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.

Keywords:

• Infinite-horizon optimal control
• two-point boundary value problem
• quasilinearization method
• exponential bernstein functions
• convergence analysis

2010 Mathematics Subject Classifications:

• 49J15
• 49Mxx
• 65Dxx
• 49M05
• 65L03

Acknowledgments

The authors are very grateful to the referees for their comments and valuable suggestions which improved the paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).