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An elastic tube connected to two rigid tanks
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Optimal control problems for a neutral integro-differential system with infinite delay
February
2022, 11(1): 199-224.
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 |
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 and Control Theory,
2022, 11
(1)
: 199-224.
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. |
| [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. |
| [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. |
| [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. |
| [5] |
H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, , Springer, New York, 2010. |
| [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. |
| [7] |
R. Courant and D. Hilbert, Methods of Mathematical Physics, II: Partial Differential Equations, Wiley Online Library, 1962. |
| [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. |
| [9] |
N. Dunford and J. T. Schwartz, Linear Operators, Part I: General Theory, Interscience Publishers, Inc., New York, 1958. |
| [10] |
K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Springer, New York, 2000. |
| [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. |
| [12] |
P. R. Garabedian, Partial Differential Equations, AMS Chelsea Publishing, 1964. |
| [13] |
O. Heaviside,
Electromagnetic induction and its propagation, The Electrician, 14 (1885), 178-180.
|
| [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. |
| [15] |
M. Miklavčič, Applied Functional Analysis and Partial Differential Equations, World Scientific Publishing Co., 1998. |
| [16] |
P. M. Morse and K. U. Ingard, Theoretical Acoustics, McGraw-Hill, New York, 1968. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [22] |
H. Rath and I. Teipel,
Der fördereffekt in ventillosen, elastischen leitungen, Z. Angew. Math. Phys., 29 (1978), 123-133.
|
| [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. |
| [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. |
| [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. |
| [26] |
M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, Birkhäuser Basel, 2009.
doi: 10.1007/978-3-7643-8994-9. |
| [27] |
K. Yosida, Functional Analysis, Grundlehren der mathematischen Wissenschaften, Springer-Verlag, Berlin Heidelberg, 1980. |
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. |
| [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. |
| [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. |
| [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. |
| [5] |
H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, , Springer, New York, 2010. |
| [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. |
| [7] |
R. Courant and D. Hilbert, Methods of Mathematical Physics, II: Partial Differential Equations, Wiley Online Library, 1962. |
| [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. |
| [9] |
N. Dunford and J. T. Schwartz, Linear Operators, Part I: General Theory, Interscience Publishers, Inc., New York, 1958. |
| [10] |
K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Springer, New York, 2000. |
| [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. |
| [12] |
P. R. Garabedian, Partial Differential Equations, AMS Chelsea Publishing, 1964. |
| [13] |
O. Heaviside,
Electromagnetic induction and its propagation, The Electrician, 14 (1885), 178-180.
|
| [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. |
| [15] |
M. Miklavčič, Applied Functional Analysis and Partial Differential Equations, World Scientific Publishing Co., 1998. |
| [16] |
P. M. Morse and K. U. Ingard, Theoretical Acoustics, McGraw-Hill, New York, 1968. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [22] |
H. Rath and I. Teipel,
Der fördereffekt in ventillosen, elastischen leitungen, Z. Angew. Math. Phys., 29 (1978), 123-133.
|
| [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. |
| [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. |
| [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. |
| [26] |
M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, Birkhäuser Basel, 2009.
doi: 10.1007/978-3-7643-8994-9. |
| [27] |
K. Yosida, Functional Analysis, Grundlehren der mathematischen Wissenschaften, Springer-Verlag, Berlin Heidelberg, 1980. |
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A waveguide terminated by oscillators
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