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A discrete hierarchy of double bracket equations and a class of negative power series
March
2017, 7(1): 53-72.
doi: 10.3934/mcrf.2017004
Control and stabilization of 2 × 2 hyperbolic systems on graphs
Université de Valenciennes et du Hainaut Cambrésis, LAMAV, FR CNRS 2956, F-59313 -Valenciennes Cedex 9, France |
We consider 2× 2 (first order) hyperbolic systems on networks subject to general transmission conditions and to some dissipative boundary conditions on some external vertices. We find sufficient but natural conditions on these transmission conditions that guarantee the exponential decay of the full system on graphs with dissipative conditions at all except one external vertices. This result is obtained with the help of a perturbation argument and an observability estimate for an associated wave type equation. An exact controllability result is also deduced.
Mathematics Subject Classification:
Primary: 35L50; Secondary: 93D15.
Citation:
Serge Nicaise. Control and stabilization of 2 × 2 hyperbolic systems on graphs. Mathematical Control and Related Fields,
2017, 7
(1)
: 53-72.
doi: 10.3934/mcrf.2017004
![]()
References:
| [1] |
F. Ali Mehmeti,
A characterisation of generalized c∞ notion on nets, Integral Eq. and Operator Theory, 9 (1986), 753-766.
doi: 10.1007/BF01202515. |
| [2] |
F. Ali Mehmeti, Nonlinear Wave in Networks, volume 80 of Math. Res. Akademie Verlag, 1994. |
| [3] |
K. Ammari and S. Nicaise, Stabilization of Elastic Systems by Collocated Feedback, volume 2124 of Lecture Notes in Mathematics, Springer, Cham, 2015.
doi: 10.1007/978-3-319-10900-8. |
| [4] |
W. Arendt and C.J.K. Batty,
Tauberian theorems and stability of one-parameter semigroups, Trans. Amer. Math. Soc., 306 (1988), 837-852.
doi: 10.1090/S0002-9947-1988-0933321-3. |
| [5] |
G. Bastin and J.-M. Coron,
On boundary feedback stabilization of non-uniform linear 2×2 hyperbolic systems over a bounded interval, Systems Control Lett., 60 (2011), 900-906.
doi: 10.1016/j.sysconle.2011.07.008. |
| [6] |
J. von Below,
A characteristic equation associated to an eigenvalue problem on c2-networks, Linear Algebra Appl., 71 (1985), 309-325.
doi: 10.1016/0024-3795(85)90258-7. |
| [7] |
J. von Below,
Classical solvability of linear parabolic equations on networks, J. Differential Equations, 72 (1988), 316-337.
doi: 10.1016/0022-0396(88)90158-1. |
| [8] |
J. von Below,
Sturm-Liouville eigenvalue problems on networks, Math. Methods Appl. Sci., 10 (1988), 383-395.
doi: 10.1002/mma.1670100404. |
| [9] |
J. von Below and D. Mugnolo,
The spectrum of the Hilbert space valued second derivative with general self-adjoint boundary conditions, Linear Algebra Appl., 439 (2013), 1792-1814.
doi: 10.1016/j.laa.2013.05.011. |
| [10] |
G. Berkolaiko and P. Kuchment, Introduction to Quantum Graphs, volume 186 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 2013. |
| [11] |
A. Bressan, Hyperbolic conservation laws: an illustrated tutorial, In Modelling and optimisation of flows on networks, volume 2062 of Lecture Notes in Math., pages 157-245. Springer, Heidelberg, 2013.
doi: 10.1007/978-3-642-32160-3_2. |
| [12] |
A. Bressan, S. Čanić, M. Garavello, M. Herty and B. Piccoli,
Flows on networks: Recent results and perspectives, EMS Surv. Math. Sci., 1 (2014), 47-111.
doi: 10.4171/EMSS/2. |
| [13] |
S. Čanić and E.H. Kim,
Mathematical analysis of the quasilinear effects in a hyperbolic model blood flow through compliant axi-symmetric vessels, Math. Methods Appl. Sci., 26 (2003), 1161-1186.
doi: 10.1002/mma.407. |
| [14] |
S. Cardanobile and D. Mugnolo,
Parabolic systems with coupled boundary conditions, J. Differential Equations, 247 (2009), 1229-1248.
doi: 10.1016/j.jde.2009.04.013. |
| [15] |
R. Carlson,
Spectral theory for nonconservative transmission line networks, Netw. Heterog. Media, 6 (2011), 257-277.
doi: 10.3934/nhm.2011.6.257. |
| [16] |
R. Dáger and E. Zuazua, Wave Propagation, Observation and Control in 1-d Flexible Multi-structures, volume 50 of Mathématiques & Applications (Berlin) [Mathematics & Applications], Springer-Verlag, Berlin, 2006.
doi: 10.1007/3-540-37726-3. |
| [17] |
A. Diagne, G. Bastin and J.-M. Coron,
Lyapunov exponential stability of 1-D linear hyperbolic systems of balance laws, Automatica J. IFAC, 48 (2012), 109-114.
doi: 10.1016/j.automatica.2011.09.030. |
| [18] |
M. Dick, M. Gugat and G. Leugering,
Classical solutions and feedback stabilization for the gas flow in a sequence of pipes, Netw. Heterog. Media, 5 (2010), 691-709.
doi: 10.3934/nhm.2010.5.691. |
| [19] |
M. Gugat, M. Dick and G. Leugering,
Gas flow in fan-shaped networks: Classical solutions and feedback stabilization, SIAM J. Control Optim., 49 (2011), 2101-2117.
doi: 10.1137/100799824. |
| [20] |
M. Gugat and M. Herty,
Existence of classical solutions and feedback stabilization for the flow in gas networks, ESAIM Control Optim. Calc. Var., 17 (2011), 28-51.
doi: 10.1051/cocv/2009035. |
| [21] |
F.L. Huang,
Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces, Ann. Differential Equations, 1 (1985), 43-56.
|
| [22] |
S. Imperiale and P. Joly,
Mathematical modeling of electromagnetic wave propagation in heterogeneous lossy coaxial cables with variable cross section, Appl. Numer. Math., 79 (2014), 42-61.
doi: 10.1016/j.apnum.2013.03.011. |
| [23] |
V. Kostrykin and R. Schrader,
Kirchhoff's rule for quantum wires, J. Phys. A, 32 (1999), 595-630.
doi: 10.1088/0305-4470/32/4/006. |
| [24] |
P. Kuchment,
Quantum graphs. Ⅰ. Some basic structures, Waves Random Media, 14 (2004), S107-S128.
doi: 10.1088/0959-7174/14/1/014. |
| [25] |
J. E. Lagnese, G. Leugering and E. Schmidt, Modeling, Analysis and Control of Dynamic Elastic Multi-Link Structures, Birkhäuser, Boston, 1994.
doi: 10.1007/978-1-4612-0273-8. |
| [26] |
J.E. Lagnese, G. Leugering and E.J. P.G. Schmidt,
On the analysis and control of hyperbolic systems associated with vibrating networks, Proc. Roy. Soc. Edinburgh Sect. A, 124 (1994), 77-104.
doi: 10.1017/S0308210500029206. |
| [27] |
G. Leugering and E.J. P.G. Schmidt,
On the modelling and stabilization of flows in networks of open canals, SIAM J. Control Optim., 41 (2002), 164-180.
doi: 10.1137/S0363012900375664. |
| [28] |
J. -L. Lions, Contrôlabilité Exacte, Perturbations et Stabilisation de Systémes Distribués. Tome 1, volume 8 of Recherches en Mathématiques Appliquées [Research in Applied Mathematics], Masson, Paris, 1988. |
| [29] |
G. Lumer, Connecting of local operators and evolution equations on networks, In Potential theory, Copenhagen 1979 (Proc. Colloq., Copenhagen, 1979), volume 787 of Lecture Notes in Math., pages 219-234. Springer, Berlin, 1980. |
| [30] |
A. Maffucci and G. Miano, A unified approach for the analysis of networks composed of transmission lines and lumped circuits, In Scientific computing in electrical engineering, volume 9 of Math. Ind., pages 3-11. Springer, Berlin, 2006.
doi: 10.1007/978-3-540-32862-9_1. |
| [31] |
D. Mercier and S. Nicaise,
Existence results for general systems of differential equations on one-dimensional networks and prewavelets approximation, Discrete Contin. Dynam. Systems, 4 (1998), 273-300.
doi: 10.3934/dcds.1998.4.273. |
| [32] |
D. Mugnolo and R. Pröpper, Gradient systems on networks, Discrete Contin. Dyn. Syst., (Dynamical systems, differential equations and applications. 8th AIMS Conference. Suppl.), 2 (2011), 1078-1090. |
| [33] |
S. Nicaise,
Spectre des réseaux topologiques finis, Bull. Sc. Math., 2ème série, 111 (1987), 401-413.
|
| [34] |
S. Nicaise,
Stability and controllability of an abstract evolution equation of hyperbolic type and concrete applications, Rendiconti di Matematica Serie Ⅶ, 23 (2003), 83-116.
|
| [35] |
S. Nicaise and O. Zaïr,
Identifiability, stability and reconstruction results of point sources by boundary measurements in heteregeneous trees, Rev. Mat. Complut., 16 (2003), 151-178.
doi: 10.5209/rev_REMA.2003.v16.n1.16865. |
| [36] |
A. Pazy,
Semigroups of Linear Operators and Applications to Partial Differential Equations, volume 44 of Applied Math. Sciences, Springer-Verlag, New York, 1983.
doi: 10.1007/978-1-4612-5561-1. |
| [37] |
V. Perrollaz and L. Rosier,
Finite-time stabilization of 2× 2 hyperbolic systems on tree-shaped networks, SIAM J. Control Optim., 52 (2014), 143-163.
doi: 10.1137/130910762. |
| [38] |
J. Prüss,
On the spectrum of C0-semigroups, Trans. Amer. Math. Soc., 284 (1984), 847-857.
doi: 10.2307/1999112. |
| [39] |
E.J. P.G. Schmidt,
On the modelling and exact controllability of networks of vibrating strings, SIAM J. Control Optim., 30 (1992), 229-245.
doi: 10.1137/0330015. |
| [40] |
S.J. Sherwin, V. Franke, J. Peiró and K. Parker,
One-dimensional modelling of a vascular network in space-time variables, J. Engrg. Math., 47 (2003), 217-250.
doi: 10.1023/B:ENGI.0000007979.32871.e2. |
| [41] |
M. Suzuki, J.-i. Imura and K. Aihara,
Analysis and stabilization for networked linear hyperbolic systems of rationally dependent conservation laws, Automatica J. IFAC, 49 (2013), 3210-3221.
doi: 10.1016/j.automatica.2013.08.016. |
| [42] |
L. Zhou and G.A. Kriegsmann,
A simple derivation of microstrip transmission line equations, SIAM J. Appl. Math., 70 (2009), 353-367.
doi: 10.1137/080737563. |
| [43] |
C. Zong and G.Q. Xu,
Observability and controllability analysis of blood flow network, Math. Control Relat. Fields, 4 (2014), 521-554.
doi: 10.3934/mcrf.2014.4.521. |
show all references
References:
| [1] |
F. Ali Mehmeti,
A characterisation of generalized c∞ notion on nets, Integral Eq. and Operator Theory, 9 (1986), 753-766.
doi: 10.1007/BF01202515. |
| [2] |
F. Ali Mehmeti, Nonlinear Wave in Networks, volume 80 of Math. Res. Akademie Verlag, 1994. |
| [3] |
K. Ammari and S. Nicaise, Stabilization of Elastic Systems by Collocated Feedback, volume 2124 of Lecture Notes in Mathematics, Springer, Cham, 2015.
doi: 10.1007/978-3-319-10900-8. |
| [4] |
W. Arendt and C.J.K. Batty,
Tauberian theorems and stability of one-parameter semigroups, Trans. Amer. Math. Soc., 306 (1988), 837-852.
doi: 10.1090/S0002-9947-1988-0933321-3. |
| [5] |
G. Bastin and J.-M. Coron,
On boundary feedback stabilization of non-uniform linear 2×2 hyperbolic systems over a bounded interval, Systems Control Lett., 60 (2011), 900-906.
doi: 10.1016/j.sysconle.2011.07.008. |
| [6] |
J. von Below,
A characteristic equation associated to an eigenvalue problem on c2-networks, Linear Algebra Appl., 71 (1985), 309-325.
doi: 10.1016/0024-3795(85)90258-7. |
| [7] |
J. von Below,
Classical solvability of linear parabolic equations on networks, J. Differential Equations, 72 (1988), 316-337.
doi: 10.1016/0022-0396(88)90158-1. |
| [8] |
J. von Below,
Sturm-Liouville eigenvalue problems on networks, Math. Methods Appl. Sci., 10 (1988), 383-395.
doi: 10.1002/mma.1670100404. |
| [9] |
J. von Below and D. Mugnolo,
The spectrum of the Hilbert space valued second derivative with general self-adjoint boundary conditions, Linear Algebra Appl., 439 (2013), 1792-1814.
doi: 10.1016/j.laa.2013.05.011. |
| [10] |
G. Berkolaiko and P. Kuchment, Introduction to Quantum Graphs, volume 186 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 2013. |
| [11] |
A. Bressan, Hyperbolic conservation laws: an illustrated tutorial, In Modelling and optimisation of flows on networks, volume 2062 of Lecture Notes in Math., pages 157-245. Springer, Heidelberg, 2013.
doi: 10.1007/978-3-642-32160-3_2. |
| [12] |
A. Bressan, S. Čanić, M. Garavello, M. Herty and B. Piccoli,
Flows on networks: Recent results and perspectives, EMS Surv. Math. Sci., 1 (2014), 47-111.
doi: 10.4171/EMSS/2. |
| [13] |
S. Čanić and E.H. Kim,
Mathematical analysis of the quasilinear effects in a hyperbolic model blood flow through compliant axi-symmetric vessels, Math. Methods Appl. Sci., 26 (2003), 1161-1186.
doi: 10.1002/mma.407. |
| [14] |
S. Cardanobile and D. Mugnolo,
Parabolic systems with coupled boundary conditions, J. Differential Equations, 247 (2009), 1229-1248.
doi: 10.1016/j.jde.2009.04.013. |
| [15] |
R. Carlson,
Spectral theory for nonconservative transmission line networks, Netw. Heterog. Media, 6 (2011), 257-277.
doi: 10.3934/nhm.2011.6.257. |
| [16] |
R. Dáger and E. Zuazua, Wave Propagation, Observation and Control in 1-d Flexible Multi-structures, volume 50 of Mathématiques & Applications (Berlin) [Mathematics & Applications], Springer-Verlag, Berlin, 2006.
doi: 10.1007/3-540-37726-3. |
| [17] |
A. Diagne, G. Bastin and J.-M. Coron,
Lyapunov exponential stability of 1-D linear hyperbolic systems of balance laws, Automatica J. IFAC, 48 (2012), 109-114.
doi: 10.1016/j.automatica.2011.09.030. |
| [18] |
M. Dick, M. Gugat and G. Leugering,
Classical solutions and feedback stabilization for the gas flow in a sequence of pipes, Netw. Heterog. Media, 5 (2010), 691-709.
doi: 10.3934/nhm.2010.5.691. |
| [19] |
M. Gugat, M. Dick and G. Leugering,
Gas flow in fan-shaped networks: Classical solutions and feedback stabilization, SIAM J. Control Optim., 49 (2011), 2101-2117.
doi: 10.1137/100799824. |
| [20] |
M. Gugat and M. Herty,
Existence of classical solutions and feedback stabilization for the flow in gas networks, ESAIM Control Optim. Calc. Var., 17 (2011), 28-51.
doi: 10.1051/cocv/2009035. |
| [21] |
F.L. Huang,
Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces, Ann. Differential Equations, 1 (1985), 43-56.
|
| [22] |
S. Imperiale and P. Joly,
Mathematical modeling of electromagnetic wave propagation in heterogeneous lossy coaxial cables with variable cross section, Appl. Numer. Math., 79 (2014), 42-61.
doi: 10.1016/j.apnum.2013.03.011. |
| [23] |
V. Kostrykin and R. Schrader,
Kirchhoff's rule for quantum wires, J. Phys. A, 32 (1999), 595-630.
doi: 10.1088/0305-4470/32/4/006. |
| [24] |
P. Kuchment,
Quantum graphs. Ⅰ. Some basic structures, Waves Random Media, 14 (2004), S107-S128.
doi: 10.1088/0959-7174/14/1/014. |
| [25] |
J. E. Lagnese, G. Leugering and E. Schmidt, Modeling, Analysis and Control of Dynamic Elastic Multi-Link Structures, Birkhäuser, Boston, 1994.
doi: 10.1007/978-1-4612-0273-8. |
| [26] |
J.E. Lagnese, G. Leugering and E.J. P.G. Schmidt,
On the analysis and control of hyperbolic systems associated with vibrating networks, Proc. Roy. Soc. Edinburgh Sect. A, 124 (1994), 77-104.
doi: 10.1017/S0308210500029206. |
| [27] |
G. Leugering and E.J. P.G. Schmidt,
On the modelling and stabilization of flows in networks of open canals, SIAM J. Control Optim., 41 (2002), 164-180.
doi: 10.1137/S0363012900375664. |
| [28] |
J. -L. Lions, Contrôlabilité Exacte, Perturbations et Stabilisation de Systémes Distribués. Tome 1, volume 8 of Recherches en Mathématiques Appliquées [Research in Applied Mathematics], Masson, Paris, 1988. |
| [29] |
G. Lumer, Connecting of local operators and evolution equations on networks, In Potential theory, Copenhagen 1979 (Proc. Colloq., Copenhagen, 1979), volume 787 of Lecture Notes in Math., pages 219-234. Springer, Berlin, 1980. |
| [30] |
A. Maffucci and G. Miano, A unified approach for the analysis of networks composed of transmission lines and lumped circuits, In Scientific computing in electrical engineering, volume 9 of Math. Ind., pages 3-11. Springer, Berlin, 2006.
doi: 10.1007/978-3-540-32862-9_1. |
| [31] |
D. Mercier and S. Nicaise,
Existence results for general systems of differential equations on one-dimensional networks and prewavelets approximation, Discrete Contin. Dynam. Systems, 4 (1998), 273-300.
doi: 10.3934/dcds.1998.4.273. |
| [32] |
D. Mugnolo and R. Pröpper, Gradient systems on networks, Discrete Contin. Dyn. Syst., (Dynamical systems, differential equations and applications. 8th AIMS Conference. Suppl.), 2 (2011), 1078-1090. |
| [33] |
S. Nicaise,
Spectre des réseaux topologiques finis, Bull. Sc. Math., 2ème série, 111 (1987), 401-413.
|
| [34] |
S. Nicaise,
Stability and controllability of an abstract evolution equation of hyperbolic type and concrete applications, Rendiconti di Matematica Serie Ⅶ, 23 (2003), 83-116.
|
| [35] |
S. Nicaise and O. Zaïr,
Identifiability, stability and reconstruction results of point sources by boundary measurements in heteregeneous trees, Rev. Mat. Complut., 16 (2003), 151-178.
doi: 10.5209/rev_REMA.2003.v16.n1.16865. |
| [36] |
A. Pazy,
Semigroups of Linear Operators and Applications to Partial Differential Equations, volume 44 of Applied Math. Sciences, Springer-Verlag, New York, 1983.
doi: 10.1007/978-1-4612-5561-1. |
| [37] |
V. Perrollaz and L. Rosier,
Finite-time stabilization of 2× 2 hyperbolic systems on tree-shaped networks, SIAM J. Control Optim., 52 (2014), 143-163.
doi: 10.1137/130910762. |
| [38] |
J. Prüss,
On the spectrum of C0-semigroups, Trans. Amer. Math. Soc., 284 (1984), 847-857.
doi: 10.2307/1999112. |
| [39] |
E.J. P.G. Schmidt,
On the modelling and exact controllability of networks of vibrating strings, SIAM J. Control Optim., 30 (1992), 229-245.
doi: 10.1137/0330015. |
| [40] |
S.J. Sherwin, V. Franke, J. Peiró and K. Parker,
One-dimensional modelling of a vascular network in space-time variables, J. Engrg. Math., 47 (2003), 217-250.
doi: 10.1023/B:ENGI.0000007979.32871.e2. |
| [41] |
M. Suzuki, J.-i. Imura and K. Aihara,
Analysis and stabilization for networked linear hyperbolic systems of rationally dependent conservation laws, Automatica J. IFAC, 49 (2013), 3210-3221.
doi: 10.1016/j.automatica.2013.08.016. |
| [42] |
L. Zhou and G.A. Kriegsmann,
A simple derivation of microstrip transmission line equations, SIAM J. Appl. Math., 70 (2009), 353-367.
doi: 10.1137/080737563. |
| [43] |
C. Zong and G.Q. Xu,
Observability and controllability analysis of blood flow network, Math. Control Relat. Fields, 4 (2014), 521-554.
doi: 10.3934/mcrf.2014.4.521. |
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