Advertisement

Controllability in Dynamics of Diffusion Processes with Nonlocal Conditions

  • Luisa Malaguti
  • Krzysztof Rykaczewski
  • Valentina TaddeiEmail author
Article
  • 102 Downloads

Abstract

The paper deals with semilinear evolution equations in Banach spaces. By means of linear control terms, the controllability problem is investigated and the solutions satisfy suitable nonlocal conditions. The Cauchy multi-point condition and the mean value condition are included in the present discussion. The final configuration is always achieved with a control with minimum norm. The results make use of fixed point techniques; two different approaches are proposed, depending on the use of norm or weak topology in the state space. The discussion is completed with some applications to dynamics of diffusion processes.

Keywords

Controllability semilinear differential inclusion nonlocal solution mild solution fixed point 

Mathematics Subject Classification

Primary 93B05 34A60 Secondary 34B15 47H04 47H09 47H10 

Notes

Acknowledgements

Funding was provided by Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni.

References

  1. 1.
    Akmerov, R.R., Kamenskiĭ, M.I., Potapov, A.S., Rodkina, A.E., Sadovskiĭ, B.N.: Measures of Noncompactness and Condensing Operators. Birkhiäuser, Basel (1992)CrossRefGoogle Scholar
  2. 2.
    Balachandran, K., Dauer, J.P.: Controllability of nonlinear systems in Banach spaces: a survey. J. Optim. Theory Appl. 115, 7–28 (2002)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Benchohra, M., Ntouyas, S.K.: Existence and controllability results for nonlinear differential inclusions with nonlocal conditions in Banach spaces. J. Appl. Anal. 8, 33–48 (2002)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Benchohra, M., Gatsori, E.P., Górniewicz, L., Ntouyas, S.K.: Controllability results for evolution inclusions with non-local conditions. Z. Anal. Anwend. 22, 411–431 (2003)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Benedetti, I., Obukhovskii, V.V., Taddei, V.: Controllability for systems governed by semilinear evolution inclusions without compactness. Nonlinear Differ. Equ. Appl. 21, 795–812 (2014)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Bochner, S., Taylor, A.E.: Linear functionals on certain spaces of abstractly-valued functions. Ann. Math. 39, 913–944 (1938)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Cardinali, T., Rubbioni, P.: On the existence of mild solutions of semilinear evolution differential inclusions. J. Math. Anal. Appl. 308, 620–635 (2005)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Cardinali, T., Rubbioni, P.: Corrigendum and addendum to On the existence of mild solutions of semilinear evolution differential inclusions. J. Math. Anal. Appl. 438, 514–517 (2016)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Carmichael, N., Quinn, M.D.: Fixed point methods in nonlinear control. In: Distributed Parameter Systems. Lecture Notes in Control and Information Sciences, vol. 75, pp. 24–51. Springer, Berlin (1984)Google Scholar
  10. 10.
    Diestel, J., Ruess, W.M., Schachermayer, W.: Weak compactness in \(L^{1}(\mu, X)\). Proc. Am. Math. Soc. 118, 447–453 (1993)zbMATHGoogle Scholar
  11. 11.
    Engel, K.J., Nagel, R.: One-parameter semigroups for linear evolution equations. In: Axler S, Ribet K (eds) Graduate Texts in Mathematics, vol. 194. Springer, New York (2000)Google Scholar
  12. 12.
    Fan, K.: Fixed-point and minimax theorems in locally convex topological linear spaces. Proc. Natl. Acad. Sci. USA 38, 121 (1952)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Hutson, V., Shen, W., Vickers, G.T.: Spectral theory for nonlocal dispersal with periodic or almost-periodic time dependence. Rocky Mt. J. Math. 38, 1147–1175 (2008)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Jang, T.S.: A new solution procedure for the nonlinear telegraph equation. Commun. Nonlinear Sci. Numer. Simul. 29, 307–326 (2015)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Jin, Y., Zhao, X.Q.: Spatial dynamics of a periodic population model with dispersal. Nonlinearity 22, 1167–1189 (2009)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Kamenskii, M.I., Obukhovskii, V.V., Zecca, P.: Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Space. W. de Gruyter, Berlin (2001)CrossRefGoogle Scholar
  17. 17.
    Kantorovich, L.V., Akilov, G.P.: Functional Analysis. Pergamon Press, Oxford (1982)zbMATHGoogle Scholar
  18. 18.
    Kmit, I.: Fredholm solvability of a periodic Neumann problem for a linear telegraph equation. Ukr. Math. J. 65, 423–434 (2013)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Li, W., Zhang, H.: Positive doubly periodic solutions of telegraph equations with delays. Bound. Value Probl. 2015, 12 (2015)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Obukhovskiĭ, V.V., Zecca, P.: Controllability for systems governed by semilinear differential inclusions in a Banach space with a noncompact semigroup. Nonlinear Anal. Theory Methods Appl. 70, 3424–3436 (2009)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, Berlin (1983)CrossRefGoogle Scholar
  22. 22.
    Pettis, B.J.: On the integration in vector spaces. Trans. Am. Math. Soc. 44, 277–304 (1938)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Rykaczewski, K.: Approximate controllability of differential inclusions in Hilbert spaces. Nonlinear Anal. Theory Methods Appl. 75, 2701–2712 (2012)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Triggiani, R.: A note on the lack of exact controllability for mild solutions in Banach spaces. SIAM J. Control Optim. 15, 407–411 (1977)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Sciences and Methods for EngineeringUniversity of Modena and Reggio EmiliaReggio EmiliaItaly
  2. 2.Faculty of Mathematics and Computer ScienceNicolaus Copernicus UniversityToruńPoland

Personalised recommendations