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Bi-Continuous semigroups for flows on infinite networks
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 |
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.
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. |
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. |
[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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