Free Access
RAIRO-Oper. Res.
Volume 54, Number 5, September-October 2020
Page(s) 1419 - 1435
Published online 23 July 2020
  • A. Akkouche, A. Maidi and M. Aidene, Optimal control of partial differential equations based on the variational iteration method. Comput. Math. Appl. 68 (2014) 622–631. [Google Scholar]
  • R.P. Agarwal and Y.M. Chow, Iterative methods for a fourth order boundary value problem. J. Comput. Appl. Math. 10 (1984) 203–217. [Google Scholar]
  • R.P. Agarwal and J. Vosmanský, Necessary and sufficient conditions for the convergence of approximate Picard’s iterates for nonlinear boundary value problems. Arch. Math. 21 (1985) 171–176. [Google Scholar]
  • U.M. Ascher, R.M. Mattheij and R.D. Russell, Numerical Solution of Boundary-value Problems for Ordinary Differential Equations. Prentice Hall (1988). [Google Scholar]
  • X. Bai and J.L. Junkins, Modified Chebyshev-Picard iteration methods for solution of boundary value problems. J. Astron. Sci. 58 (2011) 615–642. [Google Scholar]
  • J.T. Betts, Practical Methods for Optimal Control and Estimation Using Nonlinear Programming, 2nd edition. Society for Industrial and Applied Mathematics, Philadelphia (2009). [Google Scholar]
  • R. Bulirsch, A. Miele, J. Stoer and K.H. Well, Optimal Control: Calculus Of Variations, Optimal Control Theory And Numerical Methods. Springer, Boston (1993). [Google Scholar]
  • A.C. Chiang, Elements of Dynamic Optimization. McGraw-Hill, New York (1992). [Google Scholar]
  • D. Liberzon, Calculus of Variations and Optimal Control Theory: A Concise Introduction. Princeton University Press, New Jersey (2012). [Google Scholar]
  • S. Effati and S. Nik, Solving a class of linear and non-linear optimal control problems by homotopy perturbation method. IMA J. Math. Control Inf. 28 (2011) 539–553. [Google Scholar]
  • H.A. El-Arabawy and I.K. Youssef, A symbolic algorithm for solving linear two-point boundary value problems by modified Picard technique. Math. Comput. Model. 49 (2009) 344–351. [Google Scholar]
  • S. Gonzalez and A. Miele, Sequential gradien-restoration algorithm for optimal control problems with general boundary conditions. J. Optim. Theory Appl. 26 (1978) 395–425. [Google Scholar]
  • I.H. Abdel-Halim Hassan, Differential transformation technique for solving higher-order initial value problems. Appl. Math. Comput. 154 (2004) 299–311. [Google Scholar]
  • J.H. He, Homotopy perturbation technique. Comput. Method Appl. Mech. Eng. 178 (1999) 257–262. [Google Scholar]
  • J.H. He, Variational iteration method a kind of non-linear analytical technique: Some examples. Int. J. Non-Linear Mech. 34 (1999) 699–708. [Google Scholar]
  • J.H. He, Variational iteration method for autonomous ordinary differential systems. Appl. Math. Comput. 114 (2000) 115–123. [Google Scholar]
  • J.H. He, Homotopy perturbation method: A new nonlinear analytical technique. Appl. Math. Comput. 135 (2003) 73–79. [Google Scholar]
  • J.H. He, New interpretation of homotopy perturbation method. Int. J. Mod. Phys. B 20 (2006) 2561–2568. [Google Scholar]
  • H. Jaddu, Direct solution of nonlinear optimal control problems using quasilinearization and Chebyshev polynomials. J. Frankl. Inst. 339 (2002) 479–498. [Google Scholar]
  • M.J. Jang, C.L. Chen and Y.C. Liy, On solving the initial-value problems using the differential transformation method. Appl. Math. Comput. 115 (2000) 145–160. [Google Scholar]
  • H.B. Keller, Numerical Solution of Two Point Boundary Value Problems. Society for Industrial and Applied Mathematics, Philadelphia (1976). [Google Scholar]
  • W.G. Kelley, A.C. Peterson, The Theory of Differential Equations: Classical and Qualitative, 2nd edition. Springer, New York (2010). [Google Scholar]
  • M.M. Khader, On the numerical solutions to nonlinear biochemical reaction model using Picard-Padé technique. World J. Model. Simul. 9 (2013) 38–46. [Google Scholar]
  • D.E. Kirk, Optimal Control Theory. An Introduction. Prentice-Hall (1970). [Google Scholar]
  • M. Lal and D. Moffatt, Picard’s successive approximation for non-linear two-point-boundary-value problems. J. Comput. Appl. Math. 8 (1982) 233–236. [Google Scholar]
  • E.B. Lee and L. Markus, Foundations of Optimal Control Theory. John Wiley and Sons, New York (1967). [Google Scholar]
  • S. Liao, On the homotopy analysis method for nonlinear problems. Appl. Math. Comput. 147 (2004) 499–513. [Google Scholar]
  • A. Maidi and J.P. Corriou, Open-loop optimal controller design using variational iteration method. Appl. Math. Comput. 219 (2013) 8632–8645. [Google Scholar]
  • H.R. Marzban and S.M. Hoseini, A composite Chebyshev finite difference method for nonlinear optimal control problems. Commun. Nonlinear Sci. Numer. Simul. 18 (2013) 1347–1361. [Google Scholar]
  • H.S. Nik, S. Effati and A. Yildirim, Solution of linear optimal control systems by differential transform method. Neural Comput. Appl. 23 (2013) 1311–1317. [Google Scholar]
  • L.S. Pontryagin, V.G. Boltyanskii, R.V. Gamkrelidze and E.F. Mishchenko, The Mathematical Theory of Optimal Processes. Pergamon Press, New York (1964). [Google Scholar]
  • J.I. Ramos, On the variational iteration method and other iterative techniques for nonlinear differential equations. Appl. Math. Comput. 199 (2008) 39–69. [Google Scholar]
  • J.I. Ramos, On the Picard-Lindelof method for nonlinear second-order differential equations. Appl. Math. Comput. 203 (2008) 238–242. [Google Scholar]
  • J.I. Ramos, Iterative and non-iterative methods for non-linear Volterra integro-differential equations. Appl. Math. Comput. 214 (2009) 287–296. [Google Scholar]
  • J.I. Ramos, Picard’s iterative method for nonlinear advection-reaction-diffusion equations. Appl. Math. Comput. 215 (2009) 1526–1536. [Google Scholar]
  • W.A. Robin, Solving differential equations using modified Picard iteration. Int. J. Math. Edu. Sci. Technol. 41 (2010) 649–665. [Google Scholar]
  • R.W.H. Sargent, Optimal control. J. Comput. Appl. Math. 124 (2000) 361–371. [Google Scholar]
  • M. Shirazian and S. Effati, Solving a class of nonlinear optimal control problems via He’s variational iteration method. Int. J. Control Autom. Syst. 10 (2012) 249–256. [Google Scholar]
  • J. Stoer and R. Bulirsch, Introduction to Numerical Analysis. Springer-Verlag, Berlin (1980). [Google Scholar]
  • S. Titouche, P. Spiteri, F. Messine and M. Aidene, Optimal control of a large thermic process. J. Process Control 25 (2015) 50–58. [Google Scholar]
  • E. Trélat, Contrôle Optimal: Théorie et Applications. Vuibert, Collection Mathématiques Concrètes (2005). [Google Scholar]
  • S.A. Yousefi, M. Dehgan and A. Lotfi, Finding the optimal control of linear systems via He’s variational iteration method. Int. J. Comput. Math. 87 (2010) 1042–1050. [Google Scholar]
  • M.S. Zahedi and H.S. Nik, On homotopy analysis method applied to linear optimal control problems. Appl. Math. Model. 37 (2013) 9617–9629. [Google Scholar]

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