Approximate controllability of infinite dimensional systems of the n-th order

Jerzy Stefan Respondek

International Journal of Applied Mathematics and Computer Science (2008)

  • Volume: 18, Issue: 2, page 199-212
  • ISSN: 1641-876X

Abstract

top
The objective of the article is to obtain general conditions for several types of controllability at once for an abstract differential equation of arbitrary order, instead of conditions for a fixed order equation. This innovative approach was possible owing to analyzing the n-th order linear system in the Frobenius form which generates a Jordan transition matrix of the Vandermonde form. We extensively used the fact that the knowledge of the inverse of a Jordan transition matrix enables us to directly verify the controllability by Chen's theorem. We used the explicit analytical form of the inverse Vandermonde matrix. This enabled us to obtain more general conditions for different types of controllability for infinite dimensional systems than the conditions existing in the literature so far. The methods introduced can be easily adapted to the analysis of other dynamic properties of the systems considered.

How to cite

top

Jerzy Stefan Respondek. "Approximate controllability of infinite dimensional systems of the n-th order." International Journal of Applied Mathematics and Computer Science 18.2 (2008): 199-212. <http://eudml.org/doc/207877>.

@article{JerzyStefanRespondek2008,
abstract = {The objective of the article is to obtain general conditions for several types of controllability at once for an abstract differential equation of arbitrary order, instead of conditions for a fixed order equation. This innovative approach was possible owing to analyzing the n-th order linear system in the Frobenius form which generates a Jordan transition matrix of the Vandermonde form. We extensively used the fact that the knowledge of the inverse of a Jordan transition matrix enables us to directly verify the controllability by Chen's theorem. We used the explicit analytical form of the inverse Vandermonde matrix. This enabled us to obtain more general conditions for different types of controllability for infinite dimensional systems than the conditions existing in the literature so far. The methods introduced can be easily adapted to the analysis of other dynamic properties of the systems considered.},
author = {Jerzy Stefan Respondek},
journal = {International Journal of Applied Mathematics and Computer Science},
keywords = {inverse Vandermonde matrix; basic symmetrical polynomials; distributed parameter system; linear operators; controllability},
language = {eng},
number = {2},
pages = {199-212},
title = {Approximate controllability of infinite dimensional systems of the n-th order},
url = {http://eudml.org/doc/207877},
volume = {18},
year = {2008},
}

TY - JOUR
AU - Jerzy Stefan Respondek
TI - Approximate controllability of infinite dimensional systems of the n-th order
JO - International Journal of Applied Mathematics and Computer Science
PY - 2008
VL - 18
IS - 2
SP - 199
EP - 212
AB - The objective of the article is to obtain general conditions for several types of controllability at once for an abstract differential equation of arbitrary order, instead of conditions for a fixed order equation. This innovative approach was possible owing to analyzing the n-th order linear system in the Frobenius form which generates a Jordan transition matrix of the Vandermonde form. We extensively used the fact that the knowledge of the inverse of a Jordan transition matrix enables us to directly verify the controllability by Chen's theorem. We used the explicit analytical form of the inverse Vandermonde matrix. This enabled us to obtain more general conditions for different types of controllability for infinite dimensional systems than the conditions existing in the literature so far. The methods introduced can be easily adapted to the analysis of other dynamic properties of the systems considered.
LA - eng
KW - inverse Vandermonde matrix; basic symmetrical polynomials; distributed parameter system; linear operators; controllability
UR - http://eudml.org/doc/207877
ER -

References

top
  1. Alotaibi S., M. Sen, B. Goodwine and K.T. Yang (2004). Controllability of cross-flow heat exchangers, International Journal of Heat and Mass Transfer 47: 913-924. Zbl1083.76021
  2. Bellman R. (1960). Introduction to Matrix Analysis, McGrawHill, New York . Zbl0124.01001
  3. Balakrishnan A.V. (1998). Dynamics and Control of Distributed Systems, Cambridge University Press, Cambridge, pp. 121-201. 
  4. Brammer R. F. (1972). Controllability in linear autonomous systems with positive controllers, SIAM Journal on Control and Optimization 10: 339-353. Zbl0242.93007
  5. Butkowskij A. G. (1979). Characteristics of Distributed Parameter Systems, Nauka, Moscow, (in Russian). 
  6. Chen C.T. (1970). Introduction to Linear System Theory, Holt, Rinehart and Winston Inc, New York. 
  7. Chen G. and D.L. Russel (1982). A mathematical model for linear elastic systems with structural damping, Quarterly of Applied Mathematics 39: 433-454. Zbl0515.73033
  8. Chen G. and R. Triaggani (1990). Gevrey class semigroup arising from elastic systems with gentle dissipation: The case 0 < α < 1/2, Proceedings of the American Mathematical Society 100(2): 401-415. 
  9. Coleman M.P. and H. Wang (1993). Analysis of vibration spectrum of a Timoshenko beam with boundary damping by the wave method, Wave Motion 17: 223-239. Zbl0776.73036
  10. Curtain R. and H. Zwart (1995). An Introduction to InfiniteDimensional Systems Theory, Springer-Verlag, New York. Zbl0839.93001
  11. Davison E.J. and S.H. Wang (1975). New results on the controllability and observability of general composite systems, IEEE Transactions on Automatic Control 20: 123-128. Zbl0309.93005
  12. Dunford N. and J. Schwartz (1963). Linear Operators. Vols. 1 and 2, Interscience, New York. Zbl0128.34803
  13. Fattorini H.O. (1966). Some remarks on complete controllability, SIAM Journal on Control and Optimization 4: 686-694. Zbl0168.34906
  14. Fattorini H.O. (1967). On complete controllability of linear systems, Journal of Differential Equations 3: 391-402. Zbl0155.15903
  15. Fattorini H.O. and Russel D.L. (1971). Exact controllability theorem for linear parabolic equations in one space dimension, Archive for Rational Mechanics and Analysis 43: 272-292. Zbl0231.93003
  16. Górecki H. (1986). Optimization of Dynamic Systems, WNT, Warsaw (in Polish). 
  17. Huang F. (1988). On the mathematical model with analytic damping, SIAM Journal on Control Optimization 26(3): 714-724. Zbl0644.93048
  18. Ito K. and N. Kunimatsu (1988). Stabilization of non-linear distribuded parameter vibratory system, International Journal of Control 48:(2) 2389-2415. Zbl0663.93053
  19. Ito K. and N. Kunimatsu (1991). Semigroup model of structurally damped Timoshenko beam with boundary input, International Journal of Control 54: 367-391. Zbl0735.35112
  20. Kalman R.E. (1960). On the general theory of control systems, Proceedings of the 1st IFAC Congress, London, pp. 481-493. 
  21. Kim J.U. and Y. Renardy (1987). Boundary control of the Timoshenko beam, SIAM Journal on Control and Optimization 25: 1417-1429. Zbl0632.93057
  22. Kaczorek T. (1998). Vectors and Matrices in Automatic Control and Electrical Engineering, WNT Warsaw (in Polish). 
  23. Klamka J. (2000). Schauder's fixed point theorem in nonlinear controllability problems, Control Cybernetics 29: 1377-1393. Zbl1011.93001
  24. Klamka J. (2002). Constrained exact controllability of semilinear systems, Systems and Control Letters 47(2): 139-147. Zbl1003.93005
  25. Klamka J. (1992). Approximate controllability of second order dynamical systems, Applied Mathematics and Computer Sciences 2: 135-148. Zbl0762.93006
  26. Klamka J. (1991). Controllability of Dynamical Systems, Kluwer, Dordrecht. Zbl0732.93008
  27. Klamka J. (1976). Controllability of linear systems with timevariable delays in control, International Journal of Control 24: 869-878. Zbl0342.93006
  28. Klamka J. (1977). Absolute controllability of linear systems with time-variable delays in control, International Journal of Control 26: 57-63. Zbl0359.93005
  29. Labbe S. and E. Trelat (2006). Uniform controllability of semidiscrete approximations of parabolic control systems, Systems and Control Letters 55 (7): 597-609. Zbl1129.93324
  30. Mahmudov N.I. and S. Zorlu (2005). Controllability of semilinear stochastic systems, International Journal of Control 78(13): 997-1004. Zbl1097.93034
  31. Miller L. (2006). Non-structural controllability of linear elastic systems with structural damping, Journal of Functional Analysis 236(2): 592-608. Zbl1105.93014
  32. Respondek J. (2005a). Controllability of dynamical systems with constraints, Systems and Control Letters 54(4): 293-314. Zbl1129.93326
  33. Respondek J. (2005b). Numerical approach to the non-linear diofantic equations with applications to the controllability of infinite dimensional dynamical systems, International Journal of Control 78(13/10): 1017-1030. Zbl1108.93021
  34. Respondek J. (2007). Numerical analysis of controllability of diffusive-convective system with limited manipulating variables, International Communications in Heat and Mass Transfer 34(8): 934-944. 
  35. Sakawa Y. (1974). Controllability for partial differential equations of parabolic type, SIAM Journal on Control and Optimization 12: 389-400. Zbl0289.93010
  36. Sakawa Y. (1984). Feedback control of second order evolution equations with damping, SIAM Journal Control and Optimization 22: 343-361. Zbl0551.58010
  37. Sakawa Y. (1983). Feedback stabilization of linear diffusion system, SIAM Journal on Control an Optimization 21(5): 667-675. Zbl0527.93050
  38. Shi D.H., S.H. Hou and D. Feng (1998). Feedback stabilization of a Timoshenko beam with an end mass, International Journal of Control 69(2): 285-300. Zbl0943.93051
  39. Shi D.H., D. Feng and Q. Yan (2001). Feedback stabilization of rotating Timoshenko beam with adaptive gain, International Journal of Control 74(3): 239-251. Zbl1034.93032
  40. Shubov M.A. (1999). Spectral operators generated by Timoshenko beam model, Systems and Control Letters 38: 249-258. Zbl0985.93012
  41. Shubov M.A. (2000). Exact controllability of damped Timoshenko beam, IMA Journal of Mathematical Control and Information 17: 375-395. Zbl0991.93016
  42. Schmitendorf W.E. and B.R. Barmish (1980). Null controllability of linear system with constrained controls, SIAM Journal on Control and Optimization 18: 327-345. Zbl0457.93012
  43. Tanabe H. (1979). Equations of Evolution, Pitman, London. Zbl0417.35003
  44. Triggiani R. (1975). Controllability and observability in Banach space with bounded operators, SIAM Journal on Control and Optimization 13: 462-491. 
  45. Triggiani R. (1976). Extensions of rank conditions for controllability and observability to Banach spaces with unbounded operators, SIAM Journal on Control and Optimization 14: 313-338. Zbl0326.93003
  46. Triggiani R. (1978). On the relationship between first and second order controllable systems in Banach spaces, SIAM Journal Control and Optimization 16: 847-859. Zbl0395.93012
  47. Vieru A. (2005). On null controllability of linear systems in Banach spaces, Systems and Control Letters 54(4): 331-337. Zbl1129.93329
  48. Xu G.Q. (2005). Boundary feedback exponential stabilization of a Timoshenko beam with both ends free, International Journal of Control 78(4/10): 286-297. Zbl1076.93024

NotesEmbed ?

top

You must be logged in to post comments.