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Approximate Controllability for Systems Governed by Nonlinear Volterra Type Equations

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Abstract

We study the control systems governed by abstract Volterra equations without uniqueness in a Banach space. By using the technique of the theory of condensing maps and multivalued analysis tools, we obtain the existence result, investigate the topological structure of the solution set, and prove the invariance of a reachability set of the control system under nonlinear perturbations. Examples concerning fractional order differential equations and first order evolution equations with multiple delays are proposed to demonstrate the applications to approximate controllability results.

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References

  1. Agarwal, R.P., Belmekki, M., Benchohra, M.: A survey on semilinear differential equations and inclusions involving Riemann–Liouville fractional derivative. Adv. Differ. Equ. Art. ID 981728, 47 pp (2009)

    Google Scholar 

  2. Balachandran K., Dauer J.P.: Controllability of nonlinear systems in Banach spaces: a survey. J. Optim. Theory Appl. 115(1), 7–28 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bashirov A.E., Mahmudov N.I.: On concepts of controllability for deterministic and stochastic systems. SIAM J. Control Optim. 37(6), 1808–1821 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bensoussan A., Da Prato G., Delfour M.C., Mitter S.K.: Representation and Control of Infinite Dimensional Systems. Birkhaüser, Boston (2007)

    MATH  Google Scholar 

  5. Borsuk, K.: Theory of Retracts. Monografie Mat., vol. 44. PWN, Warszawa (1967)

  6. Browder F.E., Gupta C.P.: Topological degree and nonlinear mappings of analytic type in Banach spaces. J. Math. Anal. Appl. 26, 390–402 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  7. Dugundji J.: An extension of Tietze’s theorem. Pac. J. Math. 1, 353–367 (1951)

    MathSciNet  MATH  Google Scholar 

  8. Engel, K.-J., Nagel, R.: One-parameter semigroups for linear evolution equations. Graduate Texts in Mathematics, vol. 194. Springer-Verlag, New York (2000)

  9. Górniewicz L.: Topological Fixed Point Theory of Multivalued Mappings. 2nd edn. Springer, Dordrecht (2006)

    MATH  Google Scholar 

  10. Górniewicz L., Lassonde M.: Approximation and fixed points for compositions of R δ -maps. Topol. Appl. 55(3), 239–250 (1994)

    Article  MATH  Google Scholar 

  11. Granas A., Dugundji J.: Fixed Point Theory. Springer Monographs in Mathematics. Springer-Verlag, New York (2003)

    Google Scholar 

  12. Kamenskii, M., Obukhovskii, V., Zecca, P.: Condensing multivalued maps and semilinear differential inclusions in Banach spaces. de Gruyter Series in Nonlinear Analysis and Applications, vol. 7. Walter de Gruyter, Berlin (2001)

  13. Kilbas A.A., Srivastava H.M., Trujillo J.J.: Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies. Elsevier, Amsterdam (2006)

    Google Scholar 

  14. Liou Y.C., Obukhovskii V., Yao J.C.: Application of a coincidence index to some classes of impulsive control systems. Nonlinear Anal. 69(12), 4392–4411 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  15. Machado J.T., Kiryakova V., Mainardi F.: Recent history of fractional calculus. Commun. Nonlinear Sci. Numer. Simul. 16(3), 1140–1153 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  16. Naito K.: An inequality condition for approximate controllability of semilinear control system. J. Math. Anal. Appl. 138(1), 129–136 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  17. Naito K.: On controllability for a nonlinear Volterra equation. Nonlinear Anal. 18(1), 99–108 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  18. Naito K., Park J.Y.: Approximate controllability for trajectories of a delay Volterra control system. J. Optim. Theory Appl. 61(2), 271–279 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  19. Obukhovskii V., Yao J.-C.: Some existence results for fractional functional differential equations. Fixed Point Theory 11(1), 85–96 (2010)

    MathSciNet  MATH  Google Scholar 

  20. Obukhovskii V., Zecca P.: Controllability for systems governed by semilinear differential inclusions in a Banach space with a noncompact semigroup. Nonlinear Anal. 70(9), 3424–3436 (2009)

    Article  MathSciNet  Google Scholar 

  21. Podlubny, I.: Fractional differential equations. An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. Mathematics in Science and Engineering, vol. 198. Academic Press, San Diego (1999)

  22. Qin, Y.: Nonlinear parabolic-hyperbolic coupled systems and their attractors. Operator Theory: Advances and Applications, vol. 184. Advances in Partial Differential Equations (Basel). Birkhaüser Verlag, Basel (2008)

  23. Quinn M.D., Carmichael N.: An approach to nonlinear control problems using fixed-point methods, degree theory and pseudo-inverses. Numer. Funct. Anal. Optim. 7(23), 197–219 (1984/1985)

    MathSciNet  Google Scholar 

  24. Seidman T.I.: Invariance of the reachable set under nonlinear perturbations. SIAM J. Control Optim. 25(5), 1173–1191 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  25. Triggiani R.: A note on the lack of exact controllability for mild solutions in Banach spaces. SIAM J. Control Optim. 15(3), 407–411 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  26. Triggiani R.: Addendum: “A note on the lack of exact controllability for mild solutions in Banach spaces”. SIAM J. Control Optim. 18(1), 98–99 (1980)

    Article  MathSciNet  Google Scholar 

  27. Wang L.W.: Approximate controllability for integrodifferential equations with multiple delays. J. Optim. Theory. Appl. 143(1), 185–206 (2009)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Hanoi National University of Education, 136 Xuan Thuy, Cau Giay, Hanoi, Vietnam

    Tran Dinh Ke

  2. Faculty of Physics and Mathematics, Voronezh State Pedagogical University, 394043, Voronezh, Russia

    Valeri Obukhovskii

  3. Department of Applied Mathematics, National Sun Yat-sen University, 804, Kaohsiung, Taiwan, ROC

    Ngai-Ching Wong & Jen-Chih Yao

  4. Center for General Education, Kaohsiung Medical University, 807, Kaohsiung, Taiwan, ROC

    Jen-Chih Yao

Authors
  1. Tran Dinh Ke
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  2. Valeri Obukhovskii
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  3. Ngai-Ching Wong
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  4. Jen-Chih Yao
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Correspondence to Valeri Obukhovskii.

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Ke, T.D., Obukhovskii, V., Wong, NC. et al. Approximate Controllability for Systems Governed by Nonlinear Volterra Type Equations. Differ Equ Dyn Syst 20, 35–52 (2012). https://doi.org/10.1007/s12591-011-0101-7

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