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

* Corresponding author

Received  June 2015 Revised  February 2016 Published  December 2016

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.

Citation: Serge Nicaise. Control and stabilization of 2 × 2 hyperbolic systems on graphs. Mathematical Control & 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.  Google Scholar

[2]

F. Ali Mehmeti, Nonlinear Wave in Networks, volume 80 of Math. Res. Akademie Verlag, 1994.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

J. von Below, Sturm-Liouville eigenvalue problems on networks, Math. Methods Appl. Sci., 10 (1988), 383-395.  doi: 10.1002/mma.1670100404.  Google Scholar

[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.  Google Scholar

[10]

G. Berkolaiko and P. Kuchment, Introduction to Quantum Graphs, volume 186 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 2013.  Google Scholar

[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.  Google Scholar

[12]

A. BressanS. ČanićM. GaravelloM. Herty and B. Piccoli, Flows on networks: Recent results and perspectives, EMS Surv. Math. Sci., 1 (2014), 47-111.  doi: 10.4171/EMSS/2.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

R. Carlson, Spectral theory for nonconservative transmission line networks, Netw. Heterog. Media, 6 (2011), 257-277.  doi: 10.3934/nhm.2011.6.257.  Google Scholar

[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.  Google Scholar

[17]

A. DiagneG. 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.  Google Scholar

[18]

M. DickM. 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.  Google Scholar

[19]

M. GugatM. 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.  Google Scholar

[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.  Google Scholar

[21]

F.L. Huang, Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces, Ann. Differential Equations, 1 (1985), 43-56.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

P. Kuchment, Quantum graphs. Ⅰ. Some basic structures, Waves Random Media, 14 (2004), S107-S128.  doi: 10.1088/0959-7174/14/1/014.  Google Scholar

[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.  Google Scholar

[26]

J.E. LagneseG. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[33]

S. Nicaise, Spectre des réseaux topologiques finis, Bull. Sc. Math., 2ème série, 111 (1987), 401-413.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[38]

J. Prüss, On the spectrum of C0-semigroups, Trans. Amer. Math. Soc., 284 (1984), 847-857.  doi: 10.2307/1999112.  Google Scholar

[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.  Google Scholar

[40]

S.J. SherwinV. FrankeJ. 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.  Google Scholar

[41]

M. SuzukiJ.-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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

F. Ali Mehmeti, Nonlinear Wave in Networks, volume 80 of Math. Res. Akademie Verlag, 1994.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

J. von Below, Sturm-Liouville eigenvalue problems on networks, Math. Methods Appl. Sci., 10 (1988), 383-395.  doi: 10.1002/mma.1670100404.  Google Scholar

[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.  Google Scholar

[10]

G. Berkolaiko and P. Kuchment, Introduction to Quantum Graphs, volume 186 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 2013.  Google Scholar

[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.  Google Scholar

[12]

A. BressanS. ČanićM. GaravelloM. Herty and B. Piccoli, Flows on networks: Recent results and perspectives, EMS Surv. Math. Sci., 1 (2014), 47-111.  doi: 10.4171/EMSS/2.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

R. Carlson, Spectral theory for nonconservative transmission line networks, Netw. Heterog. Media, 6 (2011), 257-277.  doi: 10.3934/nhm.2011.6.257.  Google Scholar

[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.  Google Scholar

[17]

A. DiagneG. 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.  Google Scholar

[18]

M. DickM. 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.  Google Scholar

[19]

M. GugatM. 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.  Google Scholar

[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.  Google Scholar

[21]

F.L. Huang, Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces, Ann. Differential Equations, 1 (1985), 43-56.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

P. Kuchment, Quantum graphs. Ⅰ. Some basic structures, Waves Random Media, 14 (2004), S107-S128.  doi: 10.1088/0959-7174/14/1/014.  Google Scholar

[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.  Google Scholar

[26]

J.E. LagneseG. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[33]

S. Nicaise, Spectre des réseaux topologiques finis, Bull. Sc. Math., 2ème série, 111 (1987), 401-413.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[38]

J. Prüss, On the spectrum of C0-semigroups, Trans. Amer. Math. Soc., 284 (1984), 847-857.  doi: 10.2307/1999112.  Google Scholar

[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.  Google Scholar

[40]

S.J. SherwinV. FrankeJ. 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.  Google Scholar

[41]

M. SuzukiJ.-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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

Figure 1.  A tree shaped network: generations of edges
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