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Optimal control problems with a fixed terminal time in linear fractional-order systems

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Języki publikacji
EN
Abstrakty
EN
The paper deals with an optimal control problem in a dynamical system described by a linear differential equation with the Caputo fractional derivative. The goal of control is to minimize a Bolza-type cost functional, which consists of two terms: the first one evaluates the state of the system at a fixed terminal time, and the second one is an integral evaluation of the control on the whole time interval. In order to solve this problem, we propose to reduce it to some auxiliary optimal control problem in a dynamical system described by a first-order ordinary differential equation. The reduction is based on the representation formula for solutions to linear fractional differential equations and is performed by some linear transformation, which is called the informational image of a position of the original system and can be treated as a special prediction of a motion of this system at the terminal time. A connection between the original and auxiliary problems is established for both open-loop and feedback (closed-loop) controls. The results obtained in the paper are illustrated by examples.
Rocznik
Strony
721--744
Opis fizyczny
Bibliogr. 27 poz., wzory
Twórcy
  • Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Russia and Ural Federal University, Russia
Bibliografia
  • [1] R. A. Bandaliyev, I. G. Mamedov, M. J. Mardanov, and T. K. Melikov: Fractional optimal control problem for ordinary differential equation in weighted Lebesgue spaces, Optimization Letters, 14(6) (2020), 1519–1532.
  • [2] M. Bergounioux and L. Bourdin: Pontryagin maximum principle for general Caputo fractional optimal control problems with Bolza cost and terminal constraints, ESAIM: Control, Optimisation and Calculus of Variations, 26 (2020), Article Number 35, 38 pages.
  • [3] L. Bourdin: Cauchy–Lipschitz theory for fractional multi-order dynamics: State-transition matrices, Duhamel formulas and duality theorems, Differential and Integral Equations, 31(7/8) (2018), 559–594.
  • [4] K. Diethelm: The Analysis of Fractional Differential Equations: An Application-Oriented Exposition using Differential Operators of Caputo type, Springer, Berlin, 2010.
  • [5] M. I. Gomoyunov: Fractional derivatives of convex Lyapunov functions and control problems in fractional order systems, Fractional Calculus and Applied Analysis, 21(5) (2018), 1238–1261.
  • [6] M. I. Gomoyunov: Solution to a zero-sum differential game with fractional dynamics via approximations, Dynamic Games and Applications, 10(2) (2020), 417–443.
  • [7] M. I. Gomoyunov: Extremal shift to accompanying points in a positional differential game for a fractional-order system, Proceedings of the Steklov Institute of Mathematics, 308 (Suppl. 1) (2020), S83–S105.
  • [8] M.I. Gomoyunov: On representation formulas for solutions of linear differential equations with Caputo fractional derivatives, Fractional Calculus and Applied Analysis, 23(4) (2020), 1141–1160.
  • [9] M. I. Gomoyunov: On a solution of an optimal control problem for a linear fractional-order system. In: A. Bartoszewicz, J. Kabziński, J. Kacprzyk (eds.) Advanced, Contemporary Control. Advances in Intelligent Systems and Computing, vol. 1196, 837–846, Springer, Cham, 2020.
  • [10] M. I. Gomoyunov and N. Yu. Lukoyanov: Guarantee optimization in functional-differential systems with a control aftereffect, Journal of Applied Mathematics and Mechanics, 76(4) (2012), 369–377.
  • [11] M. I. Gomoyunov and N. Yu. Lukoyanov: On the numerical solution of differential games for neutral-type linear systems, Proceedings of the Steklov Institute of Mathematics, 301 (Suppl. 1) (2018), S44–S56.
  • [12] M. I. Gomoyunov and A. R. Plaksin: On a problem of guarantee optimization in time-delay systems, IFAC-PapersOnLine, 48(25) (2015) 172–177.
  • [13] D. Idczak and R. Kamocki: On the existence and uniqueness and formula for the solution of R–L fractional cauchy problem in Rn, Fractional Calculus and Applied Analysis, 14(4) (2011), 538–553.
  • [14] D. Idczak and S. Walczak: On a linear-quadratic problem with Caputo derivative, Opuscula Mathematica, 36(1) (2016), 49–68.
  • [15] T. Kaczorek: Minimum energy control of fractional positive electrical circuits with bounded inputs, Circuits, Systems, and Signal Processing, 35(6) (2016), 1815–1829.
  • [16] R. Kamocki and M. Majewski: Fractional linear control systems with Caputo derivative and their optimization, Optimal Control Applications and Methods, 36(6) (2015), 953–967.
  • [17] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo: Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006.
  • [18] A. N. Krasovskii and N. N. Krasovskii: Control under Lack of Information, Birkhäuser, Boston, 1995.
  • [19] N. N. Krasovskii: Stability of Motion: Applications of Lyapunov’s Second Method to Differential Systems and Equations with Delay, Stanford University Press, Stanford, California, 1963.
  • [20] N. N. Krasovskii and A. I. Subbotin: Game-Theoretical Control Problems, Springer, New York, 1988.
  • [21] V. A. Kubyshkin and S. S. Postnov: Optimal control problem for a linear stationary fractional order system in the form of a problem of moments: Problem setting and a study, Automation and Remote Control, 75(5) (2014), 805–817.
  • [22] E. B. Lee and L. Markus: Foundations of Optimal Control Theory, John Wiley and Sons, New York, 1967.
  • [23] N. Yu. Lukoyanov and M. I. Gomoyunov: Differential games on minmax of the positional quality index, Dynamic Games and Applications, 9(3) (2019), 780–799.
  • [24] N. Yu. Lukoyanov and T. N. Reshetova: Problems of conflict control of high dimensionality functional systems, Journal of Applied Mathematics and Mechanics, 62(4) (1998), 545–554.
  • [25] I. Matychyn and V. Onyshchenko: Optimal control of linear systems with fractional derivatives, Fractional Calculus and Applied Analysis, 21(1) (2018), 134–150.
  • [26] S. G. Samko, A. A. Kilbas, and O. I. Marichev: Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science Publishers, Yverdon, 1993.
  • [27] A. A. Shaburov: Asymptotic expansion of a solution for one singularly perturbed optimal control problem in Rn with a convex integral quality index, Ural Mathematical Journal, 3(1) (2017), 68–75.
Uwagi
This work was supported by the Russian Science Foundation, project no. 19-11-00105.
Typ dokumentu
Bibliografia
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