Symplectic algorithms with mesh refinement for a hypersensitive optimal control problem

Abstract

A symplectic algorithm with nonuniform grids is proposed for solving the hypersensitive optimal control problem using the density function. The proposed method satisfies the first-order necessary conditions for the optimal control problem that can preserve the structure of the original Hamiltonian systems. Furthermore, the explicit Jacobi matrix with sparse symmetric character is derived to speed up the convergence rate of the resulting nonlinear equations. Numerical simulations highlight the features of the proposed method and show that the symplectic algorithm with nonuniform grids is more computationally efficient and accuracy compared with uniform grid implementations. Besides, the symplectic algorithm has obvious advantages on optimality and convergence accuracy compared with the direct collocation methods using the same density function for mesh refinement.

Keywords:

• optimal control problem
• Hamiltonian system
• boundary-value problem
• hypersensitive problem
• symplectic algorithm

2000 AMS Subject Classifications:

• 49J15
• 30E25
• 37M15

Acknowledgements

The authors are grateful for the financial support of the National Science Foundation of China (11102031, 11472069, 11272076); the Project Funded by China Postdoctoral Science Foundation (2014M550155); the Fundamental Research Funds for Central Universities (DUT13LK25); the Program Funded by Liaoning Province Education Administration (L2013015); the State Key Laboratory of Mechanics and Control of Mechanical Structures (MCMS-0114G02).