Abstract
A new approach to solving terminal control problems with phase constraints, using sufficient optimality conditions, is considered. The approach is based on the Lagrangian formalism and duality theory. Linear controlled dynamics under phase constraints is studied. The section of phase constraints at certain time points (on a discrete grid) leads to new intermediate optimal control problems without phase constraints. These problems generate intermediate solutions in intermediate spaces. The combination of all intermediate problems, in turn, leads to the original problem on the entire time interval. In each intermediate space, we have a polyhedral set obtained as a result of a section of the phase constraints. Based on this set, a linear programming problem is formed. Thus, on each small interval between two points of the section, a full-scale intermediate optimal control problem with a fixed left end and a moving right end of the phase trajectory is formed. The right end generates the reachability set and, at the same time, it is a solution of an intermediate boundary-value linear programming problem. The solution obtained, in turn, is the initial condition for the next intermediate optimal control problem. To solve the intermediate optimal control problem, an extragradient saddle-point method is proposed. The convergence of the method to the solution of the optimal control problem in all variables is proved. The convergence guarantees obtaining a solution of the problem with a given accuracy.
Similar content being viewed by others
REFERENCES
A. S. Antipin and E. V. Khoroshilova, “Optimal control with connected initial and terminal conditions,” Proc. Steklov Inst. Math. 289 (1), 9–25 (2015).
A. S. Antipin and E. V. Khoroshilova, “Linear programming and dynamics,” Trudy Inst. Mat. Mekh. Ural. Otd. Ross. Akad. Nauk 19 (2), 7–25 (2015).
A. S. Antipin and E. V. Khoroshilova, “Linear programming and dynamics,” Ural Math. J. 1 (1), 3–19 (2015).
A. S. Antipin and E. V. Khoroshilova, “Saddle-point approach to solving problem of optimal control with fixed ends,” J. Global Optim. 65 (1), 3–17 (2016).
A. S. Antipin and E. V. Khoroshilova, “On methods of terminal control with boundary-value problems: Lagrange approach,” in Optimization and Applications in Control and Data Sciences, Ed. by B. Goldengorin (Springer, New York, 2016), pp. 17–49.
A. S. Antipin and E. V. Khoroshilova, “Feedback synthesis for a terminal control problem,” Comput. Math. Math. Phys. 58 (12), 1903–1918 (2018).
A. S. Antipin and E. V. Khoroshilova, “Controlled dynamic model with boundary-value problem of minimizing a sensitivity function,” Optim. Lett. 13 (3), 451–473 (2019).
A. S. Antipin and E. V. Khoroshilova, “Lagrangian as a tool for solving linear optimal control problems with state constraints,” Proceedings of the International Conference on Optimal Control and Differential Games Dedicated to L.S. Pontryagin on the Occasion of His 110th Birthday (2018), pp. 23–26.
A. Antipin, V. Jaćimović, and M. Jaćimović, “Dynamics and variational inequalities,” Comput. Math. Math. Phys. 57 (5), 784–801 (2017).
A. Antipin and O. Vasilieva, “Dynamic method of multipliers in terminal control,” Comput. Math. Math. Phys. 55 (5), 766–787 (2015).
V. M. Alekseev, V. M. Tikhomirov, and S. V. Fomin, Optimal Control (Nauka, Moscow, 1979; North-Holland, Amsterdam, 1979).
F. P. Vasil’ev, Optimization Methods (MTsNMO, Moscow, 2011), Vols. 1, 2 [in Russian].
Funding
This work was supported by the Russian Foundation for Basic Research, project no. 18-01-00312.
Author information
Authors and Affiliations
Corresponding authors
Additional information
Translated by E. Chernokozhin
Rights and permissions
About this article
Cite this article
Antipin, A.S., Khoroshilova, E.V. Dynamics, Phase Constraints, and Linear Programming. Comput. Math. and Math. Phys. 60, 184–202 (2020). https://doi.org/10.1134/S0965542520020037
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0965542520020037