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

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  • 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.

    Mathematics Subject Classification: Primary: 35L50, 47D03; Secondary: 93D20.


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

    Figure 2.  A waveguide terminated by oscillators

  • [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. 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.
    [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. 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.
    [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. 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.
    [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. 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.
    [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. 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.
    [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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