A numerical method for solving boundary optimal control problem modeled by heat transfer equation, in the presence of a scale invariance property

Document Type : Research Article

Authors

1 Department of Mathematical Sciences, Faculty of Hydrocarbons and Chemistry, University of Boumerdes, Boumerdes, Algeria

2 Laboratory of Applied Automation (LAA), Department of Automation, Faculty of Hydrocarbons and Chemistry, University of Boumerdes, Algeria

3 Laboratory of Operational Research and Mathematical decision, University of Tizi-Ouzou, Hasnaoua II 15000, Algeria

Abstract

In this paper, we present a computational approach for solving a boundary optimal control problem modeled by heat transfer equation with two-point boundary conditions, in the presence of a scale invariance property under dilation. First, we establish a scale-invariant solution. Indeed, the dependence of this solution towards a scale invariance factor naturally leads to an optimal control problem. Second, we propose a numerical approach to solve this problem. The idea consists in transforming the problem into an optimal control problem modeled by a system of ordinary differential equations invariant under dilation using the finite difference approximation. Therefor, the minimum principle of Pontryagin is applied to derive the necessary optimality conditions that are solved by the vartiational iteration method to get an approximate scale-invariant solutions for the optimal control law. Finally, to show the efficiency of this approach, a numerical example is illustrated and comparison with another method is performed.

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References

[1] F. Abraham, M. Behr, M. Heinkenschloss, Then effect of stabilization in finite element methods for
the optimal boundary control of the Oseen equation, Finite Elem. Anal. Des. 41 (2004) 229–251.
[2] R. Akbari, An investigation into the optimal control of the horizontal and vertical incidence of
communicable infectious diseases in society, J. Math. Model. 11 (2023) 605–616.
[3] H. Aleksandar, M. Verhaegen, Sparsity preserving optimal control of discretized PDE systems,
Comput. Methods Appl. Mech. Engrg. 335 (2018) 610-630.
[4] V. Barbu, N.H. Pavel, Optimal control problems with two-point boundary conditions, J. Optim.
Theory Appl. 77 (1993) 51–78.
[5] R. Bellman, Dynamic programming and a new formalism in the calculus of variations, Proc. Natl.
Acad. Sci. U.S.A. 40 (1954) 231–235.
[6] K. Benalia, C. David, B. Oukacha, An optimal time control problem for the one-dimensional, linear
heat equation, in the presence of a scaling parameter, RAIRO-Oper. Res. 51 (2017) 1289–1299.
[7] K. Benalia, B. Oukacha, Bang-Bang property of time optimal controls for the Heat equation in the
presence of a scale parameter, J. Int. Math. Virtual Institute 8 (2018) 69–85.
[8] K. Benalia, On the Controllability for the 1D Heat Equation with Dirichlet Boundary condition, in
the Presence of a Scale-Invariant Parameter, Int. J. Math. Comput. Sci. 19 (2023) 315–3018.
[9] J.T. Betts, S.L. Campbell, Discreteze Then Optimize, SIAM, 2005.
[10] E. Buckwar, Y. Luchko, Invariance of a partial differential equation of fractional order under the
Lie group of scaling transformations, J. Math. Anal. Appl. 227 (1998) 81–97.
[11] R.Y. Chang, S.Y. Yang, Solution of two-point-boundary-value problems by generalized orthogonal
polynomials and application to optimal control of lumped and distributed parameter systems, Int.
J. Control. 43 (2007) 1785–1802.
[12] A. Fakharian, M.T.H. Beheshtia, Solving linear and nonlinear optimal control problem using mod-
ified Adomian decomposition method, J. Comput. Robot. 1 (2008) 1–8.
[13] N. Govindarajan, C.C. de Visser, K. Krishnakumar, A sparse collocation method for solving time-
dependent HJB equations using multivariate B-splines, Automatica. 50 (2014) 2234–2244.
[14] J.H. He, A new approach to nonlinear partial differential equations, Commun. NonLinear Sci.
Numer. Simul. 2 (1997) 230–235.
[15] J.H. He, Variational iteration method for delay differential equations, Commun. Nonlinear Sci.
Numer. Simul. 2 (1997) 235–236.
[16] J.H. He, Variational iteration method a kind of non-linear analytical technique : some examples,
Int. J. NonLinear Mech. 34 (1999) 699–708.
[17] J.H. He, A generalized variational principle in micromorphic thermoelasticity, Mech. Res. Com-
mun. 32 (2005) 93–98.
[18] H. K. Hesse, G. Kanschat, Mesh adaptive multiple shooting for partial differential equations. Part
I: linear quadratic optimal control problems, J. Num. Mat. 17 (2009) 195–217.
[19] S. Hussain, G. Arora, R. Kumar, Semi-analytical methods for solving non-linear differential equa-
tions: A review, J. Math. Anal. Appl. 531 (2024) 127821.
[20] W. Jia, X. He, L. Guo, The optimal homotopy analysis method for solving linear optimal control
problems, Appl. Math. Model. 45 (2017) 865–880.
[21] J. Jun, X.S. Wang, Numerical optimal control of a size-structured PDE model for metastatic cancer
treatment, Math. Biosci. 314 (2019) 28–42.
[22] B. Kafash, A. Delavarkhalafi, S.M. Karbassi, Application of variational iteration method for
Hamilton-Jacobi-Bellman equations, Appl. Math. Model. 37 (2013) 3917–3928.
[23] I. Kucuk, Active optimal control of the KdV equation using the variational iteration method, Math.
Probl. Eng. 2010 (2010) 929103.
[24] H. Mirchi, D.K. Salkuyeh, A new iterative method for solving the systems arisen from finite element
discretization of a time-harmonic parabolic optimal control problems, Math. Comput. Simulation.
185 (2021) 771–782.
[25] E.I. Moiseev, A. A. Kholomeeva, an optimal boundary control problem with a dynamic boundary
condition, Diff Equat. 49 (2013) 640–644.
[26] Z.M. Odibat, A study on the convergence of variational iteration method, Math. Comput. Mod-
elling. 51 (2010) 1181–1192.
[27] P.J. Olver, Applications of Lie Groups to Differential Equations, Springer-Verlag, New Yorkr, 1986.
[28] L.S. Pontryagin, V.G. Boltyanskii, R.V. Gamkrelidze, E.F. Mishchenko, The Mathematical Theory
of Optimal Processes, Pergamon Press, New York, 1964.
[29] J.A. Rad, S. Kazem, K. Parand, Optimal Control of a parabolic distributed parameter system via
radial basis functions, Commun. Nonlinear Sci. 19 (2014) 2559–2567.
[30] J.I. Ramos, On the variational iteration method and other iterative techniques for nonlinear differ-
ential equations, Appl. Math. Comput. 199 (2008) 39–69.
[31] D.K. Salkuyeh, M. Pourbagher, On the solution of the distributed optimal control problem with
time-periodic parabolic equations, Math. Methods Appl. Sci. 46 (2023) 13671–13683.
A numerical method for solving boundary optimal control ... 137
[32] D.K. Salkuyeh, F. S. Sharafeh, On the numerical solution of the Burgers’s equation, Int. J. Comput.
Math. 86 (2009) 1334–1344.
[33] D.K. Salkuyeh, On the finite difference approximation to the convection–diffusion equation, Appl.
Math. Comput. 179 (2006) 79–86.
[34] D.K. Salkuyeh, H.R. Ghehsareh, Convergence of the variational iteration method for the telegraph
equation with integral conditions, Numer. Methods Partial Differ. Equ. 27 (2011) 1442–1455.
[35] D.K. Salkuyeh, A. Tavakoli, Interpolated variational iteration method for initial value problems,
Appl. Math. Model. 56 (2008) 2027–2033.
[36] D.K. Salkuyeh, Convergence of the variational iteration method for solving linear systems of ODEs
with constant coefficients, Comput. Math. Appl. 40 (2015) 3979–3990.
[37] M. Shehata, On differential-integral optimal control problems, J. Math. Model. 12 (2024) 451–462.
[38] M. Shirazian, The interpolated variational iteration method for solving a class of nonlinear optimal
control problems, J. Numer. Anal. Optim. 13 (2023) 224–242.
[39] M. Tatari and M. Dehghan, On the convergence of Hes variational iteration method, J. Comput.
Appl. Math. 207 (2007) 121–128.
[40] M. Visser, N. Yunes, Power Laws, Scale Invariance and the Generalized Frobenius Series: Appli-
cations to Newtonian and TOV Stars Near Criticality, Int. J. Mod. Phys. A. 18 (2003) 3427–3432.
Journal of Mathematical Modeling
Volume 13, Issue 1
March 2025
Pages 121-137

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