American Institute of Mathematical Sciences

Advanced
doi: 10.3934/eect.2020108
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

Energy method for exponential stability of coupled one-dimensional hyperbolic PDE-ODE systems

Gervy Marie Angeles 1,2, and  Gilbert Peralta 2,,

1. 

Fakultät für Mathematik, Universität Wien, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria
2. 

Department of Mathematics and Computer Science, University of the Philippines Baguio, Governor Pack Road, Baguio, 2600 Philippines

* Corresponding author

Received  June 2020 Revised  October 2020 Early access December 2020

We consider a hyperbolic system of partial differential equations on a bounded interval coupled with ordinary differential equations on both ends. The evolution is governed by linear balance laws, which we treat with semigroup and time-space methods. Our goal is to establish the exponential stability in the natural state space by utilizing the stability with respect to the first-order energy of the system. Derivation of a priori estimates plays a crucial role in obtaining energy and dissipation functionals. The theory is then applied to specific physical models.

Keywords: Hyperbolic PDE-ODE system, exponential stability, energy method, semigroup, dynamic boundary conditions.
Mathematics Subject Classification: Primary: 35L50, 47D03; Secondary: 93D20.
Citation: Gervy Marie Angeles, Gilbert Peralta. Energy method for exponential stability of coupled one-dimensional hyperbolic PDE-ODE systems. Evolution Equations & Control Theory, doi: 10.3934/eect.2020108
References:
[1]

G. Bastin and J.-M. Coron, Stability and Boundary Stabilization of 1-D Hyperbolic Systems, volume 88 of Progress in Nonlinear Differential Equations and Their Applications., Birkhäuser Basel, 2016. doi: 10.1007/978-3-319-32062-5.  Google Scholar

[2]

J. T. Beale, Spectral properties of an acoustic boundary condition, Indiana U. Math. J., 25 (1976), 895-917.  doi: 10.1512/iumj.1976.25.25071.  Google Scholar

[3]

R. Borsche, R. M. Colombo and M. Garavello, Mixed systems: ODEs - balance laws, J. Differ. Equations, 252 (2012), 2311-2338.  doi: 10.1016/j.jde.2011.08.051.  Google Scholar

[4]

A. Borzi and G. Propst, Numerical investigation of the liebau phenomenon, Z. Angew. Math. Phys., 54 (2003), 1050-1072.  doi: 10.1007/s00033-003-1108-x.  Google Scholar

[5]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, , Springer, New York, 2010.  Google Scholar

[6]

S. Čanić and E. H. Kim, Mathematical analysis of the quasilinear effects in a hyperbolic model blood flow through compliant axi-symmetric vessels, Math Method Applied Sci., 26 (2003), 1161-1186.  doi: 10.1002/mma.407.  Google Scholar

[7]

R. Courant and D. Hilbert, Methods of Mathematical Physics, II: Partial Differential Equations, Wiley Online Library, 1962.  Google Scholar

[8]

A. Curcio, M. E. Clark, M. Zhao and W. Ruan, A hyperbolic system of equations of blood flow in an arterial network, SIAM J. Applied Math., 64 (2004), 637-667.  doi: 10.1137/S0036139902415294.  Google Scholar

[9]

N. Dunford and J. T. Schwartz, Linear Operators, Part I: General Theory, Interscience Publishers, Inc., New York, 1958.  Google Scholar

[10]

K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Springer, New York, 2000.  Google Scholar

[11]

M. Á. Fernández, V. Milisic and A. Quarteroni, Analysis of a geometrical multiscale blood flow model based on the coupling of odes and hyperbolic pdes, Multiscale Model. Sim., 4 (2005), 215-236.  doi: 10.1137/030602010.  Google Scholar

[12]

P. R. Garabedian, Partial Differential Equations, AMS Chelsea Publishing, 1964.  Google Scholar

[13]

O. Heaviside, Electromagnetic induction and its propagation, The Electrician, 14 (1885), 178-180.   Google Scholar

[14]

K. Ito and G. Propst, Legendre-Tau-Padé approximations to the one-dimensional wave equation with boundary oscillators, Numer. Func. Anal. Opt., 19 (1998), 57-70.  doi: 10.1080/01630569808816814.  Google Scholar

[15]

M. Miklavčič, Applied Functional Analysis and Partial Differential Equations, World Scientific Publishing Co., 1998. Google Scholar

[16]

P. M. Morse and K. U. Ingard, Theoretical Acoustics, McGraw-Hill, New York, 1968. Google Scholar

[17]

J. C. Oostveen, R. F. Curtain and K. Ito, An approximation theory for strongly stabilizing solutions to the operator LQ Riccati equation, SIAM J. Control Optim., 38 (2000), 1909-1937.  doi: 10.1137/S0363012998339691.  Google Scholar

[18]

J. T. Ottesen, Valveless pumping in a fluid-filled closed elastic tube-system: one-dimensional theory with experimental validation, J. Math. Biol., 46 (2003), 309-332.  doi: 10.1007/s00285-002-0179-1.  Google Scholar

[19]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, volume 44 of Applied Mathematical Sciences, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[20]

G. Peralta and G. Propst, Stability and boundary controllability of a linearized model of flow in an elastic tube, ESAIM: Control Optim. Calc. Var., 21 (2015), 583-601.  doi: 10.1051/cocv/2014039.  Google Scholar

[21]

C. Prieur and J. J. Winkin, Boundary feedback control of linear hyperbolic systems: Application to Saint-Venant-Exner equations, Automatica, 89 (2018), 44-51.  doi: 10.1016/j.automatica.2017.11.028.  Google Scholar

[22]

H. Rath and I. Teipel, Der fördereffekt in ventillosen, elastischen leitungen, Z. Angew. Math. Phys., 29 (1978), 123-133.   Google Scholar

[23]

W. Ruan, A coupled system of ODEs and quasilinear hyperbolic PDEs arising in a multiscale blood flow model, J. Math. Anal. Appl., 343 (2008), 778-798.  doi: 10.1016/j.jmaa.2008.01.064.  Google Scholar

[24]

W. Ruan, M. E. Clark, M. Zhao and A. Curcio, Global solution to a hyperbolic problem arising in the modeling of blood flow in circulatory systems, J. Math. Anal. Appl., 331 (2007), 1068-1092.  doi: 10.1016/j.jmaa.2006.09.034.  Google Scholar

[25]

D. L. Russell, Controllability and stabilizability theory for linear partial differential equations: Recent progress and open questions, SIAM Review, 20 (1978), 639–739. doi: 10.1137/1020095.  Google Scholar

[26]

M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, Birkhäuser Basel, 2009. doi: 10.1007/978-3-7643-8994-9.  Google Scholar

[27]

K. Yosida, Functional Analysis, Grundlehren der mathematischen Wissenschaften, Springer-Verlag, Berlin Heidelberg, 1980.  Google Scholar

show all references

References:
[1]

G. Bastin and J.-M. Coron, Stability and Boundary Stabilization of 1-D Hyperbolic Systems, volume 88 of Progress in Nonlinear Differential Equations and Their Applications., Birkhäuser Basel, 2016. doi: 10.1007/978-3-319-32062-5.  Google Scholar

[2]

J. T. Beale, Spectral properties of an acoustic boundary condition, Indiana U. Math. J., 25 (1976), 895-917.  doi: 10.1512/iumj.1976.25.25071.  Google Scholar

[3]

R. Borsche, R. M. Colombo and M. Garavello, Mixed systems: ODEs - balance laws, J. Differ. Equations, 252 (2012), 2311-2338.  doi: 10.1016/j.jde.2011.08.051.  Google Scholar

[4]

A. Borzi and G. Propst, Numerical investigation of the liebau phenomenon, Z. Angew. Math. Phys., 54 (2003), 1050-1072.  doi: 10.1007/s00033-003-1108-x.  Google Scholar

[5]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, , Springer, New York, 2010.  Google Scholar

[6]

S. Čanić and E. H. Kim, Mathematical analysis of the quasilinear effects in a hyperbolic model blood flow through compliant axi-symmetric vessels, Math Method Applied Sci., 26 (2003), 1161-1186.  doi: 10.1002/mma.407.  Google Scholar

[7]

R. Courant and D. Hilbert, Methods of Mathematical Physics, II: Partial Differential Equations, Wiley Online Library, 1962.  Google Scholar

[8]

A. Curcio, M. E. Clark, M. Zhao and W. Ruan, A hyperbolic system of equations of blood flow in an arterial network, SIAM J. Applied Math., 64 (2004), 637-667.  doi: 10.1137/S0036139902415294.  Google Scholar

[9]

N. Dunford and J. T. Schwartz, Linear Operators, Part I: General Theory, Interscience Publishers, Inc., New York, 1958.  Google Scholar

[10]

K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Springer, New York, 2000.  Google Scholar

[11]

M. Á. Fernández, V. Milisic and A. Quarteroni, Analysis of a geometrical multiscale blood flow model based on the coupling of odes and hyperbolic pdes, Multiscale Model. Sim., 4 (2005), 215-236.  doi: 10.1137/030602010.  Google Scholar

[12]

P. R. Garabedian, Partial Differential Equations, AMS Chelsea Publishing, 1964.  Google Scholar

[13]

O. Heaviside, Electromagnetic induction and its propagation, The Electrician, 14 (1885), 178-180.   Google Scholar

[14]

K. Ito and G. Propst, Legendre-Tau-Padé approximations to the one-dimensional wave equation with boundary oscillators, Numer. Func. Anal. Opt., 19 (1998), 57-70.  doi: 10.1080/01630569808816814.  Google Scholar

[15]

M. Miklavčič, Applied Functional Analysis and Partial Differential Equations, World Scientific Publishing Co., 1998. Google Scholar

[16]

P. M. Morse and K. U. Ingard, Theoretical Acoustics, McGraw-Hill, New York, 1968. Google Scholar

[17]

J. C. Oostveen, R. F. Curtain and K. Ito, An approximation theory for strongly stabilizing solutions to the operator LQ Riccati equation, SIAM J. Control Optim., 38 (2000), 1909-1937.  doi: 10.1137/S0363012998339691.  Google Scholar

[18]

J. T. Ottesen, Valveless pumping in a fluid-filled closed elastic tube-system: one-dimensional theory with experimental validation, J. Math. Biol., 46 (2003), 309-332.  doi: 10.1007/s00285-002-0179-1.  Google Scholar

[19]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, volume 44 of Applied Mathematical Sciences, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[20]

G. Peralta and G. Propst, Stability and boundary controllability of a linearized model of flow in an elastic tube, ESAIM: Control Optim. Calc. Var., 21 (2015), 583-601.  doi: 10.1051/cocv/2014039.  Google Scholar

[21]

C. Prieur and J. J. Winkin, Boundary feedback control of linear hyperbolic systems: Application to Saint-Venant-Exner equations, Automatica, 89 (2018), 44-51.  doi: 10.1016/j.automatica.2017.11.028.  Google Scholar

[22]

H. Rath and I. Teipel, Der fördereffekt in ventillosen, elastischen leitungen, Z. Angew. Math. Phys., 29 (1978), 123-133.   Google Scholar

[23]

W. Ruan, A coupled system of ODEs and quasilinear hyperbolic PDEs arising in a multiscale blood flow model, J. Math. Anal. Appl., 343 (2008), 778-798.  doi: 10.1016/j.jmaa.2008.01.064.  Google Scholar

[24]

W. Ruan, M. E. Clark, M. Zhao and A. Curcio, Global solution to a hyperbolic problem arising in the modeling of blood flow in circulatory systems, J. Math. Anal. Appl., 331 (2007), 1068-1092.  doi: 10.1016/j.jmaa.2006.09.034.  Google Scholar

[25]

D. L. Russell, Controllability and stabilizability theory for linear partial differential equations: Recent progress and open questions, SIAM Review, 20 (1978), 639–739. doi: 10.1137/1020095.  Google Scholar

[26]

M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, Birkhäuser Basel, 2009. doi: 10.1007/978-3-7643-8994-9.  Google Scholar

[27]

K. Yosida, Functional Analysis, Grundlehren der mathematischen Wissenschaften, Springer-Verlag, Berlin Heidelberg, 1980.  Google Scholar

Figure 1.  An elastic tube connected to two rigid tanks
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  A waveguide terminated by oscillators
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Maria Laura Delle Monache, Paola Goatin. A front tracking method for a strongly coupled PDE-ODE system with moving density constraints in traffic flow. Discrete & Continuous Dynamical Systems - S, 2014, 7 (3) : 435-447. doi: 10.3934/dcdss.2014.7.435
[2]

Vanessa Baumgärtner, Simone Göttlich, Stephan Knapp. Feedback stabilization for a coupled PDE-ODE production system. Mathematical Control & Related Fields, 2020, 10 (2) : 405-424. doi: 10.3934/mcrf.2020003
[3]

Youshan Tao, J. Ignacio Tello. Nonlinear stability of a heterogeneous state in a PDE-ODE model for acid-mediated tumor invasion. Mathematical Biosciences & Engineering, 2016, 13 (1) : 193-207. doi: 10.3934/mbe.2016.13.193
[4]

George Avalos, Roberto Triggiani. Semigroup well-posedness in the energy space of a parabolic-hyperbolic coupled Stokes-Lamé PDE system of fluid-structure interaction. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 417-447. doi: 10.3934/dcdss.2009.2.417
[5]

Marita Holtmannspötter, Arnd Rösch, Boris Vexler. A priori error estimates for the space-time finite element discretization of an optimal control problem governed by a coupled linear PDE-ODE system. Mathematical Control & Related Fields, 2021, 11 (3) : 601-624. doi: 10.3934/mcrf.2021014
[6]

Davide Guidetti. On hyperbolic mixed problems with dynamic and Wentzell boundary conditions. Discrete & Continuous Dynamical Systems - S, 2020, 13 (12) : 3461-3471. doi: 10.3934/dcdss.2020239
[7]

Qigui Yang, Qiaomin Xiang. Chaotic oscillations of linear hyperbolic PDE with variable coefficients and implicit boundary conditions. Discrete & Continuous Dynamical Systems - S, 2021, 14 (9) : 3267-3284. doi: 10.3934/dcdss.2020335
[8]

Ciro D'Apice, Peter I. Kogut, Rosanna Manzo. On relaxation of state constrained optimal control problem for a PDE-ODE model of supply chains. Networks & Heterogeneous Media, 2014, 9 (3) : 501-518. doi: 10.3934/nhm.2014.9.501
[9]

Hichem Kasri, Amar Heminna. Exponential stability of a coupled system with Wentzell conditions. Evolution Equations & Control Theory, 2016, 5 (2) : 235-250. doi: 10.3934/eect.2016003
[10]

Ciprian G. Gal, M. Grasselli. On the asymptotic behavior of the Caginalp system with dynamic boundary conditions. Communications on Pure & Applied Analysis, 2009, 8 (2) : 689-710. doi: 10.3934/cpaa.2009.8.689
[11]

Raluca Clendenen, Gisèle Ruiz Goldstein, Jerome A. Goldstein. Degenerate flux for dynamic boundary conditions in parabolic and hyperbolic equations. Discrete & Continuous Dynamical Systems - S, 2016, 9 (3) : 651-660. doi: 10.3934/dcdss.2016019
[12]

Nicolas Fourrier, Irena Lasiecka. Regularity and stability of a wave equation with a strong damping and dynamic boundary conditions. Evolution Equations & Control Theory, 2013, 2 (4) : 631-667. doi: 10.3934/eect.2013.2.631
[13]

Luisa Arlotti. Explicit transport semigroup associated to abstract boundary conditions. Conference Publications, 2011, 2011 (Special) : 102-111. doi: 10.3934/proc.2011.2011.102
[14]

Laurence Cherfils, Stefania Gatti, Alain Miranville. Long time behavior of the Caginalp system with singular potentials and dynamic boundary conditions. Communications on Pure & Applied Analysis, 2012, 11 (6) : 2261-2290. doi: 10.3934/cpaa.2012.11.2261
[15]

Franck Davhys Reval Langa, Morgan Pierre. A doubly splitting scheme for the Caginalp system with singular potentials and dynamic boundary conditions. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 653-676. doi: 10.3934/dcdss.2020353
[16]

Ciprian G. Gal, Alain Miranville. Robust exponential attractors and convergence to equilibria for non-isothermal Cahn-Hilliard equations with dynamic boundary conditions. Discrete & Continuous Dynamical Systems - S, 2009, 2 (1) : 113-147. doi: 10.3934/dcdss.2009.2.113
[17]

Peter Brune, Björn Schmalfuss. Inertial manifolds for stochastic pde with dynamical boundary conditions. Communications on Pure & Applied Analysis, 2011, 10 (3) : 831-846. doi: 10.3934/cpaa.2011.10.831
[18]

El Mustapha Ait Ben Hassi, Salah-Eddine Chorfi, Lahcen Maniar, Omar Oukdach. Lipschitz stability for an inverse source problem in anisotropic parabolic equations with dynamic boundary conditions. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020094
[19]

Lei Wang, Zhong-Jie Han, Gen-Qi Xu. Exponential-stability and super-stability of a thermoelastic system of type II with boundary damping. Discrete & Continuous Dynamical Systems - B, 2015, 20 (8) : 2733-2750. doi: 10.3934/dcdsb.2015.20.2733
[20]

Monica Conti, Stefania Gatti, Alain Miranville. Asymptotic behavior of the Caginalp phase-field system with coupled dynamic boundary conditions. Discrete & Continuous Dynamical Systems - S, 2012, 5 (3) : 485-505. doi: 10.3934/dcdss.2012.5.485

2020 Impact Factor: 1.081

Tools

Metrics

  • PDF downloads (146)
  • HTML views (402)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences