Improvement of Neustadt–Eaton's method convergence

Neustadt and Eaton developed a method and an iterative algorithm for solving time-optimal control problems. According to the method, it is necessary to solve a discrete equation to find the initial value of the adjoint system. At each step of the algorithm it is needed to find the increment of the initial value of the adjoint system. This is achieved by choosing the step length, while the initial value must satisfy certain conditions. Then the system of differential equations is solved, where the obtained values are used as initial data and the trajectory is determined. Thus, the iterative process must be used on each step of solving the discrete equation. The convergence of this method was previously proved. However, the convergence is weak. To improve it, the following approach is proposed. After a few steps of Neustadt–Eaton's method the obtained initial values of the adjoint system are approximated and extrapolated at a certain interval. Then again we take a few steps of Neustadt–Eaton's method and use the extrapolated initial values of the adjoint system. Thus we propose “sliding” approximation of the discrete equation solutions for each component using an algebraic function followed by extrapolation. It is shown that such extrapolation significantly reduces computational costs.