Exponential stability and transfer functions of processes governed by symmetric hyperbolic systems

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1. [1] C.D. Benchimol, A note on weak stabilizability of contraction semigroups. SIAM J. Control Optim. 16 (1978) 373-379. Zbl0384.93035MR490298
2. [2] H. Bounit, H. Hammouri and J. Sau, Regulation of an irrigation canal system through the semigroup approach, in Proc. of the International Workshop Regulation of Irrigation Canals: State of the Art of Research and Applications. Marocco (1997) 261-267. 
3. [3] S.X. Chen, Introduction to partial differential equations. People Education Press (in Chinese) (1981). 
4. [4] V.T. Chow, Open channel hydraulics. Mac-Graw$·$Hill Book Company, New York (1985). 
5. [5] J.M. Coron, B. d’Andréa–Novel and G. Bastin, A Lyapunov approach to control irrigation canals modeled by Saint–Venant equations, in European Control Conference ECC’99. Karlsruhe (1999). 
6. [6] R.F. Curtain, Equivalence of input-output stability and exponential stability for infinite-dimensional systems. Math. Systems Theory 21 (1988) 19-48. Zbl0657.93050MR956620
7. [7] C. Foias, H. Özbay and A. Tannenbaum, Robust Control of Infinite Dimensional Systems. Frequency Domain Methods. Springer, Hong Kong, Lecture Notes in Control and Inform. Sci. 209 (1996). Zbl0839.93003MR1369772
8. [8] B.A. Francis and G. Zames, On $H^{\infty }$-optimal sensitivity theory for SISO feedback systems. IEEE Trans. Automat. Control 29 (1984) 9-16. Zbl0601.93015MR734241
9. [9] J.C. Friedly, Dynamic Behavior of Processes. Prentice-Hall, Inc., Englewood Cliffs, New Jersey (1972). 
10. [10] J.P. Gauthier and C.Z. Xu, $H^{\infty }$-control of a distributed parameter system with non-minimum phase. Int. J. Control 53 (1991) 45-79. Zbl0724.93028MR1085099
11. [11] K.M. Hangos, A.A. Alonso, J.D. Perkins and B.E. Ydstie, Thermodynamic approach to the structural stability of process plants. AIChE J. 45 (1999) 802-816. 
12. [12] H. Hoffman, Banach Spaces of Analytic Functions. Prentice-Hall Inc., Englewood Cliffs (1962). Zbl0117.34001MR133008
13. [13] F.L. Huang, Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces. Ann. Differential Equations 1 (1985) 43-56. Zbl0593.34048MR834231
14. [14] H.O. Kreiss, O.E. Ortiz and O.A. Reula, Stability of quasi-linear hyperbolic dissipative systems. J. Differential Equations 142 (1998) 78-96. Zbl0932.35024MR1492878
15. [15] P.D. Lax and R.S. Phillips, Local boundary conditions for dissipative symmetric linear differential operators. Comm. Pure Appl. Math. 13 (1960) 427-455. Zbl0094.07502MR118949
16. [16] T.S. Li, Global Classical Solutions for Quasilinear Hyperbolic Systems, Research in Applied Mathematics, edited by P.G. Ciarlet and J.-L. Lions. John Willey & Sons, New York (1994). Zbl0841.35064
17. [17] H. Logemann, E.P. Ryan and S. Townley, Integral control of infinite-dimensional linear systems subject to input saturation. SIAM J. Control Optim. 36 (1998) 1940-1961. Zbl0913.93031MR1638027
18. [18] H. Logemann and S. Townley, Low gain control of uncertain regular linear systems. SIAM J. Control Optim. 35 (1997) 78-116. Zbl0873.93044MR1430284
19. [19] K.A. Morris, Justification of input/output methods for systems with unbounded control and observation. IEEE Trans. Automat. Control 44 (1999) 81-85. Zbl0989.93046MR1665308
20. [20] O.E. Ortiz, Stability of nonconservative hyperbolic systems and relativistic dissipative fluids. J. Math. Phys. 42 (2001) 1426-1442. Zbl1053.35089MR1814698
21. [21] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer-Verlag, New York (1983). Zbl0516.47023MR710486
22. [22] S.A. Pohjolainen, Robust multivariable PI-controllers for infinite dimensional systems. IEEE Trans. Automat. Control 27 (1985) 17-30. Zbl0493.93029MR673070
23. [23] J. Prüss, On the spectrum of $C_0$-semigroups. Trans. Amer. Math. Soc. 284 (1984) 847-857. Zbl0572.47030MR743749
24. [24] J. Rauch, Symmetric positive systems with boundary characteristic of constant multiplicity. Trans. Amer. Math. Soc. 291 (1985) 167-187. Zbl0549.35099MR797053
25. [25] J. Rauch and M. Taylor, Exponential decay of solutions to hyperbolic equations in bounded domain. Indiana Univ. Math. J. 24 (1974) 79-86. Zbl0281.35012MR361461
26. [26] R. Rebarber, Conditions for the equivalence of internal and external stability for distributed parameter systems. IEEE Trans. Automat. Control 38 (1993) 994-998. Zbl0786.93087MR1227215
27. [27] D.L. Russell, Controllability and stabilizability theory for linear partial differential equations: Recent progress and open questions. SIAM Rev. 20 (1978) 639-739. Zbl0397.93001MR508380
28. [28] D. Salamon, Realization theory in Hilbert space. Math. Systems Theory 21 (1989) 147-164. Zbl0668.93018MR977021
29. [29] O.J. Staffans, Feedback representations of critical controls for well-posed linear systems. Int. J. Robust Nonlinear Control 8 (1998) 1189-1217. Zbl0951.93038MR1658797
30. [30] G. Weiss, Admissible observation operators for linear semigroups. Israel J. Math. 65 (1989) 17-43. Zbl0696.47040MR994732
31. [31] G. Weiss, Regular linear systems with feedback. Math. Control, Signals & Systems 7 (1994) 23-57. Zbl0819.93034MR1359020
32. [32] G. Weiss, Transfer functions of regular linear systems. Part I: Characterizations of regularity. Trans. Amer. Math. Soc. 342 (1994) 827-854. Zbl0798.93036MR1179402
33. [33] G. Weiss and R.F. Curtain, Dynamic stabilization of regular linear systems. IEEE Trans. Automat. Control 42 (1997) 4-21. Zbl0876.93074MR1439361
34. [34] C.Z. Xu and D.X. Feng, Linearization method to stability analysis for nonlinear hyperbolic systems. C. R. Acad. Sci. Paris Sér. I Math. 332 (2001) 809-814. Zbl1034.35069MR1836091
35. [35] C.Z. Xu and J.P. Gauthier, Analyse et commande d’un échangeur thermique à contre-courant. RAIRO APII 25 (1991) 377-396. Zbl0741.93064
36. [36] C.Z. Xu, J.P. Gauthier and I. Kupka, Exponential stability of the heat exchanger equation, in Proc. of the European Control Conference. Groningen, The Netherlands (1993) 303-307. 
37. [37] C.Z. Xu and H. Jerbi, A robust PI-controller for infinite dimensional systems. Int. J. Control 61 (1995) 33-45. Zbl0820.93036MR1619706
38. [38] C.Z. Xu, Exponential stability of a class of infinite dimensional time-varying linear systems, in Proc. of the International Conference on Control and Information. Hong Kong (1995). 
39. [39] C.Z. Xu, Exact observability and exponential stability of infinite dimensional bilinear systems. Math. Control, Signals & Systems 9 (1996) 73-93. Zbl0862.93007MR1410049
40. [40] C.Z. Xu and G. Sallet, Proportional and Integral regulation of irrigation canal systems governed by the Saint–Venant equation, in 14th IFAC World Congress. Beijing, China (1999). 
41. [41] C.Z. Xu and D.X. Feng, Symmetric hyperbolic systems and applications to exponential stability of heat exchangers and irrigation canals, in Proc. of the MTNS’2000. Perpignan (2000). 
42. [42] B.E. Ydstie and A.A. Alonso, Process systems and passivity via the Clausius–Planck inequality. Systems Control Lett. 30 (1997) 253-264. Zbl0901.93003

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