S.M. Orlov, N.V. Strelkovskii. Calculation of elements of a guiding program package for singular clusters of the set of initial states in the package guidance problem ... P. 150-165

Vol. 25, no. 1, 2019

A fixed-time package guidance problem is considered for a linear controlled dynamical system with a finite set of initial states. The control set is convex and compact and the target set is convex and closed. The paper focuses on the case where the set of initial states has singular clusters for which the existing algorithm for estimating the elements of a guiding program package is not applicable. It is suggested to consider a perturbed problem of augmented program guidance with a smoothed control set. It is proved that the motions of the original and perturbed problems are close to each other at the terminal time; the corresponding estimates are provided. In the case of an augmented target set with nonempty interior, it is also shown that a solution of the augmented program guidance problem that is precisely guiding to the target set can be obtained by applying the existing algorithm for the perturbed problem with compressed target set. The suggested theoretical constructions are illustrated with a model example.

Keywords: incomplete information, linear dynamical system, guidance problem, program package, singular cluster, smooth approximation

Received November 23, 2018;  

Revised December 27, 2018;  

Accepted January 14, 2018

Funding Agency: This work was supported by the Russian Science Foundation (project no. 14-11-00539).

Sergei Mikhailovich Orlov, Cand. Sci. (Phys.-Math.), Lomonosov Moscow State University, Moscow, 119991 Russia; Research Scholar, International Institute for Applied Systems Analysis, Laxenburg, 2361 Austria, e-mail: sergey.orlov@cs.msu.ru

Nikita Vital’evich Strelkovskii. Cand. Sci. (Phys.-Math.), International Institute for Applied Systems Analysis, Laxenburg, 2361 Austria; Lomonosov Moscow State University, Moscow, 119991 Russia, e-mail: strelkon@iiasa.ac.at

Cite this article as: S.M. Orlov, N.V. Strelkovskii. Calculation of elements of a guiding program package for singular clusters of the set of initial states in the package guidance problem, Trudy Instituta Matematiki i Mekhaniki URO RAN, 2019, vol. 25, no. 1, pp.  150-165. 

REFERENCES

1.   Osipov Yu.S. Control packages: an approach to solution of positional control problems with incomplete information. Russian Math. Surveys, 2006, vol. 61, no. 4, pp. 611–661. doi: 10.4213/rm1760 

2.   Kryazhimskiy A.V., Osipov, Y.S. Idealized program packages and problems of positional control with incomplete information. Proc. Steklov Inst. Math., 2010, vol. 268, suppl. 1, pp. 155–174. doi: 10.1134/S0081543810050123 

3.   Krasovskii N.N., Subbotin A.I. Game-theoretical control problems. N Y: Springer, 1987. 517 p. This book is substantially revised version of the monograph Pozitsionnye differentsial’nye igry, Moscow, Nauka Publ., 1974, 456 p.

4.   Kryazhimskiy A.V., Osipov Yu.S. On the solvability of problems of guaranteeing control for partially observable linear dynamical systems. Proc. Steklov Inst. Math., 2012, vol. 277, pp. 144–159. doi: 10.1134/S0081543812040104 

5.   Kryazhimskii A.V., Strelkovskii N.V. An Open-loop criterion for the solvability of a closed-loop guidance problem with incomplete information: Linear control systems. Proc. Steklov Inst. Math., 2015, vol. 291, suppl. 1, pp. 113–127. doi: 10.1134/S0081543815090084 

6.   Strelkovskii N.V. Constructing a strategy for the guaranteed positioning guidance of a linear controlled system with incomplete data. Moscow Univ. Comput. Math. Cybern., 2015, vol. 39, no. 3, pp. 126–134. doi: 10.3103/S0278641915030085 

7.   Surkov P.G. The problem of package guidance with incomplete information for a linear control system with a delay. Comput Math Model, 2017, vol. 28, no. 4, pp. 504–516. doi: 10.1007/s10598-017-9377-y 

8.   Maksimov V.I., Surkov P.G. On the solvability of the problem of guaranteed package guidance to a system of target sets. Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 2017, vol. 27, no. 3, pp. 344–354 (in Russian). doi: 10.20537/vm170305 

9.   Surkov P.G. The problem of package guidance under incomplete information and integral signal of observation. Sib. Elektron. Mat. Izv., 2018, vol. 15, pp. 373–388 (in Russian). doi: 10.17377/semi.2018.15.034 

10.   Strelkovskii N.V., Orlov S.M. Algorithm for constructing a guaranteeing program package in a control problem with incomplete information. Moscow Univ. Comput. Math. Cybern., 2018, vol. 42, no. 2, pp. 69–79. doi: 10.3103/S0278641918020061 

11.   Strelkovskii N.V., Orlov S.M. A method for calculation of program package elements for singular clusters. Proc. Int. Conf. “Systems Analysis: Modeling and Control” in memory of Academician A.V. Kryazhimskiy (Moscow, Russia, 2018), pp. 97–99. doi: 10.4213/proc20606 

12.   Gindes V.B. Singular control in optimal systems. Izv. Vyssh. Uchebn. Zaved. Mat., 1967, no. 7, pp. 34–42 (in Russian).

13.   Avvakumov S.N. Smooth approximation of convex compact sets. Trudy Inst. Mat. i Mekh. UrO RAN, 1996, vol. 4, pp. 184–200 (in Russian).

14.   Kiselev Yu.N., Avvakumov S.N., Orlov M.V. Optimal’noe upravlenie. Lineinaya teoriya i prilozheniya. [Optimal control. Linear theory and applications]. Moscow: MAKS Press, 2007, 272 p. ISBN: 5-89407-288-3 .

15.   Pontryagin L.S. Linear differential games. Proc. Steklov Inst. Math., 1990, vol. 185, pp. 225–232.

16.   Avvakumov S.N., Kiselev Yu.N., Orlov M.V., Methods of solving optimal control problems based on the Pontryagin maximum principle. Proc. Steklov Inst. Math., 1995, vol. 211, pp. 1–27.

17.   Avvakumov S.N., Kiselev Yu.N. Support functions of some special sets, constructive smoothing procedures, and geometric difference. In: Osipov Yu.S. (ed.) et al., Problems in dynamic control. No. 1. Moscow: Mosk. Gos. Univ., Fak. Vych. Mat. Kib. Publ., 2005, pp. 24–110. ISBN: 5-89407-241-7 .