June 2017, 12(2): 319-337. doi: 10.3934/nhm.2017014

Exact and positive controllability of boundary control systems

1. 

University of L'Aquila, Department of Information Engineering, Computer Science and Mathematics, Via Vetoio, Coppito, I-67100 L'Aquila (AQ), Italy

2. 

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

3. 

Institute of Mathematics, Physics and Mechanics, Jadranska 19, SI-1000 Ljubljana, Slovenia

Received  September 2016 Revised  January 2017 Published  May 2017

Fund Project: The second author was supported in part by grant P1-0222 of the Slovenian Research Agency

We characterize the space of all exactly reachable states of an abstract boundary control system using a semigroup approach. Moreover, we study the case when the controls of the system are constrained to be positive. The abstract results are then applied to study flows in networks with static as well as dynamic boundary conditions.

Citation: Klaus-Jochen Engel, Marjeta Kramar FijavŽ. Exact and positive controllability of boundary control systems. Networks & Heterogeneous Media, 2017, 12 (2) : 319-337. doi: 10.3934/nhm.2017014
References:
[1]

M. AdlerM. Bombieri and K.-J. Engel, Perturbation of analytic semigroups and applications to partial differential equations, J. Evol. Equ., (2016), 1-26. doi: 10.1007/s00028-016-0377-8.

[2]

W. Arendt, A. Grabosch, G. Greiner, U. Groh, H. P. Lotz, U. Moustakas, R. Nagel, F. Neubrander and U. Schlotterbeck, One-Parameter Semigroups of Positive Operators, vol. 1184 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1986. doi: 10.1007/BFb0074922.

[3]

W. Arendt, C. J. K. Batty, M. Hieber and F. Neubrander, Vector-Valued Laplace Transforms and Cauchy Problems, vol. 96 of Monographs in Mathematics, 2nd edition, Birkhäuser/Springer Basel AG, Basel, 2011. doi: 10.1007/978-3-0348-0087-7.

[4]

J. BanasiakA. Falkiewicz and P. Namayanja, Asymptotic state lumping in transport and diffusion problems on networks with applications to population problems, Math. Models Methods Appl. Sci., 26 (2016), 215-247. doi: 10.1142/S0218202516400017.

[5]

M. K. BandaM. Herty and A. Klar, Gas flow in pipeline networks, Netw. Heterog. Media, 1 (2006), 41-56. doi: 10.3934/nhm.2006.1.41.

[6]

A. Bátkai, M. Kramar Fijavž and A. Rhandi, Positive Operator Semigroups: From Finite to Infinite Dimensions, vol. 257 of Operator Theory: Advances and Applications, Birkhäuser Basel, 2017. doi: 10.1007/978-3-319-42813-0.

[7]

S. BouliteH. BouslousM. El Azzouzi and L. Maniar, Approximate positive controllability of positive boundary control systems, Positivity, 18 (2014), 375-393. doi: 10.1007/s11117-013-0249-1.

[8]

A. BressanS. ČanićM. GaravelloM. Herty and B. Piccoli, Flows on networks: Recent results and perspectives, EMS Surv. Math. Sci., 1 (2014), 47-111. doi: 10.4171/EMSS/2.

[9]

G. M. CocliteM. Garavello and B. Piccoli, Traffic flow on a road network, SIAM J. Math. Anal., 36 (2005), 1862-1886. doi: 10.1137/S0036141004402683.

[10]

R. F. Curtain and H. Zwart, An Introduction to Infinite-Dimensional Linear Systems Theory, vol. 21 of Texts in Applied Mathematics, Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4612-4224-6.

[11]

C. D'Apice, S. Göttlich, M. Herty and B. Piccoli, Modeling, Simulation, and Optimization of Supply Chains, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2010. doi: 10.1137/1.9780898717600.

[12]

B. Dorn, Semigroups for flows in infinite networks, Semigroup Forum, 76 (2008), 341-356. doi: 10.1007/s00233-007-9036-2.

[13]

B. DornM. Kramar FijavžR. Nagel and A. Radl, The semigroup approach to transport processes in networks, Phys. D, 239 (2010), 1416-1421. doi: 10.1016/j.physd.2009.06.012.

[14]

K.-J. Engel, Spectral theory and generator property for one-sided coupled operator matrices, Semigroup Forum, 58 (1999), 267-295. doi: 10.1007/s002339900020.

[15]

K.-J. Engel, Generator property and stability for generalized difference operators, J. Evol. Equ., 13 (2013), 311-334. doi: 10.1007/s00028-013-0179-1.

[16]

K.-J. EngelM. Kramar FijavžR. Nagel and E. Sikolya, Vertex control of flows in networks, Netw. Heterog. Media, 3 (2008), 709-722. doi: 10.3934/nhm.2008.3.709.

[17]

K.-J. EngelM. Kramar FijavžB. KlössR. Nagel and E. Sikolya, Maximal controllability for boundary control problems, Appl. Math. Optim., 62 (2010), 205-227. doi: 10.1007/s00245-010-9101-1.

[18]

K. -J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, vol. 194 of Graduate Texts in Mathematics, Springer-Verlag, New York, 2000. doi: 10.1007/b97696.

[19]

M. Garavello and B. Piccoli, Traffic Flow on Networks, vol. 1 of AIMS Series on Applied Mathematics, American Institute of Mathematical Sciences (AIMS), Springfield, MO, 2006.

[20]

G. Greiner, Perturbing the boundary conditions of a generator, Houston J. Math., 13 (1987), 213-229.

[21]

M. GugatM. HertyA. Klar and G. Leugering, Optimal control for traffic flow networks, J. Optim. Theory Appl., 126 (2005), 589-616. doi: 10.1007/s10957-005-5499-z.

[22]

M. GugatM. Herty and V. Schleper, Flow control in gas networks: Exact controllability to a given demand, Math. Methods Appl. Sci., 34 (2011), 745-757. doi: 10.1002/mma.1394.

[23]

M. Herty and A. Klar, Modeling, simulation, and optimization of traffic flow networks, SIAM J. Sci. Comput., 25 (2003), 1066-1087. doi: 10.1137/S106482750241459X.

[24]

H. Holden and N. H. Risebro, A mathematical model of traffic flow on a network of unidirectional roads, SIAM J. Math. Anal., 26 (1995), 999-1017. doi: 10.1137/S0036141093243289.

[25]

B. Klöss, Difference operators as semigroup generators, Semigroup Forum, 81 (2010), 461-482. doi: 10.1007/s00233-010-9232-3.

[26]

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.

[27]

G. Leugering and E. J. P. G. Schmidt, On the modelling and stabilization of flows in networks of open canals, SIAM J. Control Optim., 41 (2002), 164-180. doi: 10.1137/S0363012900375664.

[28]

T. Li, Exact boundary controllability of unsteady flows in a network of open canals, Math. Nachr., 278 (2005), 278-289. doi: 10.1002/mana.200310240.

[29]

D. Mugnolo, Semigroup Methods for Evolution Equations on Networks, Understanding Complex Systems, Springer, Cham, 2014. doi: 10.1007/978-3-319-04621-1.

[30]

E. Sikolya, Flows in networks with dynamic ramification nodes, J. Evol. Equ., 5 (2005), 441-463. doi: 10.1007/s00028-005-0221-z.

[31]

G. Weiss, Admissibility of unbounded control operators, SIAM J. Control Optim., 27 (1989), 527-545. doi: 10.1137/0327028.

show all references

References:
[1]

M. AdlerM. Bombieri and K.-J. Engel, Perturbation of analytic semigroups and applications to partial differential equations, J. Evol. Equ., (2016), 1-26. doi: 10.1007/s00028-016-0377-8.

[2]

W. Arendt, A. Grabosch, G. Greiner, U. Groh, H. P. Lotz, U. Moustakas, R. Nagel, F. Neubrander and U. Schlotterbeck, One-Parameter Semigroups of Positive Operators, vol. 1184 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1986. doi: 10.1007/BFb0074922.

[3]

W. Arendt, C. J. K. Batty, M. Hieber and F. Neubrander, Vector-Valued Laplace Transforms and Cauchy Problems, vol. 96 of Monographs in Mathematics, 2nd edition, Birkhäuser/Springer Basel AG, Basel, 2011. doi: 10.1007/978-3-0348-0087-7.

[4]

J. BanasiakA. Falkiewicz and P. Namayanja, Asymptotic state lumping in transport and diffusion problems on networks with applications to population problems, Math. Models Methods Appl. Sci., 26 (2016), 215-247. doi: 10.1142/S0218202516400017.

[5]

M. K. BandaM. Herty and A. Klar, Gas flow in pipeline networks, Netw. Heterog. Media, 1 (2006), 41-56. doi: 10.3934/nhm.2006.1.41.

[6]

A. Bátkai, M. Kramar Fijavž and A. Rhandi, Positive Operator Semigroups: From Finite to Infinite Dimensions, vol. 257 of Operator Theory: Advances and Applications, Birkhäuser Basel, 2017. doi: 10.1007/978-3-319-42813-0.

[7]

S. BouliteH. BouslousM. El Azzouzi and L. Maniar, Approximate positive controllability of positive boundary control systems, Positivity, 18 (2014), 375-393. doi: 10.1007/s11117-013-0249-1.

[8]

A. BressanS. ČanićM. GaravelloM. Herty and B. Piccoli, Flows on networks: Recent results and perspectives, EMS Surv. Math. Sci., 1 (2014), 47-111. doi: 10.4171/EMSS/2.

[9]

G. M. CocliteM. Garavello and B. Piccoli, Traffic flow on a road network, SIAM J. Math. Anal., 36 (2005), 1862-1886. doi: 10.1137/S0036141004402683.

[10]

R. F. Curtain and H. Zwart, An Introduction to Infinite-Dimensional Linear Systems Theory, vol. 21 of Texts in Applied Mathematics, Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4612-4224-6.

[11]

C. D'Apice, S. Göttlich, M. Herty and B. Piccoli, Modeling, Simulation, and Optimization of Supply Chains, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2010. doi: 10.1137/1.9780898717600.

[12]

B. Dorn, Semigroups for flows in infinite networks, Semigroup Forum, 76 (2008), 341-356. doi: 10.1007/s00233-007-9036-2.

[13]

B. DornM. Kramar FijavžR. Nagel and A. Radl, The semigroup approach to transport processes in networks, Phys. D, 239 (2010), 1416-1421. doi: 10.1016/j.physd.2009.06.012.

[14]

K.-J. Engel, Spectral theory and generator property for one-sided coupled operator matrices, Semigroup Forum, 58 (1999), 267-295. doi: 10.1007/s002339900020.

[15]

K.-J. Engel, Generator property and stability for generalized difference operators, J. Evol. Equ., 13 (2013), 311-334. doi: 10.1007/s00028-013-0179-1.

[16]

K.-J. EngelM. Kramar FijavžR. Nagel and E. Sikolya, Vertex control of flows in networks, Netw. Heterog. Media, 3 (2008), 709-722. doi: 10.3934/nhm.2008.3.709.

[17]

K.-J. EngelM. Kramar FijavžB. KlössR. Nagel and E. Sikolya, Maximal controllability for boundary control problems, Appl. Math. Optim., 62 (2010), 205-227. doi: 10.1007/s00245-010-9101-1.

[18]

K. -J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, vol. 194 of Graduate Texts in Mathematics, Springer-Verlag, New York, 2000. doi: 10.1007/b97696.

[19]

M. Garavello and B. Piccoli, Traffic Flow on Networks, vol. 1 of AIMS Series on Applied Mathematics, American Institute of Mathematical Sciences (AIMS), Springfield, MO, 2006.

[20]

G. Greiner, Perturbing the boundary conditions of a generator, Houston J. Math., 13 (1987), 213-229.

[21]

M. GugatM. HertyA. Klar and G. Leugering, Optimal control for traffic flow networks, J. Optim. Theory Appl., 126 (2005), 589-616. doi: 10.1007/s10957-005-5499-z.

[22]

M. GugatM. Herty and V. Schleper, Flow control in gas networks: Exact controllability to a given demand, Math. Methods Appl. Sci., 34 (2011), 745-757. doi: 10.1002/mma.1394.

[23]

M. Herty and A. Klar, Modeling, simulation, and optimization of traffic flow networks, SIAM J. Sci. Comput., 25 (2003), 1066-1087. doi: 10.1137/S106482750241459X.

[24]

H. Holden and N. H. Risebro, A mathematical model of traffic flow on a network of unidirectional roads, SIAM J. Math. Anal., 26 (1995), 999-1017. doi: 10.1137/S0036141093243289.

[25]

B. Klöss, Difference operators as semigroup generators, Semigroup Forum, 81 (2010), 461-482. doi: 10.1007/s00233-010-9232-3.

[26]

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.

[27]

G. Leugering and E. J. P. G. Schmidt, On the modelling and stabilization of flows in networks of open canals, SIAM J. Control Optim., 41 (2002), 164-180. doi: 10.1137/S0363012900375664.

[28]

T. Li, Exact boundary controllability of unsteady flows in a network of open canals, Math. Nachr., 278 (2005), 278-289. doi: 10.1002/mana.200310240.

[29]

D. Mugnolo, Semigroup Methods for Evolution Equations on Networks, Understanding Complex Systems, Springer, Cham, 2014. doi: 10.1007/978-3-319-04621-1.

[30]

E. Sikolya, Flows in networks with dynamic ramification nodes, J. Evol. Equ., 5 (2005), 441-463. doi: 10.1007/s00028-005-0221-z.

[31]

G. Weiss, Admissibility of unbounded control operators, SIAM J. Control Optim., 27 (1989), 527-545. doi: 10.1137/0327028.

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