Geodesic
On numerical solution of the optimal control problem based on a method using the second variation of a trajectory
Vestnik Sankt-Peterburgskogo universiteta. Prikladnaâ matematika, informatika, processy upravleniâ, Tome 15 (2019) no. 2, pp. 283-295 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

A new approach for the numerical solution of the optimal control problem is presented. This approach is based on the parametrization of the control function and computation of the first and the second derivatives of the cost functional over parameters. Computation of the second derivatives is carried out using an integral representation for the second variation of the trajectory of a controlled dynamical system,which is obtained in this work and includes a tensor of the third rank. The proposed approach differs from the approach being used previously, in which the second derivatives of the cost functional are expressed through the matrix momenta. A numerical method based on the second variation of the trajectory can be effective in problems with a great number of parameters. Using the matrix momenta, one should integrate a system of differential equations, the number of which is quadratic in the number of parameters. In the present approach, the number of the equation to be integrated is determined only by the dimension of the phase space and can be sufficiently smaller.
Keywords: optimal control, controlled dynamical system, numerical methods of the second order.
Mots-clés : second variation 
@article{VSPUI_2019_15_2_a10,
     author = {O. I. Drivotin},
     title = {On numerical solution of the optimal control problem based on a method using the second variation of a trajectory},
     journal = {Vestnik Sankt-Peterburgskogo universiteta. Prikladna\^a matematika, informatika, processy upravleni\^a},
     pages = {283--295},
     year = {2019},
     volume = {15},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VSPUI_2019_15_2_a10/}
}
TY  - JOUR
AU  - O. I. Drivotin
TI  - On numerical solution of the optimal control problem based on a method using the second variation of a trajectory
JO  - Vestnik Sankt-Peterburgskogo universiteta. Prikladnaâ matematika, informatika, processy upravleniâ
PY  - 2019
SP  - 283
EP  - 295
VL  - 15
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/VSPUI_2019_15_2_a10/
LA  - ru
ID  - VSPUI_2019_15_2_a10
ER  - 
%0 Journal Article
%A O. I. Drivotin
%T On numerical solution of the optimal control problem based on a method using the second variation of a trajectory
%J Vestnik Sankt-Peterburgskogo universiteta. Prikladnaâ matematika, informatika, processy upravleniâ
%D 2019
%P 283-295
%V 15
%N 2
%U http://geodesic.mathdoc.fr/item/VSPUI_2019_15_2_a10/
%G ru
%F VSPUI_2019_15_2_a10
O. I. Drivotin. On numerical solution of the optimal control problem based on a method using the second variation of a trajectory. Vestnik Sankt-Peterburgskogo universiteta. Prikladnaâ matematika, informatika, processy upravleniâ, Tome 15 (2019) no. 2, pp. 283-295. http://geodesic.mathdoc.fr/item/VSPUI_2019_15_2_a10/

[1] Hartman P., Ordinary differential equations, John Wiley Sons Publ, New York, 1964, 612 pp. | MR | Zbl

[2] Pontryagin L. S., Boltyanskiy V. G., Gamkrelidze R. V., Mischenko E. F., Mathematical theory of optimal processes, Nauka Publ, M., 1976, 392 pp. (In Russian) | MR

[3] Moiseev N. N., Numerical methods in the optimal systems theory, Nauka Publ, M., 1971, 424 pp. (In Russian)

[4] Pytlak R., Numerical methods for optimal control with state constraints, Lect. Notes. Math., 1707, Springer, Berlin, 1999, 216 pp. | DOI | MR | Zbl

[5] Bryson A. E. Jr., Ho Yu-Chi, Applied optimal control, Blaisdell Publ. Comp., Waltham, Mass., 1969, 496 pp.

[6] Chernous'ko F. L., Banichuk N. V., Variational problems of mechanics and control, Nauka Publ, M., 1973, 238 pp. (In Russian)

[7] Tsirlin V. M., Balakirev V. S., Dudnikov E. G., Variational methods of controlled objects optimization, Energiya Publ, M., 1976, 448 pp. (In Russian)

[8] Fedorenko R. P., Numerical solution of optimal control problems, Nauka Publ, M., 1978, 488 pp. (In Russian)

[9] Bock H. G., “Numerical solution of nonlinear multipoint boundary value problems with applications to optimal control”, Z. Ang. Math. Mech., 58 (1978), 407–409 | MR

[10] Bock H. G., Plitt K. J., “A multiple shooting algorithm for direct solution of optimal control problems”, Proc. 9th IFAC World Congress (Budapest, Hungary, July 2–6, 1984), 242–247

[11] Krylov I. A., Chernous'ko F. L., “On successive approximations method for solving of optimal control problems”, Journal of Calculating Mathematics and Mathematical Physics, 2:6 (1962), 1132–1139 (In Russian) | MR | Zbl

[12] Krylov I. A., Chernous'ko F. L., “Algorithm of successive approximations method for optimal control problems”, Journal of Calculating Mathematics and Mathematical Physics, 12:1 (1972), 14–34 (In Russian) | MR

[13] Bryson A. E., Denham W. E., “A steepest ascent method for solving optimum programming problems”, ASME J. Appl. Mech. Series E, 29 (1962), 247–257 | DOI | MR | Zbl

[14] Gorbunov V. K., “On reduction of optimal control problems to finite-dimensional ones”, Journal of Calculating Mathematics and Mathematical Physics, 18:5 (1978), 1083–1095 (In Russian) | MR | Zbl

[15] Gorbunov V. K., “Method of parameterization of optimal control problems”, Journal of Calculating Mathematics and Mathematical Physics, 19:2 (1979), 292–303 (In Russian) | MR | Zbl

[16] Demyanov V. F., Rubinov A. M., Numerical methods of extremal control solving, Leningrad University Publ, L., 1968, 180 pp. (In Russian) | MR

[17] Vasil'ev F. P., Numerical methods of extremal control solving, Nauka Publ, M., 1988, 552 pp. (In Russian)

[18] Miele A., “Recent advances in gradient algoritms for optimal control problems”, J. Optimization Theory and Applications, 17:5/6 (1975), 361–430 | DOI | MR | Zbl

[19] Mitter S. K., “Successive approximation method for the solution of optimal control problems”, Automatica, 3 (1966), 135–149 | DOI | Zbl

[20] Longmuir A. G., Bohn E. V., “Second-variation methods in dynamic optimization”, J. Optimization Theory and Applications, 3:3 (1969), 164–173 | DOI | MR | Zbl

[21] Gabasov R., Kirillova F. M., Special optimal controls, Nauka Publ, M., 1973, 256 pp. (In Russian) | MR

[22] Gorbunov V. K., Lutoshkin I. V., “Development and experience of application of method of parameterization in degenerate problems of dynamical optimization”, Proceeding of Russian Academy of Sciences. Theory and System of Management, 2004, no. 5, 67–84 (In Russian)

[23] Golfetto W. A., Silva Fernandes S., “A review of gradient algorithms for numerical computations of optimal trajectories”, J. Aerosp. Technol. Manag., 4:2 (2012), 131–143 | DOI

[24] Henrion R., The theory of the second variation and its application in optimal control, Memories de la classe des sciences, 41, no. 7, Academie royale de Belgique, 1975, 208 pp. | MR

[25] Ovsyannikov D. A., Mathematical methods of beam control, Leningrad University Publ, L., 1980, 228 pp. (In Russian)

[26] Ovsyannikov D. A., Modeling and optimization of charged particle beam dynamics, Leningrad University Publ, L., 1990, 312 pp. (In Russian)

[27] Ovsyannikov D. A., Drivotin O. I., Modeling of intensive charged particle beams, Saint Petersburg University Publ, Saint Petersburg, 2003, 176 pp. (In Russian)

[28] Bublik B. N., Garashchenko F. G., Kirichenko N. F., Structural-parametrical optimization and stability of beam dynamics, Naukova dumka Publ, Kiev, 1985, 304 pp. (In Russian) | MR

[29] Nikol'skiy M. S., Belyaevskih E. A., “L. S. Pontryagin maximum principle for some problems of optimal control of trajectory bundles”, Vestnik of Saint Petersburg University. Applied Mathematics. Computer Science. Control Processes, 14:1 (2018), 59–68 (In Russian) | MR