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.

[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. 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.

[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. 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.

[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.

[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.

[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. 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.

[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. 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.

[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.

[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. 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.

[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. 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.

[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.

[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.

[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. 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.

[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. 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.

[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.

[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.

Figure 1.  A tree shaped network: generations of edges
[1]

Kais Ammari, Eduard Feireisl, Serge Nicaise. Polynomial stabilization of some dissipative hyperbolic systems. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4371-4388. doi: 10.3934/dcds.2014.34.4371

[2]

Roberto Guglielmi. Indirect stabilization of hyperbolic systems through resolvent estimates. Evolution Equations & Control Theory, 2017, 6 (1) : 59-75. doi: 10.3934/eect.2017004

[3]

Kim Dang Phung. Boundary stabilization for the wave equation in a bounded cylindrical domain. Discrete & Continuous Dynamical Systems - A, 2008, 20 (4) : 1057-1093. doi: 10.3934/dcds.2008.20.1057

[4]

Behzad Azmi, Karl Kunisch. Receding horizon control for the stabilization of the wave equation. Discrete & Continuous Dynamical Systems - A, 2018, 38 (2) : 449-484. doi: 10.3934/dcds.2018021

[5]

Mapundi K. Banda, Michael Herty. Numerical discretization of stabilization problems with boundary controls for systems of hyperbolic conservation laws. Mathematical Control & Related Fields, 2013, 3 (2) : 121-142. doi: 10.3934/mcrf.2013.3.121

[6]

C.P. Walkden. Solutions to the twisted cocycle equation over hyperbolic systems. Discrete & Continuous Dynamical Systems - A, 2000, 6 (4) : 935-946. doi: 10.3934/dcds.2000.6.935

[7]

Cunming Liu, Jianli Liu. Stability of traveling wave solutions to Cauchy problem of diagnolizable quasilinear hyperbolic systems. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4735-4749. doi: 10.3934/dcds.2014.34.4735

[8]

Hans Zwart, Yann Le Gorrec, Bernhard Maschke. Relating systems properties of the wave and the Schrödinger equation. Evolution Equations & Control Theory, 2015, 4 (2) : 233-240. doi: 10.3934/eect.2015.4.233

[9]

Yanni Guo, Genqi Xu, Yansha Guo. Stabilization of the wave equation with interior input delay and mixed Neumann-Dirichlet boundary. Discrete & Continuous Dynamical Systems - B, 2016, 21 (8) : 2491-2507. doi: 10.3934/dcdsb.2016057

[10]

Martin Gugat, Günter Leugering, Ke Wang. Neumann boundary feedback stabilization for a nonlinear wave equation: A strict $H^2$-lyapunov function. Mathematical Control & Related Fields, 2017, 7 (3) : 419-448. doi: 10.3934/mcrf.2017015

[11]

Serge Nicaise, Julie Valein. Stabilization of the wave equation on 1-d networks with a delay term in the nodal feedbacks. Networks & Heterogeneous Media, 2007, 2 (3) : 425-479. doi: 10.3934/nhm.2007.2.425

[12]

Mohammad Akil, Ali Wehbe. Stabilization of multidimensional wave equation with locally boundary fractional dissipation law under geometric conditions. Mathematical Control & Related Fields, 2019, 9 (1) : 97-116. doi: 10.3934/mcrf.2019005

[13]

T. Colin, D. Lannes. Justification of and long-wave correction to Davey-Stewartson systems from quadratic hyperbolic systems. Discrete & Continuous Dynamical Systems - A, 2004, 11 (1) : 83-100. doi: 10.3934/dcds.2004.11.83

[14]

Louis Tebou. Stabilization of some coupled hyperbolic/parabolic equations. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1601-1620. doi: 10.3934/dcdsb.2010.14.1601

[15]

Xiaoyu Fu. Stabilization of hyperbolic equations with mixed boundary conditions. Mathematical Control & Related Fields, 2015, 5 (4) : 761-780. doi: 10.3934/mcrf.2015.5.761

[16]

Weishi Liu. Multiple viscous wave fan profiles for Riemann solutions of hyperbolic systems of conservation laws. Discrete & Continuous Dynamical Systems - A, 2004, 10 (4) : 871-884. doi: 10.3934/dcds.2004.10.871

[17]

Sorin Micu, Jaime H. Ortega, Lionel Rosier, Bing-Yu Zhang. Control and stabilization of a family of Boussinesq systems. Discrete & Continuous Dynamical Systems - A, 2009, 24 (2) : 273-313. doi: 10.3934/dcds.2009.24.273

[18]

Andrei Fursikov. Stabilization of the simplest normal parabolic equation. Communications on Pure & Applied Analysis, 2014, 13 (5) : 1815-1854. doi: 10.3934/cpaa.2014.13.1815

[19]

Gilbert Peralta, Karl Kunisch. Interface stabilization of a parabolic-hyperbolic pde system with delay in the interaction. Discrete & Continuous Dynamical Systems - A, 2018, 38 (6) : 3055-3083. doi: 10.3934/dcds.2018133

[20]

J. Alberto Conejero, Francisco Rodenas, Macarena Trujillo. Chaos for the Hyperbolic Bioheat Equation. Discrete & Continuous Dynamical Systems - A, 2015, 35 (2) : 653-668. doi: 10.3934/dcds.2015.35.653

2017 Impact Factor: 0.631

Metrics

  • PDF downloads (10)
  • HTML views (3)
  • Cited by (0)

Other articles
by authors

[Back to Top]