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doi: 10.3934/eect.2020108

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
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Figure 2.  A waveguide terminated by oscillators
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