Geodesic
On resolution of an extremum norm problem for the terminal state of a linear system
The Bulletin of Irkutsk State University. Series Mathematics, Tome 34 (2020), pp. 3-17 Cet article a éte moissonné depuis la source Math-Net.Ru

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We study extremum norm problems for the terminal state of a linear dynamical system using methods of parameterization of admissible controls. Piecewise continuous controls are approximated in the class of piecewise linear functions on a uniform grid of nodes of the time interval by linear combinations of special support functions. In this case, the restriction of a control of the original problem to the interval induces the same restrictions for the variables of the finite-dimensional problems. The finite-dimensional version of a minimum norm problem can effectively be resolved with the help of modern convex optimization programs. In the case of two variables, we propose an analytical method of resolution that uses a one-dimensional minimization problem for a parabola over a segment. For a non-convex norm maximization problem, the finite-dimensional version is resolved globally by exhaustive search over the vertices of a hypercube. The proposed approach provides further insights into global resolution of non-convex optimal control problems and is exemplified by some illustrative problems.
Keywords: linear control system, extremum norm problems for the terminal state, piecewise linear approximation, finite-dimensional problems. 
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V. A. Srochko; E. V. Aksenyushkina. On resolution of an extremum norm problem for the terminal state of a linear system. The Bulletin of Irkutsk State University. Series Mathematics, Tome 34 (2020), pp. 3-17. http://geodesic.mathdoc.fr/item/IIGUM_2020_34_a0/

[1] Arguchintsev A. V., Dykhta V. A., Srochko V. A., “Optimal control: nonlocal conditions, computational methods, and the variational principle of maximum”, Russian Mathematics, 53:1 (2009), 1–35 | DOI | MR | Zbl

[2] Antonik V. G., Srochko V. A., “Method for nonlocal improvement of extreme controls in the maximization of the terminal state norm”, Computational Mathematics and Mathematical Physics, 49:5 (2009), 762–775 | DOI | MR | Zbl

[3] Galyaev A. A., Lysenko P. V., “Energy-optimal control of harmonic oscillator”, Automation and Remote Control, 80 (2019), 16–29 | DOI | MR | Zbl

[4] Gorbunov V. K., “The reduction of optimal control problems for finite-dimensional”, Computational Mathematics and Mathematical Physics, 18:5 (1978), 1083–1095 | MR | Zbl

[5] Srochko V. A., Iterative methods for solving optimal control problems, Fizmatlit Publ., M., 2000, 160 pp. (in Russian)

[6] Srochko V. A., Aksenyushkina E. V., “Parameterization of Some Control Problems by Linear Systems”, The Bulletin of Irkutsk State University. Series Mathematics, 30 (2019), 83–98 (in Russian) | DOI | Zbl

[7] Strekalovsky A. S., Elements of nonconvex optimization, Nauka Publ., Novosibirsk, 2003, 356 pp. (in Russian)

[8] Strekalovsky A. S., Sharankhaeva E. V., “Global search in a nonconvex optimal control problem”, Computational Mathematics and Mathematical Physics, 45:10 (2005), 1719–1734 | MR | Zbl

[9] Sukharev A. G., Timokhov A. V., Fedorov V. V., Course of optimization methods, Nauka Publ., M., 1986, 328 pp. (in Russian) | MR

[10] Tyatyushkin A. I., Multi-method technology for optimization of control systems, Nauka Publ, Novosibirsk, 2006, 343 pp. (in Russian)

[11] Chernov A. V., “On application of Gaussian functions for numerical solution of optimal control problems”, Automation and Remote Control, 80 (2019), 1026–1040 | DOI | MR | Zbl