Well-posedness and approximate controllability of neutral network systems

Yassine El Gantouh; Said Hadd

doi:10.3934/nhm.2021018

Article Contents

2021, Volume 16, Issue 4: 569-589. Doi: 10.3934/nhm.2021018 open access

This issue Previous Article Bi-Continuous semigroups for flows on infinite networks Next Article Convex and quasiconvex functions in metric graphs

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

Received: February 2021

Revised: May 2021

Early access: July 2021

Published: December 2021

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

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Abstract

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.

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Mathematics Subject Classification: Primary: 35F46, 93B05; Secondary: 93C20.

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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.