Numerical solution of two-point BVPs in infinite-horizon optimal control theory: a combined quasilinearization method with exponential Bernstein functions

ABSTRACT This study is aimed to relate nonlinear infinite-horizon optimal control problems (NLIHOCPs) with open-loop information. The difficulties of solving the two-point boundary value problems (TPBVPs) arising from NLIHOCPs can be assigned to the nonlinearity of differential equations, the combination of split boundary conditions, and how the transversality conditions in infinite-horizon are treated. In this paper, we propose a combined quasilinearization method (QLM) with the exponential Bernstein functions (EBFs) for solving nonlinear TPBVPs on the semi-infinite domain. First, the QLM is used to reduce the nonlinear TPBVP to a sequence of linear differential equations. Then, a collocation method based on the EBFs is utilized to find the approximate solution of the resulting linear differential equations. By applying the EBFs, the transversality conditions for TPBVP on the semi-infinite domain are satisfied. The convergence of the QLM + EBFs is proved. Some numerical experiments are performed to confirm the validity of the proposed computational scheme.

[1]  G. B. Loghmani,et al.  Numerical Study of Generalized Three-Dimensional MHD Flow over a Porous Stretching Sheet , 2014 .

[2]  V. Lakshmikantham,et al.  Further improvement of generalized quasilinearization method , 1996 .

[3]  Emanuel Todorov,et al.  Optimal Control Theory , 2006 .

[4]  Abdul-Majid Wazwaz,et al.  A numerical study of electrohydrodynamic flow analysis in a circular cylindrical conduit using orthonormal Bernstein polynomials , 2017 .

[5]  I. Michael Ross,et al.  Direct Trajectory Optimization by a Chebyshev Pseudospectral Method ; Journal of Guidance, Control, and Dynamics, v. 25, 2002 ; pp. 160-166 , 2002 .

[6]  M. Mehrpouya.  An efficient pseudospectral method for numerical solution of nonlinear singular initial and boundary value problems arising in astrophysics , 2016 .

[7]  M. Heydari,et al.  Convergence analysis of an iterative algorithm to solve system of nonlinear stochastic Itô‐Volterra integral equations , 2020, Mathematical Methods in the Applied Sciences.

[8]  A. Delavarkhalafi,et al.  A numerical solution of open-loop Nash equilibrium in nonlinear differential games based on Chebyshev pseudospectral method , 2016, J. Comput. Appl. Math..

[9]  Paul Bracken,et al.  Solutions of differential equations in a Bernstein polynomial basis , 2007 .

[10]  Esmail Babolian,et al.  Solving generalized pantograph equations by shifted orthonormal Bernstein polynomials , 2016, J. Comput. Appl. Math..

[11]  E. Kreyszig.  Introductory Functional Analysis With Applications , 1978 .

[12]  G. B. Loghmani,et al.  A COMBINATION OF PSEUDO-SPECTRAL METHOD AND EXTRAPOLATION FOR SOLVING MHD FLOW AND HEAT TRANSFER ABOUT A ROTATING DISK , 2014 .

[13]  P. Williams.  Jacobi pseudospectral method for solving optimal control problems , 2004 .

[14]  R. McGill,et al.  Two-Point Boundary-Value-Problem Techniques , 1966 .

[15]  Extensions of the method of quasilinearization , 1995 .

[16]  M. Heydari,et al.  Jacobi‐Picard iteration method for the numerical solution of nonlinear initial value problems , 2019, Mathematical Methods in the Applied Sciences.

[17]  Tsuyoshi Murata,et al.  {m , 1934, ACML.

[18]  Kamal Mamehrashi,et al.  A new approach for solving infinite horizon optimal control problems using Laguerre functions and Ritz spectral method , 2020, Int. J. Comput. Math..

[19]  R. Bellman,et al.  Quasilinearization and nonlinear boundary-value problems , 1966 .

[20]  Nguyen Canh,et al.  A convergence theorem on the iterative solution of nonlinear two-point boundary-value systems , 1974, Kybernetika.

[21]  G. B. Loghmani,et al.  Exponential Bernstein functions: an effective tool for the solution of heat transfer of a micropolar fluid through a porous medium with radiation , 2017 .

[22]  V. Lakshmikantham.  An extension of the method of quasilinearization , 1994 .

[23]  Z. Nikooeinejad,et al.  Nash equilibrium approximation of some class of stochastic differential games: A combined Chebyshev spectral collocation method with policy iteration , 2019, J. Comput. Appl. Math..

[24]  A. Delavarkhalafi,et al.  Application of shifted Jacobi pseudospectral method for solving (in)finite-horizon min–max optimal control problems with uncertainty , 2018, Int. J. Control.

[25]  Gamal N. Elnagar,et al.  The pseudospectral Legendre method for discretizing optimal control problems , 1995, IEEE Trans. Autom. Control..

[26]  Mohammad Mehdi Rashidi,et al.  Numerical investigation of velocity slip and temperature jump effects on unsteady flow over a stretching permeable surface , 2017 .

[27]  M. Shamsi,et al.  Gauss pseudospectral and continuation methods for solving two-point boundary value problems in optimal control theory , 2015 .

[28]  R. Kalaba.  ON NONLINEAR DIFFERENTIAL EQUATIONS, THE MAXIMUM OPERATION, AND MONOTONE CONVERGENCE, , 1959 .

[29]  R. Dooren.  Numerical study of the controlled Van der Pol oscillator in Chebyshev series , 1987 .

[30]  Mohammad Mehdi Rashidi,et al.  A numerical simulation of MHD ow and radiation heat transfer of nano uids through a porous medium with variable surface heat ux and chemical reaction , 2019 .

[31]  I. Michael Ross,et al.  Pseudospectral Methods for Infinite-Horizon Nonlinear Optimal Control Problems , 2005 .

[32]  J. Primbs,et al.  Constrained nonlinear optimal control: a converse HJB approach , 1996 .

[33]  William W. Hager,et al.  Pseudospectral methods for solving infinite-horizon optimal control problems , 2011, Autom..

[34]  Astronomy,et al.  Quasilinearization approach to nonlinear problems in physics with application to nonlinear ODEs , 2001, physics/0102041.

[35]  Frank L. Lewis,et al.  Multi-player non-zero-sum games: Online adaptive learning solution of coupled Hamilton-Jacobi equations , 2011, Autom..

[36]  M. Shahini,et al.  Transformed Legendre spectral method for solving infinite horizon optimal control problems , 2016, IMA J. Math. Control. Inf..

[37]  Mohammad Mehdi Rashidi,et al.  Investigation of magneto-hemodynamic flow in a semi-porous channel using orthonormal Bernstein polynomials , 2017 .

[38]  Farshid Mirzaee,et al.  Numerical method for solving optimal control problem of the linear differential systems with inequality constraints , 2016 .

[39]  V. B. Mandelzweig,et al.  Quasilinearization method and its verification on exactly solvable models in quantum mechanics , 1999 .

[40]  Mohammad Mehdi Rashidi,et al.  A numerical study for off-centered stagnation flow towards a rotating disc , 2015 .