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: |
[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. |
An elastic tube connected to two rigid tanks
A waveguide terminated by oscillators