In the present paper, an efficient pseudospectral method for solving the Hamiltonian boundary value problems arising from a class of switching optimal control problems is presented. For this purpose, based on the Pontryagin's minimum principle, the first-order necessary conditions of optimality are derived. Then, by partitioning the time interval related to the problem under study into some subintervals, the states (and costates) and control functions are approximated on each subintervals with piecewise interpolating polynomials based on Legendre–Gauss–Radau points and a piecewise constant function, respectively. As a result, solution of the problem is turned into solution of a number of algebraic equations, in which the values of the states (and costates) and control functions at Legendre–Gauss–Radau points as well as switching and terminal points are allowed to be unknown. Numerical examples are presented at the end to show the efficiency of the proposed method.
PanBPanXMaY. A quadratic homotopy method for fuel-optimal low-thrust trajectory design. Proc IMechE, Part G: J Aerospace Engineering2019; 233: 1741–1757.
2.
QiuWJiaQMengX, et al.Maximum range trajectory optimization for a boost-glide vehicle using adaptive mesh refinement pseudospectral methods. Proc IMechE, Part G: J Aerospace Engineering2017; 231: 1171–1182.
3.
ZhangQFengZTangQ, et al.An adaptive wavelet collocation method for solving optimal control problem. Proc IMechE, Part G: J Aerospace Engineering2015; 229: 1640–1649.
4.
WangJCuiNWeiC. Rapid trajectory optimization for hypersonic entry using a pseudospectral-convex algorithm. Proc IMechE, Part G: J Aerospace Engineering. 2019; 233: 5227–5238.
5.
HuYWuBGengY, et al.Smooth time-optimal attitude control of spacecraft. Proc IMechE, Part G: J Aerospace Engineering2019; 233: 2331–2343.
6.
BettsJT. Practical methods for optimal control and estimation using nonlinear programming, 2nd ed. New York, NY: Cambridge University Press, 2009.
7.
TeoKGohCJWongKH. A unified computational approach to optimal control problems, Harlow, UK: Longman Scientific and Technical, 1991.
8.
KirkD. Optimal control theory, Englewood Cliffs, NJ: Prentice-Hall, 1970.
OsborneM. On shooting methods for boundary value problems. J Math Anal Appl1969; 27: 417–433.
11.
Oberle H and Grimm W. BNDSCO – a program for the numerical solution of optimal control problems. Technical Report 515-89/22, Institute for Flight Systems Dynamics, DLR, Oberpfaffenhofen, Germany, 1989.
12.
PengHGaoQWuZ, et al.Symplectic adaptive algorithm for solving nonlinear two-point boundary value problems in astrodynamics. Celes Mech Dyn Astron2011; 110: 319–342.
13.
PengHGaoQWuZ, et al.Symplectic approaches for solving two-point boundary-value problems. J Guidance Control Dyn2012; 35: 653–658.
14.
PengHjGaoQZhangHW, et al.Parametric variational solution of linear-quadratic optimal control problems with control inequality constraints. Appl Math Mech2014; 35: 1079–1098.
15.
MehrpouyaMShamsiMAzhmyakovV. An efficient solution of hamiltonian boundary value problems by combined gauss pseudospectral method with differential continuation approach. J Franklin Inst2014; 351: 4765–4785.
16.
MehrpouyaMShamsiM. Gauss pseudospectral and continuation methods for solving two-point boundary value problems in optimal control theory. Appl Math Model2015; 39: 5047–5057.
17.
HaberkornTMartinonPGergaudJ. Low-thrust minimum-fuel orbital transfer: A homotopic approach. J Guidance Control Dyn2004; 27: 1046–1060.
18.
HermantA. Optimal control of the atmospheric reentry of a space shuttle by a homotopy method. Opt Control Appl Meth2011; 32: 627–646.
19.
ElnagarGKazemiMRazzaghiM. Pseudospectral legendre method for discretizing optimal control problems. IEEE Trans Autom Control1995; 40: 1793–1796.
20.
FahrooFRossI. Direct trajectory optimization by a Chebyshev pseudospectral method. J Guidance Control Dyn2002; 25: 160–166.
21.
Benson D. A Gauss pseudospectral transcription for optimal control. PhD Thesis, Department of Aeronautics and Astronautics, Massachusetts Institute of Technology, 2004.
22.
MalekiMHashimI. Adaptive pseudospectral methods for solving constrained linear and nonlinear time-delay optimal control problems. J Franklin Inst2014; 351: 811–839.
23.
SabehZShamsiMDehghanM. Distributed optimal control of the viscous burgers equation via a legendre pseudo-spectral approach. Math Meth Sci2016; 39: 3350–3360.
24.
HoseiniSM. A composite pseudospectral method for solving multi-delay optimal control problems involving piecewise constant delay functions. J Franklin Inst2018; 355: 7895–7923.
25.
Khaksar-e OshaghMShamsiM. Direct pseudo-spectral method for optimal control of obstacle problem – an optimal control problem governed by elliptic variational inequality. Math Meth Appl Sci2017; 40: 4993–5004.
26.
FornbergB. A practical guide to pseudospectral methods, Cambridge: Cambridge University Press, 1998.
27.
RossIFahrooF. Pseudospectral knotting methods for solving optimal control problems. J Guidance Control Dyn2004; 27: 397–405.
28.
ShamsiM. A modified pseudospectral scheme for accurate solution of bang-bang optimal control problems. Opt Control Appl Meth2011; 32: 668–680.
29.
DarbyCHagerWRaoA. An hp-adaptive pseudospectral method for solving optimal control problems. Opt Control Appl Meth2011; 32: 476–502.
30.
GuoTJiangFLiJ. Homotopic approach and pseudospectral method applied jointly to low thrust trajectory optimization. Acta Astronaut2012; 71: 38–50.
31.
Boucher R, Kang W and Gong Q. Discontinuous Galerkin optimal control for constrained nonlinear problems. In: 11th IEEE International Conference on Control & Automation (ICCA), Taichung, 2014, pp. 84–89.
32.
TrefethenLN. Spectral methods in MATLAB, Philadelphia: SIAM, 2000.
33.
CanutoCHussainiMQuarteroniA, et al.Spectral methods in fluid dynamics, Berlin: Springer, 1991. Springer series in computational physics.
34.
DennisJSchnabelR. Numerical methods for unconstrained optimization and nonlinear equations. Classics in applied mathematics, Philadelphia, PA: Society for Industrial and Applied Mathematics, 1996.
35.
KayaCNoakesJ. Computational method for time-optimal switching control. J Optimizat Theory Appl2003; 117: 69–92.
