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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[12]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[20]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[25]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[29]

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

[30]

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

[31]

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

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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[12]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[20]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[25]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[29]

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

[30]

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

[31]

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

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