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doi: 10.3934/eect.2020091

Approximate controllability of network systems

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  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.

Citation: Yassine El Gantouh, Said Hadd, Abdelaziz Rhandi. Approximate controllability of network systems. Evolution Equations & Control Theory, 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.  Google Scholar

[2]

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

[3]

S. BoccalettiV. LatoraY. MorenoM. Chavez and D. U. Hwang, Complex networks: Structure and dynamics, Phys. Rep., 424 (2006), 175-308.  doi: 10.1016/j.physrep.2005.10.009.  Google Scholar

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

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

[7]

B. DornM. 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.  Google Scholar

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M. El AzzouziH. BouslousL. 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.  Google Scholar

[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.  Google Scholar

[10]

K.-J. EngelM. 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.  Google Scholar

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K.-J. EngelB. KlössM. 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.  Google Scholar

[12]

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

[13]

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

[14]

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

[15]

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

[16]

S. HaddR. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

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

[20]

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

[21]

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

[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.  Google Scholar

[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.  Google Scholar

[24]

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

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

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

[27]

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

[28]

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

[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.  Google Scholar

[30]

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

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.  Google Scholar

[2]

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

[3]

S. BoccalettiV. LatoraY. MorenoM. Chavez and D. U. Hwang, Complex networks: Structure and dynamics, Phys. Rep., 424 (2006), 175-308.  doi: 10.1016/j.physrep.2005.10.009.  Google Scholar

[4]

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

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

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

[7]

B. DornM. 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.  Google Scholar

[8]

M. El AzzouziH. BouslousL. 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.  Google Scholar

[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.  Google Scholar

[10]

K.-J. EngelM. 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.  Google Scholar

[11]

K.-J. EngelB. KlössM. 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.  Google Scholar

[12]

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

[13]

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

[14]

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

[15]

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

[16]

S. HaddR. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

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

[20]

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

[21]

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

[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.  Google Scholar

[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.  Google Scholar

[24]

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

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

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

[27]

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

[28]

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

[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.  Google Scholar

[30]

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

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