Skip to main content
SpringerLink
Log in

Interpretation of Trajectory Control and Optimization for the Nondense Fractional System

  • Original Paper
  • Published:
International Journal of Applied and Computational Mathematics Aims and scope Submit manuscript

Abstract

In this work, we have examined the trajectory controllability of the fractional delay differential equation with a nondense domain. The results are developed using fractional calculus theory and semigroup operator. Further, we drive the system for the existence of optimal pairs using precision measures. An illustration is given to verify the obtained outcomes.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

References

  1. Agarwal, P., Sidi Ammi, M.R., Asad, J.: Existence and uniqueness results on time scales for fractional nonlocal thermistor problem in the conformable sense. Adv. Diff. Equ. 2021(21), 1–11 (2021)

    MathSciNet  MATH  Google Scholar 

  2. Alattas, K.A., Mobayen, S., Din, Sami Ud, Asad, J., Fekih, A., Assawinchaichote, W., Vu, M.T.: Design of a non-singular adaptive integral-Type finite time tracking control for nonlinear systems with external disturbances. IEEE Access 9, 102091–102103 (2021)

    Article  Google Scholar 

  3. Attia, N., Akgül, A., Seba, D., Nour, A., Asad, J.: A novel method for fractal-fractional differential equations. Alexandria Eng. J. 61(12), 9733–9748 (2022)

    Article  Google Scholar 

  4. Bahaa, G.M.: Optimal control problem and maximum principle for fractional order cooperative systems. Kybernetika 55(2), 337–358 (2019)

    MathSciNet  MATH  Google Scholar 

  5. Balder, E.J.: Necessary and sufficient conditions for L1-strong-weak lower semi-continuity of integral functionals. Nonlinear Anal. 11(12), 1399–1404 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  6. Basile, G., Hamano, F.: On the smoothness of the output trajectories for a linear dynamic system. IEEE Trans. Autom. Control 27(1), 196–198 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  7. Chalishajar, D.N., George, Raju K., Nandakumar, A.K., Acharya, F.S.: Trajectory controllability of nonlinear integro-differential system. J. Franklin Inst. 347(7), 1065–1075 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  8. Cacace, F., Cusimano, V., Germani, A., Palumbo, P., Papi, M.: Optimal continuous-discrete linear filter and moment equations for nonlinear diffusions. IEEE Trans. Autom. Control 65(10), 3961–3976 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  9. Diaz-Garcia, J.A., Requejo-Lopez, R.: Use of nonlinear regression and nonlinear mathematical programming in the formulation of substrate mixtures for soil-less culture - a review. J. Soil Sci. Plant Nutrit. 12(1), 87–97 (2012)

    Article  Google Scholar 

  10. Dhayal, R., Muslim, M., Abbas, S.: Approximate and trajectory controllability of fractional neutral differential equation. Adv. Oper. Theory 4(4), 802–820 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  11. Fernando, J.A.K.M., Amarasinghe, A.D.U.S.: Drying kinetics and mathematical modeling of hot air drying of coconut coir pith. SpringerPlus 5, 1–12 (2016)

    Article  Google Scholar 

  12. Fesharaki, S.J., Kamali, M., Sheikholeslam, F., Talebi, H.A.: Robust model predictive control with sliding mode for constrained non-linear systems. IET Control Theory Appl. 14(17), 2592–2599 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  13. Fu, X.L.: On solutions of neutral nonlocal evolution equations with nondense domain. J. Math. Anal. Appl. 299, 392–410 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  14. Gatsori, E.P.: Controllability results for nondensely defined evolution differential inclusions with nonlocal conditions. J. Math. Anal. Appl. 297, 194–211 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  15. Govindaraj, V., Muslim, M., George, R.K.: Trajectory controllability of fractional dynamical systems. J. Control Decis. 4(2), 114–130 (2017)

    MathSciNet  Google Scholar 

  16. Govindaraj, V., George, R.K.: Trajectory controllability of fractional integrodifferential systems in Hilbert spaces. Asian J. Control 20(5), 1994–2004 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  17. Gu, H., Zhou, Y., Ahmad, B., Alsaedi, A.: Integral solutions of fractional evolution equations with nondense domain, Electronic. J. Diff. Equ. 145, 1–15 (2017)

    MATH  Google Scholar 

  18. Jothimani, K., Valliammal, N., Ravichandran, C.: Existence result for a neutral fractional integrodifferential equation with state dependent delay. J. Appl. Nonlinear Dyn. 7(4), 371–381 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  19. Jothimani, K., Kaliraj, K., Hammouch, Z., Ravichandran, C.: New results on controllability in the framework of fractional integrodifferential equations with non-dense domain. European Phys. J. Plus 134(441), 1–10 (2019)

    Google Scholar 

  20. Jothimani, K., Kaliraj, K., Panda, S.K., Nisar, K.S., Ravichandran, C.: Results on controllability of non-densely characterized neutral fractional delay differential system. Evolut. Equ. Control Theory 10(3), 619–631 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  21. Jiang, Y.R., Huang, N.J.: Solvability and optimal controls of fractional delay evolution inclusions with Clarke subdifferential. Math. Methods Appl. Sci. 40(8), 3026–3039 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  22. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and applications of fractional differential equations. In: North-Holland mathematics studies, vol 204. Elsevier Science, Amsterdam, (2006)

  23. Kumar, V., Malik, M., Debbouche, A.: Total controllability of neutral fractional differential equation with non-instantaneous impulsive effects. J. Comput. Appl. Math. 383, 113158 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  24. Komolafe, C.A., Ojediran, J.O., Ajao, F.O., Dada, O.A., Afolabi, Y.T., Oluwaleye, I.O., Alake, A.S.: Modelling of moisture diffusivity during solar drying of locust beans with thermal storage material under forced and natural convection mode. Case Stud. Thermal Eng. 15, 100542 (2019)

    Article  Google Scholar 

  25. Muslim, M., Kumar, A.: Trajectory controllability of fractional differential systems of order a \(\alpha \in (1,2]\) with deviated argument. J. Anal. 28, 295–304 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  26. Muslim, M., George, R.K.: Trajectory controllability of the nonlinear systems governed by fractional differential equations. Diff. Equ. Dyn. Syst. 27, 529–537 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  27. Muslim, M., Kumar, A., Agarwal, R.: Exact and trajectory controllability of second order nonlinear differential equations with deviated argument. Funct. Diff. Equ. 23(1–2), 27–41 (2016)

    MATH  Google Scholar 

  28. Nisar, K.S., Jothimani, K., Kaliraj, K., Ravichandran, C.: An analysis of controllability results for nonlinear Hilfer neutral fractional derivatives with non-dense domain. Chaos Solitons Fractals 146, 110915 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  29. Pazy, A.: Semigroups of linear operators and applications to partial differential equations. Springer-verlag, New York (1983)

    Book  MATH  Google Scholar 

  30. Pan, X., Li, X., Zhao, J.: Solvability and optimal controls of semi linear Riemann- Liouville fractional differential equations. Abstr. Appl. Anal. 2014, 216919 (2014)

    Article  MATH  Google Scholar 

  31. Podlubny, I.: Fractional Differential Equations. An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. Academic Press, San Diego (1999)

  32. Ravichandran, C., Baleanu, D.: On the controllability of fractional functional integro-differential systems with an infinite delay in Banach spaces. Adv. Diff. Equ. (2013). https://doi.org/10.1186/1687-1847-2013-291

    Article  MathSciNet  MATH  Google Scholar 

  33. Qin, H., Zuo, X., Liu, J., Liu, L.: Approximate controllability and optimal controls of fractional dynamical systems of order \(1<q<2\) in Banach space. Adv. Diff. Equ. (2015). https://doi.org/10.1186/s13662-015-0399-5

    Article  MathSciNet  MATH  Google Scholar 

  34. Raja, M.M., Vijayakumar, V., Udhayakumar, R.: Results on the existence and controllability of fractional integro-differential system of order \(1< r< 2\) via measure of noncompactness. Chaos Solitons Fractals 139, 110299 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  35. Rojsiraphisal, T., Mobayen, S., Asad, J., Vu, M.T., Chang, A., Puangmalai, J.: Fast terminal sliding control of underactuated robotic systems based on disturbance observer with experimental validation. Mathematics 9(16), 1–17 (2021)

    Article  Google Scholar 

  36. Slotine, J.J.E., Li, W.: Applied Nonlinear Control. Prentice Hall, New Jersey (1991)

    MATH  Google Scholar 

  37. Trigeassou, J.C., Maamri, N.: Optimal state control of fractional order differential systems: the infinite state approach. Fractal Fract. (2021). https://doi.org/10.3390/fractalfract5020029

    Article  MATH  Google Scholar 

  38. Valliammal, N., Ravichandran, C.: Results on fractional neutral integrodifferential systems with state dependent delay in Banach spaces. Nonlinear Stud. 25(1), 159–171 (2018)

    MathSciNet  MATH  Google Scholar 

  39. Vijayakumar, V.: Approximate controllability results for nondensely defined fractional neutral differential inclusions with Hille Yosida operators. Int. J. Control 92(9), 2210–2222 (2019)

    Article  MATH  Google Scholar 

  40. Wang, J.R., Zhou, Y., Medved, M.: On the solvability and optimal controls of fractional integrodifferential evolution systems with infinite delay. J. Optim. Theory Appl. 152, 31–50 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  41. Zhou, Y.: Basic Theory of Fractional Differential Equations. World Scientific, Singapore (2014)

    Book  MATH  Google Scholar 

Download references

Funding

None.

Author information

Authors and Affiliations

  1. Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, Vellore, Tamilnadu, 632 014, India

    K. Jothimani

  2. Department of Mathematics, Kongunadu Arts and Science College, Coimbatore, Tamilnadu, 641 029, India

    C. Ravichandran

  3. Max-Planck Institute for Dynamics of Complex Technical Systems, Sandtorstrasse 1, 39106, Magdeburg, Germany

    Vipin Kumar

  4. INSA Hauts-de-France, LAMIH UMR CNRS 8201, UPHF, Campus Mont Houy, 59300, Valenciennes, France

    Mohamed Djemai

  5. QUARTZ Laboratory EA 7393, ENSEA Cergy, 6 Avenue du Ponceau, 95000, Cergy, France

    Mohamed Djemai

  6. Department of Mathematics, College of Arts and Sciences, Prince Sattam Bin Abdulaziz University, Wadi Aldawasir, 11991, Saudi Arabia

    Kottakkaran Sooppy Nisar

Authors
  1. K. Jothimani
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. C. Ravichandran
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Vipin Kumar
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Mohamed Djemai
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Kottakkaran Sooppy Nisar
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to C. Ravichandran.

Ethics declarations

Conflicts of interest

None.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Jothimani, K., Ravichandran, C., Kumar, V. et al. Interpretation of Trajectory Control and Optimization for the Nondense Fractional System. Int. J. Appl. Comput. Math 8, 273 (2022). https://doi.org/10.1007/s40819-022-01478-z

Download citation

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s40819-022-01478-z

Keywords

Mathematics Subject Classification

Search

Navigation