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

Well-posedness and approximate controllability of neutral network systems

Yassine El Gantouh and  Said Hadd ,

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

* Corresponding author: Said Hadd, s.hadd@uiz.ac.ma

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.

Keywords: Neutral network systems, boundary control systems, approximate controllability, controllability rank condition.
Mathematics Subject Classification: Primary: 35F46, 93B05; Secondary: 93C20.
Citation: Yassine El Gantouh, Said Hadd. Well-posedness and approximate controllability of neutral network systems. Networks & 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.  Google Scholar

[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.  Google Scholar

[3]

A. Bátkai and S. Piazzera, Semigroups for Delay Equations, Research Notes in Mathematics, 10, A K Peters Ltd, Wellesley, 2005.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, 2010.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

Y. El Gantouh, S. Hadd and A. Rhandi, Approximate controllabilty of network systems, Evol. Equ. Control Theory, (2020). doi: 10.3934/eect.2020091.  Google Scholar

[13]

Y. El Gantouh, S. Hadd and A. Rhandi, Controllability of vertex delay type problems by the regular linear systems approach, Preprint. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

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

[17]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

[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.  Google Scholar

[31]

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

[32]

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

[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.  Google Scholar

[34]

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

[35]

Q. C. Zhong, Robust Control of Time-Delay Systems, Springer-Verlag Limited, London, 2006. Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[3]

A. Bátkai and S. Piazzera, Semigroups for Delay Equations, Research Notes in Mathematics, 10, A K Peters Ltd, Wellesley, 2005.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, 2010.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

Y. El Gantouh, S. Hadd and A. Rhandi, Approximate controllabilty of network systems, Evol. Equ. Control Theory, (2020). doi: 10.3934/eect.2020091.  Google Scholar

[13]

Y. El Gantouh, S. Hadd and A. Rhandi, Controllability of vertex delay type problems by the regular linear systems approach, Preprint. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

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

[17]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

[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.  Google Scholar

[31]

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

[32]

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

[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.  Google Scholar

[34]

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

[35]

Q. C. Zhong, Robust Control of Time-Delay Systems, Springer-Verlag Limited, London, 2006. Google Scholar

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