Improvement of Neustadt-Eaton’s method convergence

  • Starov Valentin G., Sobolev Institute of Mathematics, 4 Acad. Koptyug Avenue, Novosibirsk 630090, Russia
Keywords: approximation, extrapolation, iteration, optimal control, adjoint system


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.


Aleksandrov V. M., “An approximate solution of the linear time-optimality problem,” Autom. Remote Control, 59, No. 12, 1699–1707 (1999).

Boltyanskii V. G., Mathematical Methods of Optimal Control, Holt, Rinehart and Winston (1971).

Tyatyushkin A. I., Numerical Methods and Software for Controlled Systems Optimization [in Russian], Nauka, Novosibirsk (1992).

Fedorenko R. P., Approximate Solution to Optimal Control Problems [in Russian], Nauka, Moscow (1978).

Shevchenko G. V., “Numerical algorithm for a linear time-optimal control problem,” Comput. Math. Math. Phys., 42, No. 8, 1123–1134 (2002).

Aleksandrov V. M., “Sequential synthesis of time-optimal control,” Comput. Math. Math. Phys., 39, No. 9, 1402–1415 (1999).

Aleksandrov V. M., “Convergence of the method of sequential synthesis of time-optimal control,” Comput. Math. Math. Phys., 39, No. 10, 1582–1593 (1999).

Aleksandrov V. M., “A numerical method for solving a linear time-optimal control problem,” Comput. Math. Math. Phys., 38, No. 6, 881–893 (1998).

Aleksandrov V. M., “Solution of optimal control problems on the basis of the quasi-optimal control method [in Russian],” Tr. Inst. Mat., 10, pp. 18–54, Nauka, Novosibirsk (1988).

Zholudev A. I., Tyatyushkin A. I., and Ehrinchek N. M., “Numerical optimization methods for controlled processes [in Russian],” Izv. Akad. Nauk, Tekh. Kibern., No. 4, 14–31 (1989).

Pshenichnyj B. N. and Sobolenko L. A., “Accelerated method for solving time-optimal linear control problem [in Russian],” Zh. Vychisl. Mat. Mat. Fiz., 8, No. 6, 1345–1352 (1968).

Dubovitskii A. Ya. and Rubtsov V. A., ”Linear time-minimal problems,” U. S. S. R. Comput. Math. Math. Phys., 8, No. 5, 1–18 (1971).

Kiselev Yu. N., “Fast-convergence algorithms for linear time-optimal control,” Cybernetics, 26, No. 6, 848–869 (1990).

Srochko V. A., Iterative Methods for Solving Optimal Control Problems [in Russian], FIZMATLIT, Moscow (2000).

Dyurkovich E., “A numerical method for solving time-optimal problems, with an estimate of accuracy,” Sov. Math., Dokl., 26, 132–136 (1982).

Starov V. G., “Neustadt–Eaton’s method modification [in Russian],” Mat. Zametki SVFU, 21, No. 3, 107–112 (2014).
How to Cite
Starov, V. (&nbsp;) “Improvement of Neustadt-Eaton’s method convergence”, Mathematical notes of NEFU, 26(1), pp. 70-80. doi:
Mathematical Modeling