Numerical method of the exact control for the elastic string problem with moving boundary
Resumo
The objective of this article is to investigate, analytically and numerically, the exact control problem associated with a mathematical model that describes the vertical vibrations of elastic strings, with moving contours. To determine the approximate numerical solution, following the description of the HUM (Hilbert Uniqueness Method), the finite element method associated with the Newmark method was used for the adjunct problem and the backward problem. To calculate the optimal control, the conjugated gradient method is used. The properties of a numerical problem must be consistent with the theoretical mathematical one. However, in order to obtain the numerical control, the grid is of fundamental importance, so we used the Bi-Grid technique. In this way, we developed an algorithm to solve this problem. Numerical simulations were performed for different initial data and moving contours, with and without the Bi-Grid algorithm and the numerical results confirmed the consistency between the theoretical and numerical results.Downloads
Referências
Araruna, F.D.; Medeiros, L. A.; Antunes, G.: Exact controllability for the semilinear string equation in non cylindrical domains. Control and Cybernetics, [S.l.], v. 33, n. 2, p. 237-257, 2004.
Araruna, F. D. ; Chaves-Silva, F. W. ; Rojas-Medar, M. A. . Exact Controllability of Galerkin’s Approximations of Micropolar Fluids. Proceedings of the American Mathematical Society, v. 138, p. 1361-1370, 2010.
Araujo de Souza, Diego Resultados te´oricos y numericos de control para EDPs lineales y no lineales. Universidad de Sevilla. Phs Thesis (2015) http://hdl.handle.net/11441/25453
Capistrano-Filho, R, Pazoto, A., Rosier, L. Internal controllability of the Korteweg-de Vries equation on a bounded domain. ESAIM: COCV v.21, N.4, pg 1076-1107, (2015)
Carre˜no, A, N. ; Santos, M.C. . Stackelberg-Nash exact controllability for the Kuramoto-Sivashinsky equation. Journal of Differential Equations, v. 266, p. 6068-6108, 2019.
Chaves-Silva, F. W.; Araujo de Souza, Diego:. On the controllability of some equations of Sobolev-Galpern type. Journal of Differential Equations, Vol.268, No4, pg.1633-1657,2020.
Chaves-Silva, F. W.: A hyperbolic system and the cost of the null controllability for the Stokes system. Computational & Applied Mathematics, v. 34, p. 1057-1074, 2015.
Chaves-Silva, Felipe W.; Rosier, L.; Zuazua, E. : Null controllability of a system of viscoelasticity with a moving control. Journal de Math´ematiques Pures et Appliqu´ees, v. 101, Issue 2, (2014), Pages 198-222
De Carvalho, P.P. :Some numerical results for control of 3D heat equations using Nash equilibrium. Comp. Appl. Math. 40, 92 (2021).
De Carvalho, P.P.; Lımaco, J. ; Thamsten, Y. ; Menezes, D. :Local null controllability of a class of non-Newtonian incompressible viscous fluids. Evolution Equations and Control Theory, v. 1, p. 31, 2021.
Font, R.; Periago, F. Numerical simulation of the boundary exact control for the system of linear elasticity. Applied Mathematics Letters 23:1021-1026 (2010) DOI:10.1016/j.aml.2010.04.030
Glowinski, R.; Kinton,W.; Wheeler, M. F. A mixed finite element formulation for the boundary controllability of the wave equation. :International Journal for Numerical Methods in Engineering, Chichester, v. 27, n. 3, p. 623-635, 1989.
Glowinski, R.; Li, C.-H.; Lions, J.-L. A numerical approach to the exact boundary controllability of the wave equation (I). Dirichlet controls: Description of the numerical methods. :Japan Journal of Applied Mathematics, v. 7, n. 1, p. 1-76, 1990.
Glowinski, R. Ensuring well posedness by analogy; stokes problem and boundary control for the wave equation. :Journal of Computational Physics, Orlando, v. 103, n. 2, p. 189-221, 1992.
Kogut, P.I; Kupenko, O.P.; Leugering, G.On boundary exact controllability of one-dimensional waveequations with weak and strong interior degeneration. Math Meth Appl Sci. 2022;45:770–792
Borsch,V.L , Kogut, P.I.PI. The exact bounded solutiontoan initial boundary value problem for 1-D hyperbolic equation with interior degeneracy.I. Separation of variables. J Optim Diff Equ Appl (JODEA). 2020;28(2):2-20.25.
Lımaco, J.; Nu˜nez-Chavez, M. R.; Huaman, D. N. Exact controllability for nonlocal and nonlinear hyperbolic PDEs, Nonlinear Analysis 214 (112569), 2022.
Lions, J. L. :Controlabilite exacte de systemes distribuees, C. R. Acad. Sci., Paris, 302 (1986), 471-475.
Lions, J. L. :Controlabilite exacte, pertubation et estabilization de systemes distribuees: tome 1: controlabilite exacte. Paris: Masson, 1988.
Medeiros, L. A. da J.; Limaco, J.; Menezes, S. B. Vibrations of elastic strings: mathematical aspects, part 1 e part 2. :Journal of Computational Analysis and Applications. v. 4, n. 2, p. 97-127, 2002, v. 4, n. 3, p. 212-263, 2002.
Miranda, M. M. HUM and the wave equation with variable coefficients. :Asymptotic Analysis, Amsterdam, v. 11, n. 4, p. 317-341, 1995.
Moraes, C. E. O. de, Algorithm for the exact control of the elastic string problem with moving boundary. Last access: 01/29/2020. https://goo.gl/QCvUrr
Negreanu, M. : M´etodos num´ericos para el an´alisis de la propagaci´on, observaci´on y control de ondas. 2003. (PH.D Thesis) - Universidad Complutense de Madrid, Madrid, 2003.
Rincon, M. A.; Garay, M. Z.; Miranda, M. M. Numerical approximation of the exact control for the string equation. :International Journal of Pure and Applied Mathematics, [S.l.], v. 8, n. 3, p. 349-368, 2003.
Salem, A.: Numerical approximation of exact controllability for Korteweg-de Vries equation. AIP Conference Proceedings 1107, 231 (2009);
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Funding data
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Coordenação de Aperfeiçoamento de Pessoal de Nível Superior
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Conselho Nacional de Desenvolvimento Científico e Tecnológico
Grant numbers 301571/2019-8 -
Fundação Carlos Chagas Filho de Amparo à Pesquisa do Estado do Rio de Janeiro
Grant numbers CNE 2022/2025
