Abstract
A method for solving the terminal control problem with a fixed time interval and fixed initial conditions is proposed. The solution to the boundary value problem posed at the right end of the time interval determines the terminal conditions. This boundary value problem is a finite-dimensional convex programming problem. The dynamics of the terminal control problem is described by a linear controllable system of differential equations. This system is interpreted as a conventional system of linear equality constraints. Then the terminal control problem can be regarded as a dynamic convex programming problem posed in an infinite-dimensional functional Hilbert space. In this paper, the functional problem is treated as a saddle-point problem rather than optimization problem. Accordingly, a saddle-point approach to solving the problem is proposed. This approach is based on maximizing the dual function generated by the modified Lagrangian function of the convex programming problem posed in the functional space. The convergence of the proposed methods is also proved in the functional space. This convergence has the additional property of being monotone in norm with respect to controls, phase trajectories, adjoint functions, as well as finite-dimensional terminal variables.
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References
A. N. Kolmogorov and S. V. Fomin, Elements of the Theory of Functions and Functional Analysis (Dover, New York, 1999; Fizmatlit, Moscow, 2009).
F. P. Vasil’ev, Optimization Methods (MTsNMO, Moscow, 2011), Vols. 1, 2 [in Russian].
A. S. Antipin, “Equilibrium programming: Models and solution methods,” Izv. Irkutsk. Gos. Univ. Ser. Mat. 2(1), 8–36 (2009).
A. S. Antipin, “Two-person game with Nash equilibrium in optimal control problems,” Optim. Lett. 6(7), 1349–1378 (2012).
A. S. Antipin and E. V. Khoroshilova, “Linear programming and dynamics,” Tr. Inst. Mat. Mekh. Ural. Otd. Ross. Akad. Nauk 19(2), 7–25 (2013).
A. S. Antipin and E. V. Khoroshilova, “Optimal control with coupled initial and terminal conditions,” Tr. Inst. Mat. Mekh. Ural. Otd. Ross. Akad. Nauk 20(2), 13–28 (2014).
A. S. Antipin, “Terminal control of boundary models,” Comput. Math. Math. Phys. 54(2), 275–302 (2014).
A. S. Antipin and O. O. Vasilieva, “Augmented Lagrangian method for optimal control problems,” Analysis, Modeling, Optimization, and Numerical Techniques, Ser. Proc. Math. Stat., Ed. by G. Olivar and O. Vasilieva (Springer, Switzerland, 2015), Vol. 121, pp. 1–36.
A. S. Antipin, “Convex programming method using a symmetric modification of the Lagrangian functional,” Ekon. Mat. Metody 12(6), 1164–1173 (1976).
A. S. Antipin, “A method for finding saddle points of the augmented Lagrangian function,” Ekon. Mat. Met. 13(3), 560–565 (1977).
E. G. Gol’shtein and N. V. Tret’yakov, Augmented Lagrangians: Theory and Optimization Methods (Nauka, Moscow, 1989) [in Russian].
B. T. Polyak, Introduction to Optimization (Nauka, Moscow, 1983; Optimization Software, New York, 1987).
L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, and E. F. Mishchenko, The Mathematical Theory of Optimal Processes (Nauka, Moscow, 1976; Gordon and Breach, New York, 1986).
A. S. Antipin, “Balanced programming: Gradient-type methods,” Autom. Remote Control 58(8), 1337–1347 (1997).
A. S. Antipin, “Equilibrium programming: Proximal methods,” Comput. Math. Math. Phys. 37(11), 1285–1296 (1997).
A. S. Antipin, “Extra-proximal methods for solving two-person nonzero-sum games,” Math. Program. 120(1) (2009).
R. T. Rockafellar, “Augmented Lagrangians and applications of the proximal point algorithm in convex programming,” Math. Oper. Res. 1(2), 97–116 (1976).
W. W. Hager, “Multiplier methods for nonlinear optimal control,” SIAM J. Numer. Anal. 27(4), 1061–1080 (1990).
O. O. Vasilieva, “The search of equilibrium strategies for controlled boundary value problem,” Asian J. Control 3(1), 50–56 (2001).
O. O. Vasilieva, “Search for equilibrium controls in controlled boundary value problems,” Izv. Irkutsk. Gos. Univ. Ser. Mat. 1(1), 70–85 (2007).
O. O. Vasilieva and O. V. Vasil’ev, “On the search for equilibrium controls in an m-person differential game,” Russ. Math. 44(12), 7–12 (2000).
L. A. Lusternik and V. I. Sobolev, Elements of Functional Analysis (Nauka, Moscow, 1965; Gordon and Breach, New York, 1968).
R. T. Rockafellar and R. J.-B. Wets, Variational Analysis (Springer-Verlag, Berlin, 1998).
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Original Russian Text © A.S. Antipin, O.O. Vasilieva, 2015, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2015, Vol. 55, No. 5, pp. 776–797.
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Antipin, A.S., Vasilieva, O.O. Dynamic method of multipliers in terminal control. Comput. Math. and Math. Phys. 55, 766–787 (2015). https://doi.org/10.1134/S096554251505005X
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DOI: https://doi.org/10.1134/S096554251505005X