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Exact and approximate distributed controllability of processes described by KdV and Boussinesq equations: The Green’s function approach

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Języki publikacji
EN
Abstrakty
EN
In this paper, we study the constrained exact and approximate controllability of traveling wave solutions of Korteweg-de Vries (third order) and Boussinesq (fourth order) semi-linear equations using the Green’s function approach. Control is carried out by a moving external source. Representing the general solution of those equations in terms of the Frasca’s short time expansion, system of constraints on the distributed control is derived for both types of controllability. Due to the possibility of explicit solution provided by the heuristic method, the controllability analysis becomes straightforward. Numerical analysis confirms theoretical derivations.
Rocznik
Strony
177--193
Opis fizyczny
Bibliogr. 28 poz., rys., wykr., wzory
Twórcy
autor
  • Institute of Control Engineering, Silesian University of Technology, Gliwice, Poland
  • Department on Dynamics of Deformable Systems and Coupled Fields, Institute of Mechanics, National Academy of Sciences of Armenia
  • Department on Dynamics of Deformable Systems and Coupled Fields, Institute of Mechanics, National Academy of Sciences of Armenia
  • Institute of Natural Sciences, Shanghai Jiao Tong University, Shanghai, China
Bibliografia
  • [1] Klamka J.: Controllability of Dynamical Systems, Kluwer Academic, Dordrecht, 1991.
  • [2] Avdonin S.A. and Ivanov S.A.: Families of Exponentials. The Method of Moments in Controllability Problems for Distributed Parameter Systems, Cambridge University Press, New York, 1995.
  • [3] Fursikov A. and Imanuvilov O.Yu.: Controllability of Evolution Equations, Lecture Notes Series, vol. 34. Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul, 1996.
  • [4] Zuazua E.: Controllability and Observability of Partial Differential Equations: Some Results and Open Problems, Handbook of Differential Equations: Evolutionary Differential Equations, vol. 3, Elsevier/North-Holland, Amsterdam, 2006.
  • [5] Glowinski R., Lions J.-L., and He J., Exact and Approximate Controllability for Distributed Parameter Systems: A Numerical Approach, Cambridge University Press, New York, 2008.
  • [6] Avetisyan A. S. and Khurshudyan As.Zh.: Controllability of Dynamic Systems: The Green’s Function Approach, Cambridge Scholars Publishing, Cambridge, 2018.
  • [7] Khurshudyan As. Zh.: Heuristic determination of resolving controls for exact and approximate controllability of nonlinear dynamic systems, Mathematical Problems in Engineering, 2018, DOI: 10.1155/2018/9496371.
  • [8] Khurshudyan As. Zh.: Resolving controls for the exact and approximate controllability of the viscous Burgers’ equation: The Green’s function approach, International Journal of Modern Physics C, 29(6) (2018), 1850045, 14 pages.
  • [9] Khurshudyan As. Zh.: Exact and approximate controllability conditions for micro-swimmers deflection governed by electric field on a plane: The Green’s function approach, Archives of Control Sciences, 28(3) (2018), 335–347.
  • [10] Avetisyan A. S. and Khurshudyan As. Zh.: Exact and approximate controllability of nonlinear dynamic systems in infinite time: The Green’s function approach, ZAMM, 98(11) (2018), 1992–2009.
  • [11] Khurshudyan As. Zh.: Distributed controllability of heat equation in unbounded domains: The Green’s function approach, Archives of Control Sciences, 29(1) (2019), 57–71.
  • [12] Khurshudyan As. Zh. and Arakelyan Sh. Kh.: Resolving controls for approximate controllability of sandwich beams with uncertainty: The Green’s function approach, Mechanics of Composite Materials, 55(1) (2019), 85–94.
  • [13] Klamka J.: Constrained controllability of nonlinear systems, Journal of Mathematical Analysis and Applications, 201(2) (1996), 365–374.
  • [14] Klamka J.: Constrained approximate controllability. IEEE Transactions on Automatic Control, 2000, vol. 45, issue 9, pp. 1745–1749.
  • [15] Klamka J.: Constrained controllability of semilinear systems, Nonlinear Analysis, 47(6) (2001), 2939–2949.
  • [16] Klamka J.: Controllability and Minimum Energy Control, Springer, Cham, 2019.
  • [17] Russell D. L. and Zhang B.-Y.: Exact controllability and stabilizability of the Korteweg-de Vries equation. Transactions of AMS, 1996, vol. 348, pp. 3643–3672.
  • [18] Rosier L. and Zhang B.-Y.: Control and stabilization of the Korteweg–de Vries equation: Recent progresses, Journal of System Science and Complex-ity, 22(4) (2009), 647–682.
  • [19] Zhang B.-Y.: Exact controllability of the generalized Boussinesq equation, In Control and estimation of distributed parameter systems, International Series onNumerical Mathematics, vol. 126, pp. 297–310. Birkhauser, Basel, 1998.
  • [20] Frasca M. and Khurshudyan As. Zh.: Green’s function for higher order nonlinear equations: Case studies for KdV and Boussinesq equations, International Journal of Modern Physics C, 2018, vol. 29, 1850104, 13 pages.
  • [21] Khurshudyan As. Zh.: An identity for the Heaviside function and its application in representation of nonlinear Green’s function, Computational & Applied Mathematics, 39 (2020), DOI: 10.1007/s40314-019-1011-5.
  • [22] Polyanin A. D. and Zaitsev V. F.: Handbook of Ordinary Differential Equations: Exact Solutions, Methods, and Problems, 3rd edition, Chapman & Hall/CRC Press, Boca Raton, 2017.
  • [23] Polyanin A. D. and Zaitsev V.: Handbook of Nonlinear Partial Differential Equations, Chapman & Hall/CRC Press, Boca Raton, 2012.
  • [24] Kantorovich K. L. and Krylov V. I.: Approximate Methods of Higher Analysis, Interscience Publishers, New York, 1958.
  • [25] Teodorescu P. P., Kecs W. W., and Toma A.: Distribution Theory: With Applications in Engineering and Physics, WILEY-VCH Verlag, Weinheim, 2003.
  • [26] Butkovskii A.G.: Some problems of control of the distributed-parameter systems, Automation and Remote Control, 72(6) (2011), 1237–1241.
  • [27] Arakelyan Sh. Kh. and Khurshudyan As. Zh.: The Bubnov-Galerkin procedure for solving mobile control problems for systems with distributed parameters, Mechanics. Proc. Nat. Acad. Sci. Armenia, 68(3) (2015), 54–75.
  • [28] Khapalov A. Y.: Mobile Point Sensors and Actuators in the Controllability Theory of Partial Differential Equations, Springer, Cham, Switzerland, 2017.
Uwagi
EN
1. The work of the first author is supported by National Science Centre in Poland under grant: “Modelling, optimization and control for structural reduction of device noise”, no. UMO-2017/25/B/ST7/02236. The third author thankfully acknowledges the support of the State Administration of Foreign Experts Affairs of China.
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2. Opracowanie rekordu ze środków MNiSW, umowa Nr 461252 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2020).
Typ dokumentu
Bibliografia
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