, Volume 13, Issue 3, pp 157–167 | Cite as

Some recent results on the stability of linear time varying systems

  • S Pradeef
  • Shashi K Shrivastava


Some theorems derived recently by the authors on the stability of multidimensional linear time varying systems are reported in this paper. To begin with, criteria based on Liapunov’s direct method are stated. These are followed by conditions on the asymptotic behaviour and boundedness of solutions. Finally,L 2 andL ∞ stabilities of these systems are discussed. In conclusion, mention is made of some of the problems in aerospace engineering to which these theorems have been applied.


Stability linear time varying systems Liapunov’s direct method 


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  1. Bellman R 1943Duke Math. J. 10: 643–647MATHCrossRefMathSciNetGoogle Scholar
  2. Bellman R 1953Stability theory of differential equations (New York: McGraw-Hill)MATHGoogle Scholar
  3. Cesari L 1963Asymptotic behaviour and stability problems in ordinary differential equations (New York: Springer-Verlag)Google Scholar
  4. Coppel W A 1965Stability and asymptotic behaviour of differential equations (Boston: Heath)Google Scholar
  5. Desoer C A, Vidyasagar M 1975Feedback systems: Input-output properties (New York: Academic Press)MATHGoogle Scholar
  6. Gronwall T H 1918Ann. Math. 20: 212–296MathSciNetCrossRefGoogle Scholar
  7. Gunderson H, Rigas H, Van Vleck F S 1974SIAM J. Appl. Math. 26: 345–349CrossRefMathSciNetMATHGoogle Scholar
  8. Huseyin K 1978Vibrations and stability of multiple parameter systems (Netherlands: Sijthoff & Noordhoff)MATHGoogle Scholar
  9. Jankovic M S 1976 Lateral vibrations of an extending cantilever rod. University of Toronto, Institute for Aerospace Studies, Technical Note 202Google Scholar
  10. Kreider D L, Kuller R G, Ostberg D R 1968Elementary differential equations (Massachusetts: Addison Wesley)MATHGoogle Scholar
  11. Lancaster P 1966Lambda-matrices and vibrating systems (New York: Pergamon)MATHGoogle Scholar
  12. Modi V J, Brereton R C 1969CASI Trans. (Can. Aeronaut. Space Inst.) 2: 44–48Google Scholar
  13. Modi V J, Neilson J E 1968Aeronaut. J. 72: 807–810Google Scholar
  14. Pradeep S 1987On the stability of multidimensional linear systems with time varying parameters, Ph.D. dissertation, Indian Institute of Science, BangaloreGoogle Scholar
  15. Pradeep S, Shrivastava S K 1987aActa Astronaut. (to be published)Google Scholar
  16. Pradeep S, Shrivastava S K 1987bJ. Astronaut. Sci. (to be published)Google Scholar
  17. Pradeep S, Shrivastava S K 1987cJ. Guidance Control Dyn. (to be published)Google Scholar
  18. Pradeep S, Shrivastava S K 1987dMech. Res. Commun. (to be published)Google Scholar
  19. Sandberg I W 1965SIAM J. Control 2: 192–195CrossRefMathSciNetGoogle Scholar
  20. Shrivastava S K 1981J. Appl. Mech. 48: 174–176MATHMathSciNetCrossRefGoogle Scholar
  21. Shrivastava S K, Pradeep S 1985J. Guidance Control Dyn. 8: 579–583MATHGoogle Scholar
  22. Taylor J H, Narendra K S 1969SIAM J. Appl. Math. 17: 343–352MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Indian Academy of Sciences 1992

Authors and Affiliations

  • S Pradeef
    • 1
  • Shashi K Shrivastava
    • 1
  1. 1.Department of Aerospace EngineeringIndian Institute of ScienceBangaloreIndia

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