American Institute of Mathematical Sciences

Advanced
December  2008, 3(4): 709-722. doi: 10.3934/nhm.2008.3.709

Vertex control of flows in networks

Klaus-Jochen Engel 1, , Marjeta Kramar Fijavž 2, , Rainer Nagel 3, and  Eszter Sikolya 4,

1. 

University of L'Aquila, Faculty of Engineering, Piazzale E. Pontieri 2, I-67100 Monteluco di Roio (L’Aquila), Italy
2. 

University of Ljubljana, Faculty of Civil and Geodetic Engineering, Jamova 2, SI-1000 Ljubljana, Slovenia
3. 

University of Tübingen, Mathematics Institute, Auf der Morgenstelle 10, D-72076 Tübingen
4. 

Eötvös Loránd University Budapest, Department of Applied Analysis, Pázmány Péter sétány 1/c., Budapest H-1117, Hungary

Received  April 2008 Revised  June 2008 Published  October 2008

We study a transport equation in a network and control it in a single vertex. We describe all possible reachable states and prove a criterion of Kalman type for those vertices in which the problem is maximally controllable. The results are then applied to concrete networks to show the complexity of the problem.
Keywords: operator semigroups, boundary control, transport equation, networks.
Mathematics Subject Classification: 34B45, 93B05, 47N70.
Citation: Klaus-Jochen Engel, Marjeta Kramar Fijavž, Rainer Nagel, Eszter Sikolya. Vertex control of flows in networks. Networks & Heterogeneous Media, 2008, 3 (4) : 709-722. doi: 10.3934/nhm.2008.3.709
[1]

Robert Denk, Yoshihiro Shibata. Generation of semigroups for the thermoelastic plate equation with free boundary conditions. Evolution Equations & Control Theory, 2019, 8 (2) : 301-313. doi: 10.3934/eect.2019016
[2]

Fabio Camilli, Raul De Maio, Andrea Tosin. Transport of measures on networks. Networks & Heterogeneous Media, 2017, 12 (2) : 191-215. doi: 10.3934/nhm.2017008
[3]

Qinglan Xia. An application of optimal transport paths to urban transport networks. Conference Publications, 2005, 2005 (Special) : 904-910. doi: 10.3934/proc.2005.2005.904
[4]

Bopeng Rao, Laila Toufayli, Ali Wehbe. Stability and controllability of a wave equation with dynamical boundary control. Mathematical Control & Related Fields, 2015, 5 (2) : 305-320. doi: 10.3934/mcrf.2015.5.305
[5]

Xiaoshan Chen, Fahuai Yi. Free boundary problem of Barenblatt equation in stochastic control. Discrete & Continuous Dynamical Systems - B, 2016, 21 (5) : 1421-1434. doi: 10.3934/dcdsb.2016003
[6]

Alexander Arguchintsev, Vasilisa Poplevko. An optimal control problem by parabolic equation with boundary smooth control and an integral constraint. Numerical Algebra, Control & Optimization, 2018, 8 (2) : 193-202. doi: 10.3934/naco.2018011
[7]

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
[8]

Umberto Biccari. Internal control for a non-local Schrödinger equation involving the fractional Laplace operator. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021014
[9]

Muhammad I. Mustafa. On the control of the wave equation by memory-type boundary condition. Discrete & Continuous Dynamical Systems, 2015, 35 (3) : 1179-1192. doi: 10.3934/dcds.2015.35.1179
[10]

Peter I. Kogut. On approximation of an optimal boundary control problem for linear elliptic equation with unbounded coefficients. Discrete & Continuous Dynamical Systems, 2014, 34 (5) : 2105-2133. doi: 10.3934/dcds.2014.34.2105
[11]

Jésus Ildefonso Díaz, Tommaso Mingazzini, Ángel Manuel Ramos. On the optimal control for a semilinear equation with cost depending on the free boundary. Networks & Heterogeneous Media, 2012, 7 (4) : 605-615. doi: 10.3934/nhm.2012.7.605
[12]

Proscovia Namayanja. Chaotic dynamics in a transport equation on a network. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 3415-3426. doi: 10.3934/dcdsb.2018283
[13]

Ioana Ciotir, Nicolas Forcadel, Wilfredo Salazar. Homogenization of a stochastic viscous transport equation. Evolution Equations & Control Theory, 2021, 10 (2) : 353-364. doi: 10.3934/eect.2020070
[14]

Horst R. Thieme. Positive perturbation of operator semigroups: growth bounds, essential compactness and asynchronous exponential growth. Discrete & Continuous Dynamical Systems, 1998, 4 (4) : 735-764. doi: 10.3934/dcds.1998.4.735
[15]

Frank Neubrander, Koray Özer, Lee Windsperger. On subdiagonal rational Padé approximations and the Brenner-Thomée approximation theorem for operator semigroups. Discrete & Continuous Dynamical Systems - S, 2020, 13 (12) : 3565-3579. doi: 10.3934/dcdss.2020238
[16]

Peter I. Kogut, Olha P. Kupenko. On optimal control problem for an ill-posed strongly nonlinear elliptic equation with $p$-Laplace operator and $L^1$-type of nonlinearity. Discrete & Continuous Dynamical Systems - B, 2019, 24 (3) : 1273-1295. doi: 10.3934/dcdsb.2019016
[17]

Luisa Arlotti. Explicit transport semigroup associated to abstract boundary conditions. Conference Publications, 2011, 2011 (Special) : 102-111. doi: 10.3934/proc.2011.2011.102
[18]

Jean-François Coulombel, Frédéric Lagoutière. The Neumann numerical boundary condition for transport equations. Kinetic & Related Models, 2020, 13 (1) : 1-32. doi: 10.3934/krm.2020001
[19]

Fabio Ancona, Laura Caravenna, Annalisa Cesaroni, Giuseppe M. Coclite, Claudio Marchi, Andrea Marson. Analysis and control on networks: Trends and perspectives. Networks & Heterogeneous Media, 2017, 12 (3) : i-ii. doi: 10.3934/nhm.201703i
[20]

Fabio Ancona, Laura Caravenna, Annalisa Cesaroni, Giuseppe M. Coclite, Claudio Marchi, Andrea Marson. Analysis and control on networks: Trends and perspectives. Networks & Heterogeneous Media, 2017, 12 (2) : i-ii. doi: 10.3934/nhm.201702i

2019 Impact Factor: 1.053

Tools

Metrics

  • PDF downloads (118)
  • HTML views (0)
  • Cited by (24)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences