Abstract

Control theory is a section of modern mathematics being actively developed at present time. The class of problems investigated within the framework of this theory is quite extensive and includes issues related to the existence of solutions as well as issues related to the effective methods for constructing controls. One of the approaches to solving control problems under lack of information was suggested by Yu.S. Osipov in the fundamental paper published in the Russian Mathematical Surveys in 2006. Later, this approach, called the method of program packages, was developed, in particular, in the articles cited in this paper. This approach is based on a suitable modification of the method of nonanticipatory strategies (quasistrategies) for solving control problems with unknown initial states. As is known, quasistrategies reflecting the Volterra properties of program realizations of closedloop controls in corresponding program disturbances are oriented to the investigation of problems with known initial states under the presence of unknown dynamical disturbances. Such disturbances are usually absent in standard control problems with incomplete information and incompleteness of information is due to a lack of information about the initial state of the system. So, program packages became an analogue of the properties of nonanticipativeness for problems with unknown initial states. It should be noted that in all previous works related to the method of program packages, the guidance problems to one single target set were considered.
In the present paper the guaranteed guidance problem to a collection of target sets under incomplete information about the initial state is considered for a linear autonomous control dynamical system. The criterion for the solvability of that problem is established. It is based on the method of program packages. An illustrative example is given.

References

 Krasovskii A.N., Krasovskii N.N. Control under lack of information, Boston: Birkhauser, 1995, 324 p.
 Krasovskii N.N., Subbotin A.I. Gametheoretical control problems, New York: Springer, 1988, 517 p. Original Russian text published in Krasovskii N.N., Subbotin A.I. Pozitsionnye differentsial'nye igry, Moscow: Nauka, 1974, 456 p.
 Osipov Yu.S. Control packages: an approach to solution of positional control problems with incomplete information, Russian Mathematical Surveys, 2006, vol. 61, issue 4, pp. 611661. DOI: 10.1070/RM2006v061n04ABEH004342
 Kryazhimskiy A.V., Osipov Yu.S. On the solvability of problems of guaranteeing control for partially observable linear dynamical systems, Proceedings of the Steklov Institute of Mathematics, 2012, vol. 277, pp. 144159. DOI: 10.1134/S0081543812040104
 Kryazhimskii A.V., Strelkovskii N.V. An openloop criterion for the solvability of a closedloop guidance problem with incomplete information. Linear control systems, Proceedings of the Steklov Institute of Mathematics, 2015, vol. 291, suppl. 1, pp. 113127. DOI: 10.1134/S0081543815090084
 Grigorenko N.L. A control problem with dominating uncertainty, Proceedings of the Steklov Institute of Mathematics, 2014, vol. 287, suppl. 1, pp. 6876. DOI: 10.1134/S0081543814090077
 Maksimov V.I. Differential guidance game with incomplete information on the state coordinates and unknown initial state, Differential Equations, 2015, vol. 51, issue 12, pp. 16561665. DOI: 10.1134/S0012266115120137
 Rozenberg V.L. A control problem under incomplete information for a linear stochastic differential equation, Proceedings of the Steklov Institute of Mathematics, 2016, vol. 295, suppl. 1, pp. 145155. DOI: 10.1134/S0081543816090157
 Surkov P.G. The problem of package guidance by a given time for a linear control system with delay, Proceedings of the Steklov Institute of Mathematics, 2017, vol. 296, suppl. 1, pp. 218227. DOI: 10.1134/S0081543817020201
 Maksimov V.I. Tracking a given solution of a nonlinear distributed secondorder equation, Differential Equations, 2016, vol. 52, issue 1, pp. 128132. DOI: 10.1134/S0012266116010110
 Strelkovskii N.V. Constructing a strategy for the guaranteed positioning guidance of a linear controlled system with incomplete data, Moscow University Computational Mathematics and Cybernetics, 2015, vol. 39, issue 3, pp. 126134. DOI: 10.3103/S0278641915030085
