Boundary control, stabilization and zero-pole dynamics for a non-linear distributed parameter system

International Journal of Robust and Nonlinear Control

Volumn 9, Issue 11, 1999, Pages 737-768

  Byrnes, Christopher I ^a   Gilliam, David S ^b   Shubov, Victor I ^b  

^a WASHINGTON UNIVERSITY   (United States)

^b TEXAS TECH UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

BOUNDARY CONDITIONS; CLOSED LOOP CONTROL SYSTEMS; CONTROL SYSTEM SYNTHESIS; CONVERGENCE OF NUMERICAL METHODS; FEEDBACK CONTROL; NONLINEAR CONTROL SYSTEMS; PERTURBATION TECHNIQUES; POLES AND ZEROS; ROBUSTNESS (CONTROL SYSTEMS); ROOT LOCI; SYSTEM STABILITY;

BURGERS' EQUATION; NEUMANN BOUNDARY CONDITIONS;

DISTRIBUTED PARAMETER CONTROL SYSTEMS;

EID: 0032684683     PISSN: 10498923     EISSN: None     Source Type: Journal    
DOI: 10.1002/(SICI)1099-1239(199909)9:11<737::AID-RNC432>3.0.CO;2-3     Document Type: Article

References (35)

1. Pohjolainen, S. A., 'Computation of Transmission Zeros for Distributed Parameter Systems', Int. J. Control, 33(2), 199-212 (1981).
2. Curtain R. F. and H. J. Zwart, An Introduction to Infinite-Dimensional Linear Systems, Springer, Berlin, 1995.
3. Byrnes, C. I., D. S. Gilliam and J. He, 'Root locus and boundary feedback design for a class of distributed parameter systems', SIAM J. Control Optim., 32(5), 1364-1427 (1994).
4. Byrnes, C. I. and A. Isidori, 'Asymptotic stabilization of minimum phase nonlinear systems', IEEE Trans. Automat. Control, 36, 1122-1137 (1991).
5. Isidori, A. and A. Krener, 'Nonlinear Zero Distributions', in Proc. 19th IEEE Conf. Dec. and Control, 1980.
6. Isidori, A. and C. H. Moog, 'On the Nonlinear Equivalent of the Notion of Transmission Zeroes', in Modelling and Adaptive Control (Proc. IIASA Conf. Sopron (1986)), Springer, Berlin, 1988.
7. Isidori, A., Nonlinear Control Systems, 3rd edn, Springer, Berlin, 1995.
8. Isidori, A. and C. I. Byrnes, 'Output regulation of nonlinear systems', IEEE Trans. Automat. Control, AC-35, 131-140 (1990).
9. Fletcher, C. A. J., 'Burgers' equation: a model for all reasons', in Numerical Solutions of Partial Differential Equations, J. Noye (Ed), North-Holland Publ. Co., Amsterdam, 1982.
10. Bateman, H., 'Some recent researches on the motion of fluids', Mon. Weather Rev., 43, 163-170 (1915).
11. Burgers, J., 'Application of a model system to illustrate some points of the statistical theory of free turbulence', Nederl. Akad. Wefensh. Proc., 43, 2-12 (1940).
12. Lighthill, M. J., 'Viscosity effects in sound waves of finite amplitude', Surveys in Mechanics, G. K. Batchelor and R. M. Davies (Eds), Cambridge University Press, Cambridge, 1956, pp. 250-351.
13. Burns, J. A. and S. Kang, 'A control problem for Burgers' equation with bounded input/output', Nonlinear Dyn., 2, 235-262 (1991).
14. Burns, J. A. and S. Kang, 'A Stabilization problem for Burgers' equation with unbounded control and observation', Int. Ser. Numer. Math., 100, 51-72 (1991).
15. xx, Comm. Pure Appl. Math., 3, 201-230 (1950).
16. Kiselev, A. A. and O. A. Ladyzheskaya, 'On the existence and uniqueness of the solution of the nonstationary problem for a viscous incompressible fluid', Izv. Akad. Nauk SSSR, Ser. Mat., 21, 655-680 (1957).
17. Byrnes, C. I., D. S. Gilliam and V. Shubov, 'On the global dynamics of a controlled viscous Burgers' equation', J. Dyn. Control Systems, 4, 457-519 (1998).
18. Van Ly, H., K. D. Mease and E. S. Titi, 'Some remarks on distributed and boundary control of the viscous Burgers' equation', Numer. Funct. Anal. Optim., 18(1-2), 143-188 (1997).
19. Ito, K. and Y. Yan, 'Viscous scalar conservation laws with nonlinear flux feedback and global attractor', ICAM Report 95-10-01, 1995.
20. Kreiss, H.-O. and J. Lorenz, Initial-Boundary Value Problems and the Navier-Stokes Equations, Academic Press, New York, 1989.
21. Byrnes, C. I., D. S. Gilliam and V. I. Shubov, 'High gain limit of trajectories and attractors for a boundary controlled viscous burgers equation', J. Math. Systems, Estimation Control, 6(4) (Summary 485-489), http://www.birkhauser. com/journals/jmsec/jmsec.html.
22. Byrnes, C. I., D. S. Gilliam and V. I. Shubov, 'Convergence of trajectories for a controlled viscous Burgers' equation', in Control and Estimation of Distributed Parameter Systems: Nonlinear Phenomena, Birkhäuser, Boston, 1994, pp. 61-77, with C. I. Byrnes and V. I. Shubov.
23. Byrnes, C. I., D. S. Gilliam and V. I. Shubov, 'Zero and pole dynamics for a controlled Burger's equation', in Proc. 33rd IEEE Conf. on Decision and Control, Vol. 1, December 1994, pp. 295-298, Lake Buena Vista, F1, with C. I. Byrnes and V. I. Shubov.
24. Byrnes, C. I., D. S. Gilliam and V. I. Shubov, 'Boundary control, for a viscous Burgers' equation', in Identification and Control in Systems Governed by Partial Differential Equations, H. T. Banks, R. H. Fabiano and K. Ito (Eds), SIAM Philadelphia, PA, 1993, pp. 171-185.
25. Ladyzhenskaya, O. A., V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, Translations of AMS, Vol. 23, 1968.
26. Ladyzhenskaya, O. A., The Boundary Value Problems of Mathematical Physics, Applied Mathematical Sciences, Vol. 49, Springer, Berlin, 1985.
27. Ladyzhenskaya, O. A., The Mathematical Theory of Viscous Incompressible Flow, 2nd edn, Gordon and Breach, New York, 1989.
28. Byrnes, C. I. and A. Isidori, 'A frequency domain philosophy for nonlinear systems, with applications to stabilization and to adaptive control', in Proc. 23rd IEEE Conf. on Decision and Control, Las Vegas, 1984.
29. Sussman, H. J. and P. V. Kokotovic, 'The peaking phenomenon and the global stabilization of nonlinear systems', IEEE Trans. Automat. Control, 36(4), 424-440 (1991).
30. Tsinias, J., 'Planar nonlinear systems: practical stabilization and Hermes controllability condition', Systems Control Lett., 17(4), 291-296 (1996).
31. Byrnes, C. I. and D. S. Gilliam, 'Boundary feedback stabilization of a viscous Burgers' equation', in Computation and Control III: Proc. 3rd Bozeman Conf., 1992, K. L. Bowers and J. L. Lund (Eds), Birkhäuser, Boston, 1993.
32. Ladyzhenskaya, O. A., 'The dynamical system that is generated by the Navier-Stokes equations', J. Sov. Math., 3, 458-479 (1975).
33. Shubov, V. I., 'Long time behavior of infinite dimensional dissipative dynamical systems', J. Estimation Control, 2, 381-427 (1992).
34. Temam, R., Infinite Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Sciences, Vol. 68, Springer, Berlin, 1988.
35. Smirnov, M. M., Problems on the Equations of Mathematical Physics, Noordhoff, Groningen, The Netherlands, 1966.