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Schauder’s fixed-point theorem in approximate controllability problems Cover

Schauder’s fixed-point theorem in approximate controllability problems

International Journal of Applied Mathematics and Computer Science
Volume 26 (2016): Issue 2 (June 2016)
By: Artur Babiarz,  Jerzy Klamka and  Michał Niezabitowski  
Open Access
|Jul 2016

References

  1. Arapostathis, A., George, R.K. and Ghosh, M.K. (2001). On the controllability of a class of nonlinear stochastic systems, Systems & Control Letters 44(1): 25-34.10.1016/S0167-6911(01)00123-2
    Search in Google Scholar
  2. Babiarz, A., Czornik, A., Klamka, J. and Niezabitowski, M. (2015a). Controllability of discrete-time linear switched systems with constrains on switching signal, in N.T. Nguyen et al. (Eds.), Intelligent Information and Database Systems, Lecture Notes in Computer Science, Vol. 9011, Springer International Publishing, Berlin, pp. 304-312.
    Search in Google Scholar
  3. Babiarz, A., Czornik, A., Klamka, J. and Niezabitowski, M. (2015b). The selected problems of controllability of discrete-time switched linear systems with constrained switching rule, Bulletin of the Polish Academy of Sciences: Technical Sciences 63(3): 657-666.10.1515/bpasts-2015-0077
    Search in Google Scholar
  4. Babiarz, A., Czornik, A. and Niezabitowski, M. (2016). Output controllability of the discrete-time linear switched systems, Nonlinear Analysis: Hybrid Systems 21: 1-10.10.1016/j.nahs.2015.12.004
    Search in Google Scholar
  5. Bader, R., Gabor, G. and Kryszewski, W. (1996). On the extension of approximations for set-valued maps and the repulsive fixed points, Bollettino della Unione Matematica Italiana B 10(2): 399-416.
    Search in Google Scholar
  6. Bader, R. and Kryszewski, W. (1994). Fixed-point index for compositions of set-valued maps with proximally ∞-connected values on arbitrary ANR’s, Set-Valued Analysis 2(3): 459-480.10.1007/BF01026835
    Search in Google Scholar
  7. Balachandran, K. and Dauer, J. (2002). Controllability of nonlinear systems in Banach spaces: A survey, Journal of Optimization Theory and Applications 115(1): 7-28.10.1023/A:1019668728098
    Search in Google Scholar
  8. Balachandran, K. and Sakthivel, R. (2001). Controllability of integrodifferential systems in Banach spaces, Applied Mathematics and Computation 118(1): 63-71.10.1016/S0096-3003(00)00040-0
    Search in Google Scholar
  9. Bashirov, A.E. and Kerimov, K.R. (1997). On controllability conception for stochastic systems, SIAM Journal on Control and Optimization 35(2): 384-398.10.1137/S0363012994260970
    Search in Google Scholar
  10. Bashirov, A.E. and Mahmudov, N.I. (1999). On concepts of controllability for deterministic and stochastic systems, SIAM Journal on Control and Optimization 37(6): 1808-1821.10.1137/S036301299732184X
    Search in Google Scholar
  11. Benchohra, M., Gorniewicz, L., Ntouyas, S. and Ouahab, A. (2004). Controllability results for impulsive functional differential inclusions, Reports on Mathematical Physics 54(2): 211-228.10.1016/S0034-4877(04)80015-6
    Search in Google Scholar
  12. Benchohra, M. and Ouahab, A. (2005). Controllability results for functional semilinear differential inclusions in Fr´echet spaces, Nonlinear Analysis: Theory, Methods & Applications 61(3): 405-423.10.1016/j.na.2004.12.002
    Search in Google Scholar
  13. Bian, W. (1999). Constrained controllability of some nonlinear systems, Applicable Analysis 72(1-2): 57-73.10.1080/00036819908840730
    Search in Google Scholar
  14. Chang, Y.-K. (2007). Controllability of impulsive functional differential systems with infinite delay in Banach spaces, Chaos, Solitons & Fractals 33(5): 1601-1609.10.1016/j.chaos.2006.03.006
    Search in Google Scholar
  15. Curtain, R.F. and Zwart, H. (1995). An Introduction to Infinitedimensional Linear Systems Theory, Springer Science & Business Media, Berlin.10.1007/978-1-4612-4224-6
    Search in Google Scholar
  16. Czornik, A. and ´Swierniak, A. (2001). On controllability with respect to the expectation of discrete time jump linear systems, Journal of the Franklin Institute 338(4): 443-453.10.1016/S0016-0032(01)00014-X
    Search in Google Scholar
  17. Czornik, A. and ´Swierniak, A. (2004). On direct controllability of discrete time jump linear system, Journal of the Franklin Institute 341(6): 491-503.10.1016/j.jfranklin.2004.05.005
    Search in Google Scholar
  18. Czornik, A. and ´Swierniak, A. (2005). Controllability of discrete time jump linear systems, Dynamics of Continuous Discrete and Impulsive Systems B: Applications & Algorithms 12(2): 165-189.
    Search in Google Scholar
  19. Dauer, J. and Mahmudov, N. (2002). Approximate controllability of semilinear functional equations in Hilbert spaces, Journal of Mathematical Analysis and Applications 273(2): 310-327.10.1016/S0022-247X(02)00225-1
    Search in Google Scholar
  20. Dauer, J., Mahmudov, N. and Matar, M. (2006). Approximate controllability of backward stochastic evolution equations in Hilbert spaces, Journal of Mathematical Analysis and Applications 323(1): 42-56.10.1016/j.jmaa.2005.09.089
    Search in Google Scholar
  21. Do, V. (1989). A note on approximate controllability of semilinear systems, Systems & Control Letters 12(4): 365-371.10.1016/0167-6911(89)90047-9
    Search in Google Scholar
  22. Dubov, M. and Mordukhovich, B. (1978). Theory of controllability of linear stochastic systems, Differential Equations 14: 1609-1612.
    Search in Google Scholar
  23. George, R.K. (1995). Approximate controllability of nonautonomous semilinear systems, Nonlinear Analysis: Theory, Methods & Applications 24(9): 1377-1393.10.1016/0362-546X(94)E0082-R
    Search in Google Scholar
  24. Gorniewicz, L., Granas, A. and Kryszewski, W. (1991). On the homotopy method in the fixed point index theory of multi-valued mappings of compact absolute neighborhood retracts, Journal of Mathematical Analysis and Applications 161(2): 457-473.10.1016/0022-247X(91)90345-Z
    Search in Google Scholar
  25. Gorniewicz, L., Ntouyas, S. and O’Regan, D. (2005). Controllability of semilinear differential equations and inclusions via semigroup theory in Banach spaces, Reports on Mathematical Physics 56(3): 437-470.10.1016/S0034-4877(05)80096-5
    Search in Google Scholar
  26. Henríquez, H.R. (2008). Approximate controllability of linear distributed control systems, Applied Mathematics Letters 21(10): 1041-1045.10.1016/j.aml.2007.10.024
    Search in Google Scholar
  27. Hong, X.Z. (1982). A note on approximate controllability for semilinear one-dimensional heat equation, Applied Mathematics and Optimization 8(1): 275-285.10.1007/BF01447763
    Search in Google Scholar
  28. Jeong, J.-M. and Roh, H.-H. (2006). Approximate controllability for semilinear retarded systems, Journal of Mathematical Analysis and Applications 321(2): 961-975.10.1016/j.jmaa.2005.09.005
    Search in Google Scholar
  29. Klamka, J. (2000). Constrained approximate controllability, IEEE Transactions on Automatic Control 45(9): 1745-1749.10.1109/9.880640
    Search in Google Scholar
  30. Kryszewski, W. and Zezza, P. (1994). Remarks on the relay controllability of control systems, Journal of Mathematical Analysis and Applications 188(1): 45-65.10.1006/jmaa.1994.1410
    Search in Google Scholar
  31. Kumlin, P. (2004). A note on fixed point theory, Functional Analysis Lecture, Mathematics, Chalmers & GU, Gothenburg.
    Search in Google Scholar
  32. Lasiecka, I. and Triggiani, R. (1991). Exact controllability of semilinear abstract systems with application to waves and plates boundary control problems, Applied Mathematics and Optimization 23(1): 109-154.10.1007/BF01442394
    Search in Google Scholar
  33. Li, M., Wang, M. and Zhang, F. (2006). Controllability of impulsive functional differential systems in Banach spaces, Chaos, Solitons & Fractals 29(1): 175-181.10.1016/j.chaos.2005.08.041
    Search in Google Scholar
  34. Mahmudov, N.I. (2001a). Controllability of linear stochastic systems, IEEE Transactions on Automatic Control 46(5): 724-731.10.1109/9.920790
    Search in Google Scholar
  35. Mahmudov, N.I. (2001b). Controllability of linear stochastic systems in Hilbert spaces, Journal of Mathematical Analysis and Applications 259(1): 64-82.10.1006/jmaa.2000.7386
    Search in Google Scholar
  36. Mahmudov, N.I. (2002). On controllability of semilinear stochastic systems in Hilbert spaces, IMA Journal of Mathematical Control and Information 19(4): 363-376.10.1093/imamci/19.4.363
    Search in Google Scholar
  37. Mahmudov, N.I. (2003). Approximate controllability of semilinear deterministic and stochastic evolution equations in abstract spaces, SIAM Journal on Control and Optimization 42(5): 1604-1622.10.1137/S0363012901391688
    Search in Google Scholar
  38. Mahmudov, N.I. (2008). Approximate controllability of evolution systems with nonlocal conditions, Nonlinear Analysis: Theory, Methods & Applications 68(3): 536-546.10.1016/j.na.2006.11.018
    Search in Google Scholar
  39. Mahmudov, N. and Zorlu, S. (2003). Controllability of non-linear stochastic systems, International Journal of Control 76(2): 95-104.10.1080/0020717031000065648
    Search in Google Scholar
  40. Naito, K. (1987). Controllability of semilinear control systems dominated by the linear part, SIAM Journal on Control and Optimization 25(3): 715-722.10.1137/0325040
    Search in Google Scholar
  41. Naito, K. (1989). Approximate controllability for trajectories of semilinear control systems, Journal of Optimization Theory and Applications 60(1): 57-65.10.1007/BF00938799
    Search in Google Scholar
  42. Narayanamoorthy, S. and Sowmiya, S. (2015). Approximate controllability result for nonlinear impulsive neutral fuzzy stochastic differential equations with nonlocal conditions, Advances in Difference Equations 2015(1): 1-16.10.1186/s13662-015-0454-2
    Search in Google Scholar
  43. Pazy, A. (2012). Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer Science & Business Media, Berlin.
    Search in Google Scholar
  44. Przeradzki, B. (2012). A generalization of Krasnoselskii fixed point theorem for sums of compact and contractible maps with application, Open Mathematics 10(6): 2012-2018.10.2478/s11533-012-0102-y
    Search in Google Scholar
  45. Radhakrishnan, B. and Balachandran, K. (2011). Controllability of impulsive neutral functional evolution integrodifferential systems with infinite delay, Nonlinear Analysis: Hybrid Systems 5(4): 655-670.10.1016/j.nahs.2011.05.001
    Search in Google Scholar
  46. Sakthivel, R., Mahmudov, N. and Kim, J. (2007). Approximate controllability of nonlinear impulsive differential systems, Reports on Mathematical Physics 60(1): 85-96.10.1016/S0034-4877(07)80100-5
    Search in Google Scholar
  47. Sakthivel, R., Nieto, J.J. and Mahmudov, N. (2010). Approximate controllability of nonlinear deterministic and stochastic systems with unbounded delay, Taiwanese Journal of Mathematics 14(5): 1777-1797.10.11650/twjm/1500406016
    Search in Google Scholar
  48. Shen, L. and Sun, J. (2011). Approximate controllability of stochastic impulsive systems with control-dependent coefficients, IET Control Theory & Applications 5(16): 1889-1894.10.1049/iet-cta.2010.0422
    Search in Google Scholar
  49. Sikora, B. and Klamka, J. (2012). On constrained stochastic controllability of dynamical systems with multiple delays in control, Bulletin of the Polish Academy of Sciences: Technical Sciences 60(2): 301-305.10.2478/v10175-012-0040-7
    Search in Google Scholar
  50. Sırbu, M. and Tessitore, G. (2001). Null controllability of an infinite dimensional SDE with state-and control-dependent noise, Systems & Control Letters 44(5): 385-394.10.1016/S0167-6911(01)00158-X
    Search in Google Scholar
  51. Wang, L. (2006). Approximate controllability and approximate null controllability of semilinear systems, Communications on Pure and Applied Analysis 5(4): 953-962.10.3934/cpaa.2006.5.953
    Search in Google Scholar
  52. Zang, Y. and Li, J. (2013). Approximate controllability of fractional impulsive neutral stochastic differential equations with nonlocal conditions, Boundary Value Problems 2013(1): 1-13.10.1186/1687-2770-2013-193
    Search in Google Scholar
  53. Zhou, H.X. (1983). Approximate controllability for a class of semilinear abstract equations, SIAM Journal on Control and Optimization 21(4): 551-565. 10.1137/0321033
    Search in Google Scholar
DOI: https://doi.org/10.1515/amcs-2016-0018 | Journal eISSN: 2083-8492 | Journal ISSN: 1641-876X
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Language: English
Page range: 263 - 275
Submitted on: Sep 14, 2015
Accepted on: Jan 29, 2016
Published on: Jul 2, 2016
Published by: University of Zielona Góra
In partnership with: Paradigm Publishing Services
Publication frequency: 4 issues per year
Keywords:
approximate controllability,
Banach space,
Hilbert space,
Schauder’s fixed-point theorem,
infinite-dimensional space.
Related subjects:
Mathematics,
Applied mathematics

© 2016 Artur Babiarz, Jerzy Klamka, Michał Niezabitowski, published by University of Zielona Góra
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

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