátrai and E. Sikolya, Asymptotic behavior of flows in networks, Forum Math., 19 (2007), 429-461. doi: 10.1515/FORUM.2007.018.
[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, Control and Dynamics, 27 (2004), 737-744.
[18]	H. Minc, Nonnegative Matrices, John Wiley & Sons, 1988.
[19]	R. Nagel, Semigroup methods for nonautonomous Cauchy problems, Evolution Equations, Lecture Notes in Pure and Appl. Math., 168 (1995), 301-316.
[20]	R. Nagel and G. Nickel, Well-posedness for nonautonomous abstract Cauchy problems, Prog. Nonlinear Differential Equations Appl., 50 (2002), 279-293.
[21]	A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, 1983. doi: 10.1007/978-1-4612-5561-1.
[22]	A. Radl, Transport processes in networks with scattering ramification nodes, J. Appl. Funct. Anal., 3 (2008), 461-483.
[23]	P. I. Richards, Shock waves on the highway, Oper. Res., 4 (1956), 42-51. doi: 10.1287/opre.4.1.42.
[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), 5257-5262. doi: 10.1109/ACC.2006.1657558.
[25]	H. H. Schaefer, Banach Lattices and Positive Operators, Grundlehren Math. Wiss., 215, Springer-Verlag, 1974.
[26]	E. Sikolya, Flows in networks with dynamic ramification nodes, J. Evol. Equ., 5 (2005), 441-463. doi: 10.1007/s00028-005-0221-z.
[27]	B. Sridhar and P. K. Menon, Comparison of linear dynamic models for air traffic flow management, Proc. 16th IFAC World Congress, (2005).
[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), 569-595. doi: 10.3934/nhm.2007.2.569.

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References:

[1]	F. Bayazit, Positive evolution families solving nonautonomous difference equations, Positivity, 16 (2012), 653-684. doi: 10.1007/s11117-011-0139-3.
[2]	F. Bayazit, On the Asymptotic Behavior of Periodic Evolution Families on Banach Spaces, Ph.D thesis, Eberhard Karls Universität Tübingen, 2012.
[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), 5520-5526.
[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), 5539-5545.
[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), 804-818. doi: 10.1109/TCST.2006.876904.
[6]	B. Dorn, Semigroups for flows in infinite networks, Semigroup Forum, 76 (2008), 341-356. doi: 10.1007/s00233-007-9036-2.
[7]	B. Dorn, Flows in Infinite Networks - A Semigroup Approach, Ph.D thesis, Eberhard Karls Universität Tübingen, 2008.
[8]	B. Dorn, V. Keicher and E. Sikolya, Asymptotic periodicity of recurrent flows in infinite networks, Math. Z., 263 (2009), 69-87. doi: 10.1007/s00209-008-0410-x.
[9]	B. Dorn, M. Kramar Fijavž, R. Nagel and A. Radl, The semigroup approach to flows in networks, Physica D, 239 (2010), 1416-1421. doi: 10.1016/j.physd.2009.06.012.
[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), 205-227. doi: 10.1007/s00245-010-9101-1.
[11]	K.-J. Engel, M. Kramar Fijavž, R. Nagel and E. Sikolya, Vertex control of flows in networks, J. Networks Heterogeneous Media, 3 (2008), 709-722. doi: 10.3934/nhm.2008.3.709.
[12]	K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Math. 194, Springer-Verlag, 2000.
[13]	M. Garavello and B. Piccoli, Traffic Flow on Networks, American Institute of Mathematical Sciences, 2006.
[14]	M. Kramar and E. Sikolya, Spectral properties and asymptotic periodicity of flows in networks, Math. Z., 249 (2005), 139-162. doi: 10.1007/s00209-004-0695-3.
[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), 317-345. doi: 10.1098/rspa.1955.0089.
[16]	T. Mátrai and E. Sikolya, Asymptotic behavior of flows in networks, Forum Math., 19 (2007), 429-461. doi: 10.1515/FORUM.2007.018.
[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, Control and Dynamics, 27 (2004), 737-744.
[18]	H. Minc, Nonnegative Matrices, John Wiley & Sons, 1988.
[19]	R. Nagel, Semigroup methods for nonautonomous Cauchy problems, Evolution Equations, Lecture Notes in Pure and Appl. Math., 168 (1995), 301-316.
[20]	R. Nagel and G. Nickel, Well-posedness for nonautonomous abstract Cauchy problems, Prog. Nonlinear Differential Equations Appl., 50 (2002), 279-293.
[21]	A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, 1983. doi: 10.1007/978-1-4612-5561-1.
[22]	A. Radl, Transport processes in networks with scattering ramification nodes, J. Appl. Funct. Anal., 3 (2008), 461-483.
[23]	P. I. Richards, Shock waves on the highway, Oper. Res., 4 (1956), 42-51. doi: 10.1287/opre.4.1.42.
[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), 5257-5262. doi: 10.1109/ACC.2006.1657558.
[25]	H. H. Schaefer, Banach Lattices and Positive Operators, Grundlehren Math. Wiss., 215, Springer-Verlag, 1974.
[26]	E. Sikolya, Flows in networks with dynamic ramification nodes, J. Evol. Equ., 5 (2005), 441-463. doi: 10.1007/s00028-005-0221-z.
[27]	B. Sridhar and P. K. Menon, Comparison of linear dynamic models for air traffic flow management, Proc. 16th IFAC World Congress, (2005).
[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), 569-595. doi: 10.3934/nhm.2007.2.569.

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