A Boundary Value Problem of Terminal Control with a Quadratic Criterion of Quality
In a Hilbert space, we consider the problem of terminal control with linear dynamics, fixed left end and moving right end of the trajectories. On the reachability set(under additional linear constraints) the objective functional as the sum of integral and terminal components of the quadratic form is minimized. To solve the problem, we do not use the classical approach based on the consideration of the optimal control problem as an optimization problem. Instead, the saddle-point method for solving the problem is proposed. We prove its convergence.
1. Antipin A. S. One method of finding a saddle point of a modified Lagrange function (in Russian). Economics and Mathematical Methods, 1977, vol. XIII, issue 3, pp. 560–565.
2. Antipin A. S., Khoroshilova E. V. On extragradient type methods for solving optimal control problem with linear constraints (in Russian). Proceedings of ISU. Mathematics, 2010, vol. 3, № 3, pp. 2–20.
3. Antipin A. S. Modified Lagrange function method for optimal control problems with free right end (in Russian). Proceedings of ISU. Mathematics, 2011, vol. 4, № 2, pp. 27–44.
4. Antipin A. S., Khoroshilova E. V. Terminal control boundary value problems of convex programming (in Russian). Optimization and application, 2013, issue 3, pp. 17–55.
5. Antipin A. S. Terminal control boundary models (in Russian). Computational Mathematics and Mathematical Physics, 2014, vol. 54, № 2, pp. 257–285.
6. Antipin A. S., Khoroshilova E. V. Optimal control related initial and terminal conditions (in Russian). Proceedings of the Institute of Mathematics and Mechanics UB RAS, 2014, vol. 20, № 2, pp. 7–22.
7. Vasil’ev F. P., Khoroshilova E. V. Extra-gradient method for finding a saddle point in the optimal control (in Russian). Bulletin of Lomonosov Moscow State University. Series 15. Computational Mathematics and Cybernetics, 2010, № 3, pp. 18–23.
8. Vasil’ev F. P., Khoroshilova E. V., Antipin A. S. Extragradient regularized method for finding a saddle point in the optimal control problem (in Russian). Proceedings of Institute of Mathematics and Mechanics, UB RAS, 2011, vol. 17, №1, pp. 27–37.
9. Vasil’ev F. P. Optimization Methods. In 2 books (in Russian). Moscow, 2011.
10. Ioffe A. D., Tikhomirov V. M. Theory of extremal problems (in Russian). Moscow, 1974, 479 p.
11. Konnov I. V. Nonlinear optimization and variational inequalities (in Russian). Kazan, 2013, 508 p.
12. Korpelevich G. M. Extragradient method for finding saddle points and other problems (in Russian) Economics and Mathematical Methods, 1976, vol. XII, issue 6, pp. 747–756.
13. Lyusternik L. A., Sobolev V. I. Elements of functional analysis (in Russian). Moscow, Nauka, 1965.
14. Natanson I. P. Theory of functions of a real variable (in Russian). Moscow, 1957, 552 p.
15. Srochko V. A., Aksenyushkina E. V. Linear-quadratic optimal control problem: rationale and convergence of nonlocal methods for solving (in Russian). Proceedings of ISU. Mathematics, 2013, vol. 6, № 1, pp. 89–100.
16. Khoroshilova E. V. Extragradient method in optimal control problem with terminal constraints (in Russian). Automation and Remote Control, 2012, issue 3, pp. 117–133.
17. Facchinei F., Pang J.-S. Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer-Verlag, 2003, Vol. 1.
18. Khoroshilova E. V. Extragradient-type method for optimal control problem with linear constraints and convex objective function. Optim. Lett., 2013, vol. 7, № 6, pp. 1193–1214.