American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Uniform stability in a vectorial full Von Kármán thermoelastic system with solenoidal dissipation and free boundary conditions
  • EECT Home
  • This Issue
  • Next Article
    S-asymptotically $ \omega $-periodic mild solutions and stability analysis of Hilfer fractional evolution equations
December  2021, 10(4): 749-766. doi: 10.3934/eect.2020091

Approximate controllability of network systems

Yassine El Gantouh 1, , Said Hadd 1, and  Abdelaziz Rhandi 2,,

1. 

Faculty of Sciences, Department of Mathematics, Ibn Zohr University, Hay Dakhla, BP8106, 80000–Agadir, Morocco
2. 

Dipartimento di Ingegneria dell'Informazione, Ingegneria Elettrica e Matematica Applicata, Università degli Studi di Salerno, Via Giovanni Paolo Ⅱ, 132, 84084 Fisciano (Sa), Italy

* Corresponding author: Abdelaziz Rhandi

Dedicated to Rainer Nagel on the occasion of his 80th-Birthday

Received  October 2019 Revised  July 2020 Published  December 2021 Early access  September 2020

Fund Project: This work has been supported by the COST Action CA18232. The third author is a member of the Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM)

In this paper, the rich feedback theory of regular linear systems in the Salamon-Weiss sense as well as some advanced tools in semigroup theory are used to formulate and solve control problems for network systems. In fact, we derive necessary and sufficient conditions for approximate controllability of such systems. These criteria, in some particular cases, are given by the well-known Kalman's controllability rank condition.

Keywords: Boundary systems, boundary perturbations, controllability, network systems, Kalman's controllability rank condition.
Mathematics Subject Classification: Primary:93B05, 93B70;Secondary:93C73, 93C05, 93B28.
Citation: Yassine El Gantouh, Said Hadd, Abdelaziz Rhandi. Approximate controllability of network systems. Evolution Equations and Control Theory, 2021, 10 (4) : 749-766. doi: 10.3934/eect.2020091
References:
[1]

R. K. Ahuja, T. L. Magnanti and J. B. Orlin, Network Flows: Theory, Algorithms and Applications, Prentice Hall, Inc., Englewood Cliffs, NJ, 1993.

[2]

F. Bayazit, B. Dorn and A. Rhandi, Flows in networks with delay in the vertices, Math. Nachr., 285 (2012), 1603-1615.  doi: 10.1002/mana.201100163.

[3]

S. Boccaletti, V. Latora, Y. Moreno, M. Chavez and D. U. Hwang, Complex networks: Structure and dynamics, Phys. Rep., 424 (2006), 175-308.  doi: 10.1016/j.physrep.2005.10.009.

[4]

B. Bolobás, Modern Graph Theory, Springer-Verlag, New York, 1998.

[5] J. Casti, Linear Dynamical Systems, Academic Press, Orlando, Florida, 1987. 
[6]

B. Dorn, Flows in Infinite Networks - A Semigroup Approach, Ph.D thesis, Tuebingen University, Germany, 2008.

[7]

B. Dorn, M. Kramar Fijav$\tilde z$, R. 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.

[8]

M. El Azzouzi, H. Bouslous, L. Maniar and S. Boulite, Constrained approximate controllability of boundary control systems, IMA J. Math. Control Inform., 33 (2016), 669-683.  doi: 10.1093/imamci/dnv002.

[9]

K.-J. Engel and M. Kramar Fijav$\tilde z$, Exact and positive controllability of boundary control systems, Netw. Heterog. Media, 12 (2017), 319-337.  doi: 10.3934/nhm.2017014.

[10]

K.-J. Engel, M. Kramar Fijav$\tilde z$, R. Nagel and E. Sikolya, Vertex control of flows in networks, Netw. Heterog. Media, 3 (2008), 709-722.  doi: 10.3934/nhm.2008.3.709.

[11]

K.-J. Engel, B. Klöss, M. Kramar Fijav$\tilde z$, 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.

[12]

K.-J. Engel and R. Nagel, One-parameter Semigroups for Linear Evolution Equations, Springer-Verlag, New York, 2000.

[13]

H. O. Fattorini, Boundary control systems, SIAM J. Control, 6 (1968), 349-385.  doi: 10.1137/0306025.

[14]

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

[15]

S. Hadd, An Evolution Equation Approach, Ph.D thesis, Cadi Ayyad University, Marrakech, 2005.

[16]

S. Hadd, R. Manzo and A. Rhandi, Unbounded perturbations of the generator domain, Discrete Contin. Dyn. Syst., 35 (2015), 703-723.  doi: 10.3934/dcds.2015.35.703.

[17]

U. Knauer, Algeberaic Graph Theory. Morphisms, Monoids and Matrices, De Gruyter Studies in Mathematics, Vol. 41, Walter de Gruyter & Co., Berlin, 2011. doi: 10.1515/9783110255096.

[18]

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.

[19]

C.-T. Lin, Structural controllability, IEEE Trans. Automatic Control, AC-19 (1974), 201-208.  doi: 10.1109/tac.1974.1100557.

[20]

Y.-Y. Liu, J.-J. Slotine and A.-L. Barabasi, Controllability of complex networks, Nature, 473 (2011), 167-173.  doi: 10.1038/nature10011.

[21]

T. Matrai and E. Sikolya, Asymptotic behavior of flows in networks, Forum Math., 19 (2007), 429-461.  doi: 10.1515/FORUM.2007.018.

[22]

D. Salamon, Infinite-dimensional linear system with unbounded control and observation: A functional analytic approach, Trans. Amer. Math. Soc., 300 (1987), 383-431.  doi: 10.2307/2000351.

[23]

R. W. Shields and J. B. Pearson, Structural controllability of multi-input linear systems, IEEE Trans. Automat. Control, AC-21 (1976), 203-212.  doi: 10.1109/tac.1976.1101198.

[24]

E. Sikolya, Semigroups for Flows in Networks, Ph.D thesis, Tuebingen University, Germany, 2004.

[25] O. J. Staffans, Well-posed Linear Systems, Cambridge Univ. Press, Cambridge, 2005.  doi: 10.1017/CBO9780511543197.
[26]

M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, Birkhäuser Verlag, Basel, 2009. doi: 10.1007/978-3-7643-8994-9.

[27]

G. Weiss, Admissible observation operators for linear semigoups, Israel J. Math., 65 (1989), 17-43.  doi: 10.1007/BF02788172.

[28]

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

[29]

G. Weiss, Transfer functions of regular linear systems. Part Ⅰ: Characterizations of regularity, Trans. Amer. Math. Soc., 342 (1994), 827-854.  doi: 10.2307/2154655.

[30]

G. Weiss, Regular linear systems with feedback, Math. Control Signals Systems, 7 (1994), 23-57.  doi: 10.1007/BF01211484.

show all references

References:
[1]

R. K. Ahuja, T. L. Magnanti and J. B. Orlin, Network Flows: Theory, Algorithms and Applications, Prentice Hall, Inc., Englewood Cliffs, NJ, 1993.

[2]

F. Bayazit, B. Dorn and A. Rhandi, Flows in networks with delay in the vertices, Math. Nachr., 285 (2012), 1603-1615.  doi: 10.1002/mana.201100163.

[3]

S. Boccaletti, V. Latora, Y. Moreno, M. Chavez and D. U. Hwang, Complex networks: Structure and dynamics, Phys. Rep., 424 (2006), 175-308.  doi: 10.1016/j.physrep.2005.10.009.

[4]

B. Bolobás, Modern Graph Theory, Springer-Verlag, New York, 1998.

[5] J. Casti, Linear Dynamical Systems, Academic Press, Orlando, Florida, 1987. 
[6]

B. Dorn, Flows in Infinite Networks - A Semigroup Approach, Ph.D thesis, Tuebingen University, Germany, 2008.

[7]

B. Dorn, M. Kramar Fijav$\tilde z$, R. 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.

[8]

M. El Azzouzi, H. Bouslous, L. Maniar and S. Boulite, Constrained approximate controllability of boundary control systems, IMA J. Math. Control Inform., 33 (2016), 669-683.  doi: 10.1093/imamci/dnv002.

[9]

K.-J. Engel and M. Kramar Fijav$\tilde z$, Exact and positive controllability of boundary control systems, Netw. Heterog. Media, 12 (2017), 319-337.  doi: 10.3934/nhm.2017014.

[10]

K.-J. Engel, M. Kramar Fijav$\tilde z$, R. Nagel and E. Sikolya, Vertex control of flows in networks, Netw. Heterog. Media, 3 (2008), 709-722.  doi: 10.3934/nhm.2008.3.709.

[11]

K.-J. Engel, B. Klöss, M. Kramar Fijav$\tilde z$, 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.

[12]

K.-J. Engel and R. Nagel, One-parameter Semigroups for Linear Evolution Equations, Springer-Verlag, New York, 2000.

[13]

H. O. Fattorini, Boundary control systems, SIAM J. Control, 6 (1968), 349-385.  doi: 10.1137/0306025.

[14]

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

[15]

S. Hadd, An Evolution Equation Approach, Ph.D thesis, Cadi Ayyad University, Marrakech, 2005.

[16]

S. Hadd, R. Manzo and A. Rhandi, Unbounded perturbations of the generator domain, Discrete Contin. Dyn. Syst., 35 (2015), 703-723.  doi: 10.3934/dcds.2015.35.703.

[17]

U. Knauer, Algeberaic Graph Theory. Morphisms, Monoids and Matrices, De Gruyter Studies in Mathematics, Vol. 41, Walter de Gruyter & Co., Berlin, 2011. doi: 10.1515/9783110255096.

[18]

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.

[19]

C.-T. Lin, Structural controllability, IEEE Trans. Automatic Control, AC-19 (1974), 201-208.  doi: 10.1109/tac.1974.1100557.

[20]

Y.-Y. Liu, J.-J. Slotine and A.-L. Barabasi, Controllability of complex networks, Nature, 473 (2011), 167-173.  doi: 10.1038/nature10011.

[21]

T. Matrai and E. Sikolya, Asymptotic behavior of flows in networks, Forum Math., 19 (2007), 429-461.  doi: 10.1515/FORUM.2007.018.

[22]

D. Salamon, Infinite-dimensional linear system with unbounded control and observation: A functional analytic approach, Trans. Amer. Math. Soc., 300 (1987), 383-431.  doi: 10.2307/2000351.

[23]

R. W. Shields and J. B. Pearson, Structural controllability of multi-input linear systems, IEEE Trans. Automat. Control, AC-21 (1976), 203-212.  doi: 10.1109/tac.1976.1101198.

[24]

E. Sikolya, Semigroups for Flows in Networks, Ph.D thesis, Tuebingen University, Germany, 2004.

[25] O. J. Staffans, Well-posed Linear Systems, Cambridge Univ. Press, Cambridge, 2005.  doi: 10.1017/CBO9780511543197.
[26]

M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, Birkhäuser Verlag, Basel, 2009. doi: 10.1007/978-3-7643-8994-9.

[27]

G. Weiss, Admissible observation operators for linear semigoups, Israel J. Math., 65 (1989), 17-43.  doi: 10.1007/BF02788172.

[28]

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

[29]

G. Weiss, Transfer functions of regular linear systems. Part Ⅰ: Characterizations of regularity, Trans. Amer. Math. Soc., 342 (1994), 827-854.  doi: 10.2307/2154655.

[30]

G. Weiss, Regular linear systems with feedback, Math. Control Signals Systems, 7 (1994), 23-57.  doi: 10.1007/BF01211484.

[1]

Sergei Avdonin, Jeff Park, Luz de Teresa. The Kalman condition for the boundary controllability of coupled 1-d wave equations. Evolution Equations and Control Theory, 2020, 9 (1) : 255-273. doi: 10.3934/eect.2020005
[2]

Guillaume Olive. Boundary approximate controllability of some linear parabolic systems. Evolution Equations and Control Theory, 2014, 3 (1) : 167-189. doi: 10.3934/eect.2014.3.167
[3]

Klaus-Jochen Engel, Marjeta Kramar FijavŽ. Exact and positive controllability of boundary control systems. Networks and Heterogeneous Media, 2017, 12 (2) : 319-337. doi: 10.3934/nhm.2017014
[4]

Hamid Maarouf. Local Kalman rank condition for linear time varying systems. Mathematical Control and Related Fields, 2022, 12 (2) : 433-446. doi: 10.3934/mcrf.2021029
[5]

Tatsien Li (Daqian Li). Global exact boundary controllability for first order quasilinear hyperbolic systems. Discrete and Continuous Dynamical Systems - B, 2010, 14 (4) : 1419-1432. doi: 10.3934/dcdsb.2010.14.1419
[6]

Kuntal Bhandari, Franck Boyer. Boundary null-controllability of coupled parabolic systems with Robin conditions. Evolution Equations and Control Theory, 2021, 10 (1) : 61-102. doi: 10.3934/eect.2020052
[7]

Tatsien Li, Bopeng Rao, Zhiqiang Wang. A note on the one-side exact boundary controllability for quasilinear hyperbolic systems. Communications on Pure and Applied Analysis, 2009, 8 (1) : 405-418. doi: 10.3934/cpaa.2009.8.405
[8]

Matthias Eller. A remark on Littman's method of boundary controllability. Evolution Equations and Control Theory, 2013, 2 (4) : 621-630. doi: 10.3934/eect.2013.2.621
[9]

Tatsien Li, Bopeng Rao, Zhiqiang Wang. Exact boundary controllability and observability for first order quasilinear hyperbolic systems with a kind of nonlocal boundary conditions. Discrete and Continuous Dynamical Systems, 2010, 28 (1) : 243-257. doi: 10.3934/dcds.2010.28.243
[10]

Orazio Arena. A problem of boundary controllability for a plate. Evolution Equations and Control Theory, 2013, 2 (4) : 557-562. doi: 10.3934/eect.2013.2.557
[11]

Peter Šepitka. Riccati equations for linear Hamiltonian systems without controllability condition. Discrete and Continuous Dynamical Systems, 2019, 39 (4) : 1685-1730. doi: 10.3934/dcds.2019074
[12]

Ait Ben Hassi El Mustapha, Fadili Mohamed, Maniar Lahcen. On Algebraic condition for null controllability of some coupled degenerate systems. Mathematical Control and Related Fields, 2019, 9 (1) : 77-95. doi: 10.3934/mcrf.2019004
[13]

Yassine El Gantouh, Said Hadd. Well-posedness and approximate controllability of neutral network systems. Networks and Heterogeneous Media, 2021, 16 (4) : 569-589. doi: 10.3934/nhm.2021018
[14]

Damien Allonsius, Franck Boyer. Boundary null-controllability of semi-discrete coupled parabolic systems in some multi-dimensional geometries. Mathematical Control and Related Fields, 2020, 10 (2) : 217-256. doi: 10.3934/mcrf.2019037
[15]

Monica De Angelis, Pasquale Renno. Asymptotic effects of boundary perturbations in excitable systems. Discrete and Continuous Dynamical Systems - B, 2014, 19 (7) : 2039-2045. doi: 10.3934/dcdsb.2014.19.2039
[16]

Umberto De Maio, Akamabadath K. Nandakumaran, Carmen Perugia. Exact internal controllability for the wave equation in a domain with oscillating boundary with Neumann boundary condition. Evolution Equations and Control Theory, 2015, 4 (3) : 325-346. doi: 10.3934/eect.2015.4.325
[17]

John E. Lagnese. Controllability of systems of interconnected membranes. Discrete and Continuous Dynamical Systems, 1995, 1 (1) : 17-33. doi: 10.3934/dcds.1995.1.17
[18]

Lianwen Wang. Approximate controllability and approximate null controllability of semilinear systems. Communications on Pure and Applied Analysis, 2006, 5 (4) : 953-962. doi: 10.3934/cpaa.2006.5.953
[19]

Mohammed Aassila. Exact boundary controllability of a coupled system. Discrete and Continuous Dynamical Systems, 2000, 6 (3) : 665-672. doi: 10.3934/dcds.2000.6.665
[20]

Larissa V. Fardigola. Transformation operators in controllability problems for the wave equations with variable coefficients on a half-axis controlled by the Dirichlet boundary condition. Mathematical Control and Related Fields, 2015, 5 (1) : 31-53. doi: 10.3934/mcrf.2015.5.31

2020 Impact Factor: 1.081

Tools

Metrics

  • PDF downloads (348)
  • HTML views (536)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy