Opuscula Mathematica
Opuscula Math. 36, no. 1 (2016), 49-68

On a linear-quadratic problem with Caputo derivative

Dariusz Idczak
Stanislaw Walczak
Abstract. In this paper, we study a linear-quadratic optimal control problem with a fractional control system containing a Caputo derivative of unknown function. First, we derive the formulas for the differential and gradient of the cost functional under given constraints. Next, we prove an existence result and derive a maximum principle. Finally, we describe the gradient and projection of the gradient methods for the problem under consideration.
Keywords: fractional Caputo derivative, linear-quadratic problem, existence and uniqueness of a solution, maximum principle, gradient method, projection of the gradient method.
Mathematics Subject Classification: 26A33, 49J15, 49K15, 49M37.
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References ▼
  1. [1] O.P. Agrawal, General formulation for the numerical solution of optimal control problems, Internat. J. Control 50 (1989), 368-379.
  2. [2] O.P. Agrawal, A general formulation and solution scheme for fractional optimal control problems, Nonlinear Dynam. 38 (2004) 1, 323-337.
  3. [3] R. Almeida, S. Pooseh, D.F.M. Torres, Fractional order optimal control problems with free terminal time, J. Ind. Manag. Optim. 10 (2014) 2, 363-381.
  4. [4] L. Bourdin, D. Idczak, Fractional fundamental lemma and fractional integration by parts formula - Applications to critical points of Bolza functionals and to linear boundary value problems, Advances in Differential Equations 20 (2015) 3-4, 213-232.
  5. [5] T.L. Guo, The necessary conditions of fractional optimal control in the sense of Caputo, J. Optim. Theory Appl. 156 (2013), 115-126.
  6. [6] D. Idczak, Optimal control of a coercive Dirichlet problem, SIAM J. Control Optim. 36 (1998) 4, 1250-1267.
  7. [7] D. Idczak, R. Kamocki, On the existence and unqueness and formula for the solution of R-L fractional Cauchy problem in \(R^{n}\), Fract. Calc. Appl. Anal. 14 (2011) 4, 538-553.
  8. [8] D. Idczak, R. Kamocki, Fractional differential repetitive processes with Riemann-Liouville and Caputo derivatives, Multidim. Syst. and Sign. Process. 26 (2015), 193-206.
  9. [9] D. Idczak, S. Walczak, Compactness of fractional imbeddings, Proceedings of the 17th International Conference on Methods & Models in Automation and Robotics (2012), 585-588.
  10. [10] D. Idczak, S. Walczak, Optimization of a fractional Mayer problem - existence of solutions, maximum principle, gradient methods, Opuscula Math. 34 (2014) 4, 763-775.
  11. [11] H. Górecki, S. Fuksa, A. Korytowski, W. Mitkowski, Optimal control in linear systems with quadratic performance index, PWN, Warszawa, 1983 [in Polish].
  12. [12] R. Kamocki, Some ordinary and distributed parameters fractional control systems and their optimization, Doctoral Thesis, University of Lodz, Lodz, 2012.
  13. [13] R. Kamocki, Pontryagin maximum principle for fractional ordinary optimal control problems, Math. Methods Appl. Sci. 37 (2014) 11, 1668-1686.
  14. [14] R. Kamocki, On the existence of optimal solutions to fractional optimal control problems, Appl. Math. Comput. 235 (2014), 94-104.
  15. [15] R. Kamocki, M. Majewski, Fractional linear control systems with Caputo derivative and their optimization, Optim. Control Appl. Meth. (2014), DOI: 10.1002/oca.2150. http://dx.doi.org/10.1002/oca.2150
  16. [16] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006.
  17. [17] Q. Lin, R. Loxton, K.L. Teo, Y.H. Wu, Optimal control computation for nonlinear systems with state-dependent stopping criteria, Automatica Journal IFAC 48 (2012), 2116-2129.
  18. [18] A.B. Malinowska, D.F.M. Torres, Introduction to the Fractional Calculus of Variations, Imperial College Press, London, 2012.
  19. [19] S. Pooseh, R. Almeida, D.F.M. Torres, Free time fractional optimal control problems, European Control Conference (ECC), Zürich, Switzerland, 2013, 3985-3990.
  20. [20] S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives - Theory and Applications, Gordon and Breach, Amsterdam, 1993.
  21. [21] C. Tricaud, Y. Chen, An approximate method for numerically solving fractional order optimal control problems of general form, Comput. Math. Appl. 59 (2010) 5, 1644-1655.
  22. [22] F.P. Vasiliev, Methods of Solving of Extreme Problems, Science, Moscov, 1981 [in Russian].
  23. [23] F.P. Vasiliev, Numerical Methods of Solving of Extreme Problems, Science, Moscov, 1988 [in Russian].
About the authors ▼
Dariusz Idczak
University of Lodz, Faculty of Mathematics and Computer Science, Banacha 22, 90-238 Lodz, Poland
Stanislaw Walczak
University of Lodz, Faculty of Mathematics and Computer Science, Banacha 22, 90-238 Lodz, Poland
About the article ▼
Communicated by Marek Galewski.
Received: 2014-11-25.
Revised: 2015-03-09.
Accepted: 2015-04-10.
Published online: 2015-09-19.
Cite this article as ▼
Dariusz Idczak, Stanislaw Walczak, On a linear-quadratic problem with Caputo derivative, Opuscula Math. 36, no. 1 (2016), 49-68, http://dx.doi.org/10.7494/OpMath.2016.36.1.49.
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