# American Institute of Mathematical Sciences

December  2021, 16(4): 569-589. doi: 10.3934/nhm.2021018

## Well-posedness and approximate controllability of neutral network systems

 Department of Mathematics, Faculty of Sciences Agadir, Ibn Zohr University, Hay Dakhla, BP. 8106, 80000–Agadir, Morocco

Received  February 2021 Revised  May 2021 Published  December 2021 Early access  July 2021

Fund Project: This work has been supported by COST Action CA18232. The authors would like to thank Prof. A. Rhandi for discussing some part of this paper

In this paper, we study the concept of approximate controllability of retarded network systems of neutral type. On one hand, we reformulate such systems as free-delay boundary control systems on product spaces. On the other hand, we use the rich theory of infinite-dimensional linear systems to derive necessary and sufficient conditions for the approximate controllability. Moreover, we propose a rank condition for which we can easily verify the conditions of controllability. Our approach is mainly based on the feedback theory of regular linear systems in the Salamon-Weiss sense.

Citation: 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
##### References:
 [1] J. Banasiak and P. Namayanja, Asymptotic behaviour of flows on reducible networks, J. Networks Heterogeneous Media, 9 (2014), 197-216.  doi: 10.3934/nhm.2014.9.197. [2] J. Banasiak and A. Puchalska, A Transport on networks - a playground of continuous and discrete mathematics in population dynamics, In Mathematics Applied to Engineering, Modelling, and Social Issues, Springer, Cham, 2019,439–487. [3] A. Bátkai and S. Piazzera, Semigroups for Delay Equations, Research Notes in Mathematics, 10, A K Peters Ltd, Wellesley, 2005. [4] F. Bayazit, B. Dorn and M. K. Fijavž, Asymptotic periodicity of flows in time-depending networks, J. Networks Heterogeneous Media, 8 (2013), 843-855.  doi: 10.3934/nhm.2013.8.843. [5] 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. [6] 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. [7] H. Bounit and S. Hadd, Regular linear systems governed by neutral FDEs., J. Math. Anal. Appl., 320 (2006), 836-858.  doi: 10.1016/j.jmaa.2005.07.048. [8] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, 2010. [9] R. F. Curtain and H. Zwart, An Introduction to Infinite-Dimensional Linear Systems Theory, Texts in Applied Mathematics, 21 Spinger-Verlag, New York, 1995. doi: 10.1007/978-1-4612-4224-6. [10] J. Diblík, D. Ya. Khusainov and M. Røužičková, Controllability of linear discrete systems with constant coefficients and pure delay, SIAM J. Control Optim., 47 (2008), 1140-1149.  doi: 10.1137/070689085. [11] B. Dorn, M. F. Kramar, R. Nagel and A. Radl, The semigroup approach to transport processes in networks, Physica D: Nonlinear Phenomena, 239 (2010), 1416-1421.  doi: 10.1016/j.physd.2009.06.012. [12] Y. El Gantouh, S. Hadd and A. Rhandi, Approximate controllabilty of network systems, Evol. Equ. Control Theory, (2020). doi: 10.3934/eect.2020091. [13] Y. El Gantouh, S. Hadd and A. Rhandi, Controllability of vertex delay type problems by the regular linear systems approach, Preprint. [14] K.-J. Engel and M. K. Fijavž, Exact and positive controllability of boundary control systems, networks, J. Networks Heterogeneous Media, 12 (2017), 319-337.  doi: 10.3934/nhm.2017014. [15] K.-J. Engel, M. K. Fijavž, B. Klöss, 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. [16] K.-J. Engel and R. Nagel, One-parameter Semigroups for Linear Evolution Equations, Springer-Verlag, New York, 2000. [17] G. Greiner, Perturbing the boundary conditions of a generator, Houston J. Math., 13 (1987), 213-229. [18] S. Hadd, Unbounded perturbations of $C_0$-semigroups on Banach spaces and Applications, Semigroup Forum, 70 (2005), 451-465.  doi: 10.1007/s00233-004-0172-7. [19] S. Hadd, A. Idrissi and A. Rhandi, The regular linear systems associated to the shift semigroups and application to control delay systems with delay, Math. Control Signals Sys., 18 (2006), 272-291.  doi: 10.1007/s00498-006-0002-4. [20] S. Hadd, R. Manzo and A. Rhandi, Unbounded perturbations of the generator domain, Discrete and Continuous Dynamical Sys., 35 (2015), 703-723.  doi: 10.3934/dcds.2015.35.703. [21] S. Hadd, H. Nounou and M. Nounou, Eventual norm continuity for neutral semigroups on Banach spaces, J. Math. Anal. Appl., 375 (2011), 543-552.  doi: 10.1016/j.jmaa.2010.09.065. [22] J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Applied Math. Sciences Series, 99, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4342-7. [23] T. Matrai and E. Sikolya, Asymptotic behavior of flows in networks, Forum Math., 19 (2007), 429-461.  doi: 10.1515/FORUM.2007.018. [24] R. Rabah and G. M. Sklyar, The analysis of exact controllability of neutral-type systems by the moment problem approach, SIAM J. Control Optim., 46 (2007), 2148-2181.  doi: 10.1137/060650246. [25] D. Salamon, On controllability and observability of time delay systems, IEEE Trans. Automat. Control, 29 (1984), 432-439.  doi: 10.1109/TAC.1984.1103560. [26] 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. [27] N. K. Son, D. D. Thuan and N. T. Hong, Radius of approximate controllability oflinear retarded systems under structured perturbations, Systems Control Lett., 84 (2015), 13-20.  doi: 10.1016/j.sysconle.2015.07.006. [28] O. Staffans, Well-posed Linear Systems, Press, Cambridge Univ, 2005.  doi: 10.1017/CBO9780511543197. [29] Y. Sun, P. W. Nelson and A. G. Ulsoy, Controllability and observability of systems of linear delay differential equations via the matrix Lambert $W$ function, IEEE Trans. Automat. Control, 53 (2008), 854-860.  doi: 10.1109/TAC.2008.919549. [30] M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, Birkhäuser, Basel, Boston, Berlin, 2009. doi: 10.1007/978-3-7643-8994-9. [31] G. Weiss, Admissible observation operators for linear semigoups, Israel J. Math., 65 (1989), 17-43.  doi: 10.1007/BF02788172. [32] G. Weiss, Admissibility of unbounded control operators, SIAM J. Control Optim., 27 (1989), 527-545.  doi: 10.1137/0327028. [33] G. Weiss, Transfer functions of regular linear systems. Part I: Characterization of regularity, Trans. Amer. Math. Soc., 342 (1994), 827-854.  doi: 10.2307/2154655. [34] G. Weiss, Regular linear systems with feedback, Math. Control Signals Systems, 7 (1994), 23-57.  doi: 10.1007/BF01211484. [35] Q. C. Zhong, Robust Control of Time-Delay Systems, Springer-Verlag Limited, London, 2006.

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##### References:
 [1] J. Banasiak and P. Namayanja, Asymptotic behaviour of flows on reducible networks, J. Networks Heterogeneous Media, 9 (2014), 197-216.  doi: 10.3934/nhm.2014.9.197. [2] J. Banasiak and A. Puchalska, A Transport on networks - a playground of continuous and discrete mathematics in population dynamics, In Mathematics Applied to Engineering, Modelling, and Social Issues, Springer, Cham, 2019,439–487. [3] A. Bátkai and S. Piazzera, Semigroups for Delay Equations, Research Notes in Mathematics, 10, A K Peters Ltd, Wellesley, 2005. [4] F. Bayazit, B. Dorn and M. K. Fijavž, Asymptotic periodicity of flows in time-depending networks, J. Networks Heterogeneous Media, 8 (2013), 843-855.  doi: 10.3934/nhm.2013.8.843. [5] 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. [6] 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. [7] H. Bounit and S. Hadd, Regular linear systems governed by neutral FDEs., J. Math. Anal. Appl., 320 (2006), 836-858.  doi: 10.1016/j.jmaa.2005.07.048. [8] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, 2010. [9] R. F. Curtain and H. Zwart, An Introduction to Infinite-Dimensional Linear Systems Theory, Texts in Applied Mathematics, 21 Spinger-Verlag, New York, 1995. doi: 10.1007/978-1-4612-4224-6. [10] J. Diblík, D. Ya. Khusainov and M. Røužičková, Controllability of linear discrete systems with constant coefficients and pure delay, SIAM J. Control Optim., 47 (2008), 1140-1149.  doi: 10.1137/070689085. [11] B. Dorn, M. F. Kramar, R. Nagel and A. Radl, The semigroup approach to transport processes in networks, Physica D: Nonlinear Phenomena, 239 (2010), 1416-1421.  doi: 10.1016/j.physd.2009.06.012. [12] Y. El Gantouh, S. Hadd and A. Rhandi, Approximate controllabilty of network systems, Evol. Equ. Control Theory, (2020). doi: 10.3934/eect.2020091. [13] Y. El Gantouh, S. Hadd and A. Rhandi, Controllability of vertex delay type problems by the regular linear systems approach, Preprint. [14] K.-J. Engel and M. K. Fijavž, Exact and positive controllability of boundary control systems, networks, J. Networks Heterogeneous Media, 12 (2017), 319-337.  doi: 10.3934/nhm.2017014. [15] K.-J. Engel, M. K. Fijavž, B. Klöss, 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. [16] K.-J. Engel and R. Nagel, One-parameter Semigroups for Linear Evolution Equations, Springer-Verlag, New York, 2000. [17] G. Greiner, Perturbing the boundary conditions of a generator, Houston J. Math., 13 (1987), 213-229. [18] S. Hadd, Unbounded perturbations of $C_0$-semigroups on Banach spaces and Applications, Semigroup Forum, 70 (2005), 451-465.  doi: 10.1007/s00233-004-0172-7. [19] S. Hadd, A. Idrissi and A. Rhandi, The regular linear systems associated to the shift semigroups and application to control delay systems with delay, Math. Control Signals Sys., 18 (2006), 272-291.  doi: 10.1007/s00498-006-0002-4. [20] S. Hadd, R. Manzo and A. Rhandi, Unbounded perturbations of the generator domain, Discrete and Continuous Dynamical Sys., 35 (2015), 703-723.  doi: 10.3934/dcds.2015.35.703. [21] S. Hadd, H. Nounou and M. Nounou, Eventual norm continuity for neutral semigroups on Banach spaces, J. Math. Anal. Appl., 375 (2011), 543-552.  doi: 10.1016/j.jmaa.2010.09.065. [22] J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Applied Math. Sciences Series, 99, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4342-7. [23] T. Matrai and E. Sikolya, Asymptotic behavior of flows in networks, Forum Math., 19 (2007), 429-461.  doi: 10.1515/FORUM.2007.018. [24] R. Rabah and G. M. Sklyar, The analysis of exact controllability of neutral-type systems by the moment problem approach, SIAM J. Control Optim., 46 (2007), 2148-2181.  doi: 10.1137/060650246. [25] D. Salamon, On controllability and observability of time delay systems, IEEE Trans. Automat. Control, 29 (1984), 432-439.  doi: 10.1109/TAC.1984.1103560. [26] 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. [27] N. K. Son, D. D. Thuan and N. T. Hong, Radius of approximate controllability oflinear retarded systems under structured perturbations, Systems Control Lett., 84 (2015), 13-20.  doi: 10.1016/j.sysconle.2015.07.006. [28] O. Staffans, Well-posed Linear Systems, Press, Cambridge Univ, 2005.  doi: 10.1017/CBO9780511543197. [29] Y. Sun, P. W. Nelson and A. G. Ulsoy, Controllability and observability of systems of linear delay differential equations via the matrix Lambert $W$ function, IEEE Trans. Automat. Control, 53 (2008), 854-860.  doi: 10.1109/TAC.2008.919549. [30] M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, Birkhäuser, Basel, Boston, Berlin, 2009. doi: 10.1007/978-3-7643-8994-9. [31] G. Weiss, Admissible observation operators for linear semigoups, Israel J. Math., 65 (1989), 17-43.  doi: 10.1007/BF02788172. [32] G. Weiss, Admissibility of unbounded control operators, SIAM J. Control Optim., 27 (1989), 527-545.  doi: 10.1137/0327028. [33] G. Weiss, Transfer functions of regular linear systems. Part I: Characterization of regularity, Trans. Amer. Math. Soc., 342 (1994), 827-854.  doi: 10.2307/2154655. [34] G. Weiss, Regular linear systems with feedback, Math. Control Signals Systems, 7 (1994), 23-57.  doi: 10.1007/BF01211484. [35] Q. C. Zhong, Robust Control of Time-Delay Systems, Springer-Verlag Limited, London, 2006.
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