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Research Article

Error analysis of Haar wavelet-based Galerkin numerical method with application to various nonlinear optimal control problems

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Received 01 Sep 2023, Accepted 03 Apr 2024, Published online: 18 Apr 2024

References

  • D. Aldila, M. Shahzad, S.H.A. Khoshnaw, M. Ali, F. Sultan, A. Islamilova, Y. Rais Anwar and B.M. Samiadji, Optimal control problem arising from covid-19 transmission model with rapid-test, Res. Phys. 37 (2022), p. 105501.
  • S. Ali Yousefi, M. Dehghan and A. Lotfi, Finding the optimal control of linear systems via he's variational iteration method, Int. J. Comput. Math.87(5) (2010), pp. 1042–1050.
  • S. Aniţa, V. Arnăutu, V. Capasso and V. Capasso, An Introduction to Optimal Control Problems in Life Sciences and Economics: From Mathematical Models to Numerical Simulation with MATLAB®, Vol. 2, Springer, New York, 2011.
  • E. Babolian and A. Shahsavaran, Numerical solution of nonlinear fredholm integral equations of the second kind using haar wavelets, J. Comput. Appl. Math. 225(1) (2009), pp. 87–95.
  • D. Benson, A Gauss pseudospectral transcription for optimal control, PhD thesis, Massachusetts Institute of Technology, 2005.
  • A. Bokhari, A. Amir, S.M. Bahri and F.B.M. Belgacem, A generalized bernoulli wavelet operational matrix of derivative applications to optimal control problems, Nonlinear Stud. 24(4) (2017), pp. 775–90.
  • A.H. Borzabadi and S. Asadi, A wavelet collocation method for optimal control of non-linear time-delay systems via haar wavelets, IMA J. Math. Control Inf. 32(1) (2015), pp. 41–54.
  • F. Bulut, Ö. Oruç and A. Esen, Higher order haar wavelet method integrated with strang splitting for solving regularized long wave equation, Math. Comput. Simul. 197 (2022), pp. 277–290.
  • J. Carl Panetta and K. Renee Fister, Optimal control applied to competing chemotherapeutic cell-kill strategies, SIAM. J. Appl. Math. 63(6) (2003), pp. 1954–1971.
  • S.B. Chae, Lebesgue Integration, Springer Science & Business Media, New York, 2012.
  • C.F. Chen and C.H. Hsiao, Haar wavelet method for solving lumped and distributed-parameter systems, IEE Proc.-Control Theory Appl. 144(1) (1997), pp. 87–94. 0880.
  • V. Cichella, I. Kaminer, C. Walton, N. Hovakimyan and A.M. Pascoal, Consistent approximation of optimal control problems using bernstein polynomials, in 2019 IEEE 58th Conference on Decision and Control (CDC), IEEE, 2019, pp. 4292–4297.
  • R. Dai and J.E. Cochran, Wavelet collocation method for optimal control problems, J. Optim. Theory. Appl. 143 (2009), pp. 265–278.
  • C.L. Darby, W.W. Hager and A.V. Rao, An hp-adaptive pseudospectral method for solving optimal control problems, Optim. Control Appl. Methods 32(4) (2011), pp. 476–502.
  • R. Dehghan, A numerical method based on cardan polynomials for solving optimal control problems, Afr. Mat. 29(5) (2018), pp. 677–687.
  • A. Fakharian, M.T.H. Beheshti and A. Davari, Solving the hamilton–jacobi–bellman equation using adomian decomposition method, Int. J. Comput. Math.87(12) (2010), pp. 2769–2785.
  • A.A.A. Haya, Solving optimal control problem via chebyshev wavelet, PhD thesis, Masters thesis, Islamic University of Gaza, 2011.
  • S. Hosseinpour and A. Nazemi, Solving fractional optimal control problems with fixed or free final states by haar wavelet collocation method, IMA J. Math. Control Inf. 33(2) (2016), pp. 543–561.
  • S. Hosseinpour, A. Nazemi and E. Tohidi, Müntz–legendre spectral collocation method for solving delay fractional optimal control problems, J. Comput. Appl. Math. 351 (2019), pp. 344–363.
  • B. Kafash, A. Delavarkhalafi and S.M. Karbassi, Application of variational iteration method for hamilton–jacobi–bellman equations, Appl. Math. Model. 37(6) (2013), pp. 3917–3928.
  • H.R. Karimi, B. Lohmann, P.J. Maralani and B. Moshiri, A computational method for solving optimal control and parameter estimation of linear systems using haar wavelets, Int. J. Comput. Math. 81(9) (2004), pp. 1121–1132.
  • A. Kheirabadi, A.M. Vaziri and S. Effati, Solving optimal control problem using hermite wavelet, Numer. Algebra Control Optim. 9(1) (2019), pp. 101.
  • S. Lenhart and J.T. Workman, Optimal Control Applied to Biological Models, CRC Press, New York, 2007.
  • U. Lepik, Solution of optimal control problems via haar wavelets, Int. J. Pure. Appl. Math 55 (2009), pp. 81–94.
  • Ü. Lepik and H. Hein, Haar wavelets, in Haar Wavelets, Springer, New York, 2014, pp. 7–20.
  • M. Maleki and A. Hadi-Vencheh, Combination of non-classical pseudospectral and direct methods for the solution of brachistochrone problem, Int. J. Comput. Math. 87(8) (2010), pp. 1847–1856.
  • K. Maleknejad and A. Ebrahimzadeh, The use of rationalized haar wavelet collocation method for solving optimal control of volterra integral equations, J. Vib. Control 21(10) (2015), pp. 1958–1967.
  • M. Martcheva, An Introduction to Mathematical Epidemiology, Vol. 61, Springer, New York, 2015.
  • H.R. Marzban and M. Razzaghi, Rationalized haar approach for nonlinear constrained optimal control problems, Appl. Math. Model. 34(1) (2010), pp. 174–183.
  • M. Matinfar and M. Dosti, Solving linear optimal control problems using cubic b-spline quasi-interpolation, MATEMATIKA: Malaysian J. Indust. Appl. Math. 34(2) (2018), pp. 313–324.
  • M. Mesterton-Gibbons, A Primer on the Calculus of Variations and Optimal Control Theory, Vol. 50, American Mathematical Soc., Providence, RI, 2009.
  • A. Nazemi and N. Mahmoudy, Solving infinite-horizon optimal control problems using the haar wavelet collocation method, The ANZIAM J. 56(2) (2014), pp. 179–191.
  • H.S. Nik, S. Effati and M. Shirazian, An approximate-analytical solution for the hamilton–jacobi–bellman equation via homotopy perturbation method, Appl. Math. Model. 36(11) (2012), pp. 5614–5623.
  • E.M. Njeru, Use of haar wavelets to solve optimal control problem, PhD thesis, University of Nairobi, 2020.
  • Ö. Oruç, A non-uniform haar wavelet method for numerically solving two-dimensional convection-dominated equations and two-dimensional near singular elliptic equations, Comput. Math. Appl. 77(7) (2019), pp. 1799–1820.
  • K. Rabiei and Y. Ordokhani, A new operational matrix based on boubaker wavelet for solving optimal control problems of arbitrary order, Trans. Inst. Measur. Control 42(10) (2020), pp. 1858–1870.
  • Z. Rafiei, B. Kafash and S.M. Karbassi, A new approach based on using chebyshev wavelets for solving various optimal control problems, Comput. Appl. Math. 37 (2018), pp. 144–157.
  • M. Ramezani, Numerical solution of optimal control problems by using a new second kind chebyshev wavelet, Comput. Methods Differ. Equ. 4(2) (2016), pp. 162–169.
  • N. Razmjooy and M. Ramezani, Analytical solution for optimal control by the second kind chebyshev polynomials expansion, Iran. J. Sci. Technol. Trans. A: Sci. 41(4) (2017), pp. 1017–1026.
  • S.P. Sethi and S.P. Sethi, What is Optimal Control Theory? Springer, Dallas, 2019.
  • K. Sydsæter, P. Hammond, A. Seierstad and A. Strom, Further Mathematics for Economic Analysis, Pearson Education, London, 2008.
  • E. Tohidi and H.S. Nik, A bessel collocation method for solving fractional optimal control problems, Appl. Math. Model. 39(2) (2015), pp. 455–465.
  • E. Tohidi and O.R.N. Samadi, Optimal control of nonlinear volterra integral equations via legendre polynomials, IMA J. Math. Control Inf. 30(1) (2013), pp. 67–83.
  • M.L. Víchez, F. Velasco and I. Herrero, An optimal control problem with hopf bifurcations: An application to the striped venus fishery in the gulf of cádiz, Fish. Res. 67(3) (2004), pp. 295–306.
  • X. Xu, L. Xiong and F. Zhou, Solving fractional optimal control problems with inequality constraints by a new kind of chebyshev wavelets method, J. Comput. Sci. 54 (2021), p. 101412.
  • W. Zhang and H. Ma, The chebyshev–legendre collocation method for a class of optimal control problems, Int. J. Comput. Math. 85(2) (2008), pp. 225–240.

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