• Previous Article
    Stability and stabilization for the three-dimensional Navier-Stokes-Voigt equations with unbounded variable delay
  • EECT Home
  • This Issue
  • Next Article
    Multiscale homogenization of integral convex functionals in Orlicz Sobolev setting
doi: 10.3934/eect.2020108

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

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 Published  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.

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. BorscheR. 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. CurcioM. E. ClarkM. 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. FernándezV. 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. OostveenR. 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. RuanM. E. ClarkM. 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. BorscheR. 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. CurcioM. E. ClarkM. 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. FernándezV. 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. OostveenR. 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. RuanM. E. ClarkM. 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 2.  A waveguide terminated by oscillators
[1]

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

[2]

Fang Li, Bo You. On the dimension of global attractor for the Cahn-Hilliard-Brinkman system with dynamic boundary conditions. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021024

[3]

Izumi Takagi, Conghui Zhang. Existence and stability of patterns in a reaction-diffusion-ODE system with hysteresis in non-uniform media. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020400

[4]

Xiaoxiao Li, Yingjing Shi, Rui Li, Shida Cao. Energy management method for an unpowered landing. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020180

[5]

Mark F. Demers. Uniqueness and exponential mixing for the measure of maximal entropy for piecewise hyperbolic maps. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 217-256. doi: 10.3934/dcds.2020217

[6]

João Marcos do Ó, Bruno Ribeiro, Bernhard Ruf. Hamiltonian elliptic systems in dimension two with arbitrary and double exponential growth conditions. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 277-296. doi: 10.3934/dcds.2020138

[7]

Xinfu Chen, Huiqiang Jiang, Guoqing Liu. Boundary spike of the singular limit of an energy minimizing problem. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3253-3290. doi: 10.3934/dcds.2020124

[8]

Chao Xing, Zhigang Pan, Quan Wang. Stabilities and dynamic transitions of the Fitzhugh-Nagumo system. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 775-794. doi: 10.3934/dcdsb.2020134

[9]

Marcos C. Mota, Regilene D. S. Oliveira. Dynamic aspects of Sprott BC chaotic system. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1653-1673. doi: 10.3934/dcdsb.2020177

[10]

Tengfei Yan, Qunying Liu, Bowen Dou, Qing Li, Bowen Li. An adaptive dynamic programming method for torque ripple minimization of PMSM. Journal of Industrial & Management Optimization, 2021, 17 (2) : 827-839. doi: 10.3934/jimo.2019136

[11]

Soonki Hong, Seonhee Lim. Martin boundary of brownian motion on gromov hyperbolic metric graphs. Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021014

[12]

Guoliang Zhang, Shaoqin Zheng, Tao Xiong. A conservative semi-Lagrangian finite difference WENO scheme based on exponential integrator for one-dimensional scalar nonlinear hyperbolic equations. Electronic Research Archive, 2021, 29 (1) : 1819-1839. doi: 10.3934/era.2020093

[13]

Roland Schnaubelt, Martin Spitz. Local wellposedness of quasilinear Maxwell equations with absorbing boundary conditions. Evolution Equations & Control Theory, 2021, 10 (1) : 155-198. doi: 10.3934/eect.2020061

[14]

Kuntal Bhandari, Franck Boyer. Boundary null-controllability of coupled parabolic systems with Robin conditions. Evolution Equations & Control Theory, 2021, 10 (1) : 61-102. doi: 10.3934/eect.2020052

[15]

Qianqian Hou, Tai-Chia Lin, Zhi-An Wang. On a singularly perturbed semi-linear problem with Robin boundary conditions. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 401-414. doi: 10.3934/dcdsb.2020083

[16]

Wenrui Hao, King-Yeung Lam, Yuan Lou. Ecological and evolutionary dynamics in advective environments: Critical domain size and boundary conditions. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 367-400. doi: 10.3934/dcdsb.2020283

[17]

Manil T. Mohan. First order necessary conditions of optimality for the two dimensional tidal dynamics system. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020045

[18]

Lekbir Afraites, Chorouk Masnaoui, Mourad Nachaoui. Shape optimization method for an inverse geometric source problem and stability at critical shape. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021006

[19]

Chao Xing, Jiaojiao Pan, Hong Luo. Stability and dynamic transition of a toxin-producing phytoplankton-zooplankton model with additional food. Communications on Pure & Applied Analysis, 2021, 20 (1) : 427-448. doi: 10.3934/cpaa.2020275

[20]

Xiaofeng Ren, David Shoup. The impact of the domain boundary on an inhibitory system: Interior discs and boundary half discs. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3957-3979. doi: 10.3934/dcds.2020048

2019 Impact Factor: 0.953

Metrics

  • PDF downloads (28)
  • HTML views (64)
  • Cited by (0)

Other articles
by authors

[Back to Top]