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Reseach Article

Stability of Conditional Invariant Sets of Control Systems

by Mahadevaswamy B.S.
International Journal of Computer Applications
Foundation of Computer Science (FCS), NY, USA
Volume 184 - Number 40
Year of Publication: 2022
Authors: Mahadevaswamy B.S.
10.5120/ijca2022922482

Mahadevaswamy B.S. . Stability of Conditional Invariant Sets of Control Systems. International Journal of Computer Applications. 184, 40 ( Dec 2022), 1-13. DOI=10.5120/ijca2022922482

@article{ 10.5120/ijca2022922482,
author = { Mahadevaswamy B.S. },
title = { Stability of Conditional Invariant Sets of Control Systems },
journal = { International Journal of Computer Applications },
issue_date = { Dec 2022 },
volume = { 184 },
number = { 40 },
month = { Dec },
year = { 2022 },
issn = { 0975-8887 },
pages = { 1-13 },
numpages = {9},
url = { https://ijcaonline.org/archives/volume184/number40/32576-2022922482/ },
doi = { 10.5120/ijca2022922482 },
publisher = {Foundation of Computer Science (FCS), NY, USA},
address = {New York, USA}
}
%0 Journal Article
%1 2024-02-07T01:23:38.453764+05:30
%A Mahadevaswamy B.S.
%T Stability of Conditional Invariant Sets of Control Systems
%J International Journal of Computer Applications
%@ 0975-8887
%V 184
%N 40
%P 1-13
%D 2022
%I Foundation of Computer Science (FCS), NY, USA
Abstract

In this paper we attempted to extend the work of study the stability properties of conditional invariant sets for a control system in Rn. Here necessary and sufficient conditions for relative to the given control system are determined. Theorems are based on the work of Ladde, an optional control problem maximising a performance inertia is proved.

References
  1. H.A. Antosiewicz: A survey of Lyapunov’s second method. Ann. Math. Studies 41 (1958) pp.141-166, MR Vol.21 # 1432 (1960).
  2. E.A. Barbashin: vc’ en zap M.G.V. no.135 pp.110-133 (1949) Russian.
  3. R. Bellman: Vector Lyapunov functions. J. SIAM Centro Ser. A1 (1962) pp.32-34, MR, Vol.26 (1963).
  4. N.P. Bhatia: Stability and Lyapunov functions in Dynamical Systems, Contributions to Differential equations, 3 (1964), pp.175-188, MR Vol.29 (1965).
  5. N.P. Bhatia and O. Hajek: Local semi-dynamical systems, Lecture notes in Mathematics (1969), Springer-verlag.
  6. N.P. Bhatia and G.P. Szego: Stability theory of Dynomical systems, Springer-Verlag, (1970).
  7. G.D. Birkhoff: Dynamical systems - American Mathematical Society Colloquium, Vol.IX, Providence R.I. (1927).
  8. L. Cesari: Asymptotic behaviour and stability problems in ordinary differential equations - Springer - Verlag, Heidelberg (1959), MR Vol.29 # 1973 (1961).
  9. A.S.N. Charlu, A.A. Kayande and V. Lakshmikantham: Stability in tube-like domains (See 31).
  10. C. Corduneanu: Symp. Math. 6. Meccanica nonlineare Stability, Feb. (1970), L1. New York, Acad. Press (1971).
  11. C. Corduneanu: Applications of differential inequalities to stability theory (Russian) (1960).
  12. W.J. Cunningham: Introduction to nonlinear Analysis, Mc- Graw Hill, New York, (1958).
  13. S.G. Deo: Boll. della Unione. Mate Italiana 6, (1972).
  14. S.G. Deo: On vector Lyapunov functions: Proc. Amer. Math. Soc. 29, (1971), pp. 575-580, MR Vol. 43 # 7725 (1972).
  15. W. Hahn: Stability of Motion, Springer-Verlag (1967) Translated by Arne P. Baartsz, MR Vol.36 # 6716 (1968).
  16. W. Hahn: Theory and applications of Lyapunov’s direct method, MR Vol.26, (1962).
  17. A. Halany: For and against the Lyapunov function, Symposia Mathematica, Vol. VI (INDAM, Rome, 1970), pp.167-175, Acad. Press, London (1971). MR Vol.44 # 1889 (1972)
  18. A. Halany: Diff. eqn., stability, oscillations and time lag, Acad. Press, Maths on Science and Engg. Vol.23, (1966).
  19. P. Habets and K. Peiffer: Classification of stability like concepts and their study using vector Lyapunov functions. Inst. Math. Pure et Appl. Univ. Catholique di Liouvain. Rapport No.43 Nov. (1971); J. Math. Anal. Appl. 43 (1973) pp.537- 570, MR Vol. 48 # 11696 (1974).
  20. O. Hajek: Dynamical systems in the Plane, Acad. Press (1968), MR Vol. 39, (1970).
  21. R.E. Kalman: Algebraic aspects of the theory of dynamical systems; Differential equations and dynamical systems, Acad. Press, (1967).
  22. R.E. Kalman: Mathematical discription of Linear Dynamical Systems, J. SIAM Control, (1963).
  23. R.E. Kalman and J.E. Bertram: Control system Analysis and design via the second method of Lyapunov, A.S.M.E.J. of Basic Engineering (1960).
  24. A.A. Kayande and V. Lakshmikantam; General Dynamical systems and conditional stability, Proc. Cambridge Philos, Soc. 63 (1967), pp. 199-207, MR Vol. 34 # 6258 (1967).
  25. A.A. Kayande and V. Lakshmikantham: Conditional invariant sets and vector Lyapunov functions - J. Math. Anal. Appl. 14 (1966), pp.285-293, MR Vol. 32 # 7880 (1966).
  26. A.A. Kayande and V. Lakshmikantham: Complex differential systems and extensions of Lyapunov’s method. J. Math. Anal. Appl. 13 (1966), pp.337-347, MR Vol.32 # 2682(1966).
  27. A.A. Kayande and V. Lakshmikantham: General dynamical systems and differential inequalities. Technical report, U.R.I. No.2 (1968).
  28. V. Lakshmikantham: Conditional stability and vector Lyapunov functions, J. Math. Anal. Appl. 10 (1965), pp. 368-377, MR Vol. 32 # 257 (1966).
  29. G.S. Ladde and V. Lakshmikantham: On flow-invariant sets Pacific J. Math. 51 (1974), pp. 215-220, MR Vol. 49 # 10972 (1975).
  30. V. Lakshmikantham and S. Leela; Asymptotically selfinvariant sets and conditional stability - Dynamical systems - An International Symposium, Vol.2, Edited by L. Cesari, Jack. Hale, and J.P. Lasalle, Acad. Press, pp.363-373, (1967), MR Vol. 36 # 2910 (1968).
  31. V. Lakshmikantham and S. Leela: Differential and Integral Inequalities, Acad. press, Vol.55-I, Math. in Sc. & Engg. (1969).
  32. V. Lakshmikantham and S. Leela: Rev. Roun. de-Math-Pure et. appl. 12 (1967) pp.969-976.
  33. V. Lakshmikantham, S. Leela and G.S. Ladde: Conditionally asymptotically Invariant sets and perturbed systems, Annali Di Mate Pure et appl. Bolgona.
  34. V. Lakshmikantham, S. Leela and T. Sastry: Converse theorems for conditional stability, J. Math. Anal. Appl.19(1967), pp.444-456, MR Vol.(35) (1968)
  35. G.S. Ladde and S. Leela: Analysis of Invariant sets, Ann. Mat. Pura. Appl. (4) 94 (1972) pp.283-289, MR Vol.(47) # 3777 (1974).
  36. J.P. Lasalle: Stability and Control, J.SIAM Control Ser. A1 (1962) pp.3-15; MR Vol.26 (1963).
  37. J.P. Lasalle and S. Lefschets: Stability by Lyapunov’s direct method with applications, Acad, Press, New York, Vol.4, Maths in Sc. & Engg. (1961), MR Vol.23 (1962).
  38. S. Leela: Analele Stii Univ. “API Cuza” (1971)
  39. S. Lefschetz: Differential equations-geometric theory, Interscience, New York, 1957, MR Vol.22 # 12257 (1961).
  40. H. Leiphols: Stability theory, Acad. Press, New York (1970) (Translation from German edition - Stuttgar, 1968).
  41. A.M. Liapunov: Probleme general de stabilite de mouvement. Ann. Fac. Sci. Toulouse (1907), French translation of the original paper published in 1893 in Comm. Math. Kharkow; reprinted as Vol. 17 in Ann. Math. Studies, Princeton (1949).
  42. A.M. Liapunov: Stability of Motion (English translation), New York, (1966).
  43. A.A. Markov: On a general property of minimal sets (Russian) Rusk. Astron. Zn. (1932).
  44. G. Malkin: Theorie der stabilitat einer Bewegung Verlag R. olden beurg Munchen (1959), German.
  45. A.S. Oziraner: Vest Mosk Gos. Univ. Mat. Mekh. No.1 (1971), Mekh No; 1. (1972).
  46. A.S. Oziraner: Vest Mosk Gos. Univ. Mat. Mekh No.3 (1971).
  47. A.S. Oziraner: PMM 36 (1972), pp.396-404, English translation in J. Appl. Math. Mech. Oct. (1972).
  48. A.S. Oziraner and V.V. Rumiantsev: PMM 36 (1972). pp.341- 362, English translation J. Appl. Math. Mech.
  49. B.G. Pachpatte: Strict stability in dynamical systems, J. Diff. eqns. 11 (1972), pp.494-473, MR Vol.45 # 7219 (1973).
  50. K. Peiffer and N. Rouche: Liapunov’s Second method applied to partial stability (French summary), J. Mechanique (1969), pp.323-324, MR Vol.40(1970).
  51. C. Risoto: Anali di Math Pure et Appl. Sect. 6 Vol.84 (1970).
  52. V.V. Rumiantsev: Symp. Math. 6. Meccanica non-lineare et stabilita Feb (1970) L.1. New York, Acad. Press (1971), pp.243-265.
  53. V.V. Rumiantsev: PMM 35 (1971), pp.138-143 (English translation J. Appl. Math. Mech).
  54. V.V. Rumiantsev: Method of Liapunov Functions in the stability theory of Motion - Fifty years of Mechanics in USSR Vol.1 Nauka, Moscow (1968).
  55. V.V. Rumiantsev: Vest Moskov. Ges. Univ. No.4 (1957).
  56. V.V. Rumiantsev: PMM 34, No.4 (1970).
  57. E.O. Roxin: Stability in general control systems, J. Diff. equations. Vol.1 (1965), pp.115-150.
  58. N. Rouche, P. Habets and M. Laloy: Stability by Liapunov’s Direct method, Appl. Maths. Sciences, 22, Springer-Verlag (1977), MR Vol.56 # 9008 (1978).
  59. G. Sansone and R. Conti: Equazion Differensiali Non-Linear, Roma (1956), Chap.IX.
  60. P. Seibert: Liapunov Functions and Comparison Principles; Dynamical systems - An International Symposium, Vol.2, Edited by L. Cesari; J. Hale, J.P. Lasalle, Acad. Press (1976), pp.181-185.
  61. G.R. Sell: Topological dynamics and Ord. Diff. equns. Van Nostrand Reinhold Coy. Lond. (1971).
  62. G.R. Sell: On the fundamental theory of ord. diff. eqns., J. Diff. eqns. July (1965), vol.1, No.3.
  63. H. Whitney: Proc. Nat. Acad. Sci. USA 18/1932 pp.275-278 and 340-342.
  64. T. Yoshizawa: Funkcialaj Ekvacioj, 5 (1963), pp.1-11.
  65. T. Yoshizawa: Funkcialaj Ekvacioj, 6 (1964).
  66. T. Yoshizawa: Stability theory and the existence of periodic solutions and almost periodic solutions. Appl. Math. Sciences, Springer-Verlag (1975).
  67. T. Yoshizawa: Asymptotic behaviour of solutions of a system of differential equations, Contrib. Diff. eqns. 1(1963), pp.371- 387.
  68. T. Yoshizawa: Stability theory of Liapunov’s second method Publication No.9, The Math, Soc. of Japan, Tokyo, (1966).
  69. V.I. Zubov: Mathematical Methods for Investigation of Automatic control systems, Leningrad Supromgiz (1959), MR Vol.21 # 5791 (1960).
  70. V.I. Zubov: The methods of A.M. Lyapunov and their applications (English translation) Noordhoff, (1964).
Index Terms

Computer Science
Information Sciences

Keywords

Invariant Set Asymptotical Control System Perturbed System