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December  2021, 10(4): 749-766. 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  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.

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.
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