Nothing Special   »   [go: up one dir, main page]
Skip to main content
Springer Nature Link
Log in

An Efficient Approximation Technique for Solving a Class of Fractional Optimal Control Problems

  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

In this paper, we discuss a class of fractional optimal control problems, where the system dynamical constraint comprises a combination of classical and fractional derivatives. The necessary optimality conditions are derived and shown that the conditions are sufficient under certain assumptions. Additionally, we design a well-organized algorithm to obtain the numerical solution of the proposed problem by exercising Laguerre polynomials. The key motive associated with the present approach is to convert the concerned fractional optimal control problem to an equivalent standard quadratic programming problem with linear equality constraints. Given examples illustrate the computational technique of the method together with its efficiency and accuracy. Graphical representations are provided to analyze the performance of the state and control variables for distinct prescribed fractions.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

References

  1. Bhatt, S.K.: An existence theorem for a fractional control problem. J. Optim. Theory Appl. 11, 379–385 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  2. Agrawal, O.P., Baleanu, D.: A Hamiltonian formulation and a direct numerical scheme for fractional optimal control problems. J. Vib. Control 13, 1269–1281 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  3. Baleanu, D., Defterli, O., Agrawal, O.P.: A central difference numerical scheme for fractional optimal control problems. J. Vib. Control 15, 583–597 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  4. Dehghan, M., Hamedi, E.A., Khosravian-Arab, H.: A numerical scheme for the solution of a class of fractional variational and optimal control problems using the modified Jacobi polynomials. J. Vib. Control 22, 1547–1559 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  5. Pooseh, S., Almieda, R., Torres, D.F.M.: Fractional order optimal control problems with free terminal time. J. Ind. Manag. Optim. 10, 363–381 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  6. Kafash, B., Delavarkhalafi, A., Karbassi, S.M.: Application of Chebyshev polynomials to derive efficient algorithms for the solution of optimal control problems. Sci. Iran. 19, 795–805 (2012)

    Article  MATH  Google Scholar 

  7. Doha, E.H., Bhrawy, A.H., Baleanu, D., Ezz-Eldien, S.S., Hafez, R.M.: An efficient numerical scheme based on the shifted orthonormal Jacobi polynomials for solving fractional optimal control problems. Adv. Differ. Equ. 12, 1–17 (2015)

    MathSciNet  MATH  Google Scholar 

  8. Ezz-Eldien, S.S., Doha, E.H., Baleanu, D., Bhrawy, A.H.: A numerical approach based on Legendre orthonormal polynomials for numerical solutions of fractional optimal control problems. J. Vib. Control 20, 16–30 (2015)

    MathSciNet  MATH  Google Scholar 

  9. Agrawal, O.P.: Formulation of Euler–Lagrange equations for fractional variational Problems. J. Math. Anal. Appl. 272, 368–379 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  10. Agrawal, O.P.: A general formulation and solution scheme for fractional optimal control problems. Nonlinear Dyn. 38, 323–337 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  11. Agrawal, O.P.: General formulation for the numerical solution of optimal control problems. Int. J. Control 50, 627–638 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  12. Lotfi, A., Dehghan, M., Yousefi, S.A.: A numerical technique for solving fractional optimal control problems. Comput. Math. Appl. 62, 1055–1067 (2011)

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  14. Miller, K.S., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York (1993)

    MATH  Google Scholar 

  15. Oldham, K.B., Spanier, J.: The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order. Academic Press, New York (1974)

    MATH  Google Scholar 

  16. Aizenshtadt, V.S., Krylov, V.I., Metel’skii, A.S.: Tables of Laguerre Polynomials and Functions. Mathematical Tables Series. Pergamon Press, Oxford-New York (1966)

    Google Scholar 

  17. Almeida, R., Pooseh, S., Torres, D.F.M.: Fractional variational problems depending on indefinite integrals. Nonlinear Anal. 75, 1009–1025 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  18. Rudin, W.: Principles of Mathematical Analysis. McGraw-Hill, New York (1976)

    MATH  Google Scholar 

  19. Kafash, B., Delavarkhalafi, A., Karbassi, S.M., Boubaker, K.: A Numerical approach for solving optimal control problems using the Boubaker polynomials expansion scheme. J. Interpolat. Approx. Sci. Comput. 3, 1–18 (2014)

    MathSciNet  Google Scholar 

  20. Bryson Jr., A.E., Ho, Y.C.: Applied Optimal Control. Optimization, Estimation, and Control. Hemisphere Publishing Corp., Washington (1975)

    Google Scholar 

Download references

Acknowledgements

The authors are grateful to the anonymous reviewers for their valuable comments and suggestions, which improved the presentation of the paper.

Author information

Authors and Affiliations

  1. Department of Mathematics, Indian Institute of Technology Kharagpur, Kharagpur, West Bengal, 721302, India

    Neelam Singha & Chandal Nahak

Authors
  1. Neelam Singha
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Chandal Nahak
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Chandal Nahak.

Appendix: Singular Fractional Optimal Control Problem

Appendix: Singular Fractional Optimal Control Problem

We wish to address some control problems where the performance index is either linear or independent of the control function u. For example, consider the FOCP to find an optimal control u(t) that minimizes the performance index

$$\begin{aligned} J[u]=\frac{1}{2}\int _0^1 \,x^2(t)\,\mathrm{d}t \end{aligned}$$
(36)

subject to the system dynamic constraints

$$\begin{aligned} {}^{C}_0D_t^{\alpha }\,x(t)= u(t),\quad \alpha \in ]0,1[\,, \end{aligned}$$
(37)

and the initial condition \(x(0)=1\). One can clearly observe that:

  • The integrand of J is independent of the control function u.

  • Necessary optimality conditions (Theorem 3.1) reduce to \({}_0D_t^{\alpha }x=u\), \({}_tD_1^{\alpha }\lambda =x(t)\) and \(\lambda (t)=0\). The condition \(\lambda (t)=0\) complicates the system to further look for some control function, and hence the control problem is termed as singular.

If \(\alpha =1\), the problem stated above corresponds to the classical singular optimal control problem (see [20] and references therein). An independent investigation by researchers has to be carried out for such problems.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Singha, N., Nahak, C. An Efficient Approximation Technique for Solving a Class of Fractional Optimal Control Problems. J Optim Theory Appl 174, 785–802 (2017). https://doi.org/10.1007/s10957-017-1143-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-017-1143-y

Keywords

Mathematics Subject Classification

Search

Navigation