Geodesic
On a linear control problem under interference
Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematika, mehanika, fizika, Tome 9 (2017) no. 2, pp. 36-46 Cet article a éte moissonné depuis la source Math-Net.Ru

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This paper considers a linear control problem under the action of uncontrolled interference. The control process occurs in a given time interval. The possible values of interference belong to a compact set. The control is sought as the product of a scalar function and a vector function. Values of the vector function belong to a connected symmetric compact. This definition of control arises in control problems for mechanical systems of variable composition. For example, the law of variation of a reaction mass is defined as a function of time, and the control affects the direction of relative velocity in which the mass is separated. The terminal part of the board depends on the modulus of a linear function of the state vector. The integral part of the board is an integral over the interval of a degree of the scalar function. The control problem is considered within the theory of guaranteed result optimization. With the help of a linear change of variables, the control problem comes down to a one-type differential game. An optimal control existence theorem is proved under rather wide constraints on the class of problems. Necessary and sufficient conditions are found, under which an admissible control is optimal.
Keywords: control, interference, differential game. 
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V. I. Ukhobotov. On a linear control problem under interference. Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematika, mehanika, fizika, Tome 9 (2017) no. 2, pp. 36-46. http://geodesic.mathdoc.fr/item/VYURM_2017_9_2_a4/

[1] Krasovskij N. N., Subbotin A. I., Positional differential games, Nauka Publ., M., 1974, 456 pp. (in Russ.)

[2] Isaacs R., Differential games: A Mathematical Theory with Applications to Warfare and Pursuit, Control and Optimization, John Wiley and Sons, Inc., New York, 1965, 384 pp. | MR | Zbl

[3] Pontryagin L. S., “Linear differential games of pursuit”, Math. USSR-Sb., 40:3 (1981), 285–303 | DOI | Zbl | Zbl

[4] Ukhobotov V. I., Method of one-dimensional design in linear differential games with integral constraints, textbook, Chelyabinskiy gosudarstvennyy universitet Publ., Chelyabinsk, 2005, 124 pp. (in Russ.)

[5] Ukhobotov V. I., “The same type of differential games with convex purpose”, Trudy Instituta Matematiki I Mehaniki Uro RAN, 16, no. 5, 2010, 196–204

[6] Ukhobotov V. I., Gushchin D. V., “Single-type differential games with convex integral”, Proceedings of the Steklov Institute of Mathematics, 275, suppl. 1 (2011), 178–185 | DOI | MR | Zbl

[7] Krasovskii N. N., Motion control theory, Nauka Publ., M., 1968, 475 pp. (in Russ.)

[8] Kolmogorov A. N., Fomin S. V., Elements of function theory and functional analysis, Nauka Publ., M., 1972, 496 pp. (in Russ.)

[9] Kudryavtsev L. D., Course of mathematical analysis, v. 1, Vysshaya shkola Publ., M., 1981, 687 pp. (in Russ.)

[10] Pshenichnyy B. N., Convex analysis and extremum problems, Nauka Publ., M., 1980, 319 pp. (in Russ.)

[11] Ljusternik L. A., Sobolev V. I., Elements of functional analysis, Nauka Publ., M., 1965, 520 pp. (in Russ.)

[12] Riesz F., Sz.-Nagy B., Lecons d'analyse fonctionnelle, Akademiai Kiado, Budapest, 1972, 588 pp. | MR

[13] Ioffe A. D., Tihomirov V. M., Theory of extremum problems, Nauka Publ., M., 1974, 479 pp. (in Russ.)