ISSN 0278-6419, Moscow University Computational Mathematics and Cybernetics, 2018, Vol. 42, No. 2, pp. 69–79.
Allerton Press, Inc., 2018.
Original Russian Text
N.V. Strelkovskii, S.M. Orlov, 2018, published in Vestnik Moskovskogo Universiteta, Seriya 15: Vychislitel’naya Matematika i Kibernetika,
2018, No. 2, pp. 20–30.
Algorithm for Constructing a Guaranteeing Program Package
in a Control Problem with Incomplete Information
N. V. Strelkovskii
IIASA, Laxenburg, Austria
Faculty of Computational Mathematics and Cybernetics,
Moscow State University, Moscow, 119991 Russia
Received December 2, 2016
Abstract—A package control problem is considered for a target set at a moment of time. The
dynamic system under control is described by linear diﬀerential equations, the control area is a
convex compact, and the target set is convex and closed. A version of the subsequent approximations
method in extended space is proposed for constructing elements of a guaranteeing program package
in the case of regular clusters.
Keywords: incomplete information, linear dynamic systems, control problem for a target set,
subsequent approximations method.
The program packages method was proposed in [1, 2] as an instrument for studying position control
problems with incomplete information. It was developed for linear dynamic systems [3–7]. This method
was naturally included in the guaranteeing control theory developed by N.N. Krasovskii et al. .
The method is based on the statement on equivalence of guaranteeing control problems in the class
of positional strategies, and in the class of program packages. A program package is a family of
nonanticipatory program responses for all admissible initial states.
The authors of [4, 6] studied a class of problems in which the controlled system and the signal were
linear, the set of admissible initial states was ﬁnite, and the target set was convex and closed. In ,
there was a natural generalization of the program packages method to a compact set of initial points
for a linear system with delay. For the sake of simplicity, in this work we assume there is a ﬁnite set of
initial points and no delay in the linear dynamic system. The obtained algorithm can be generalized to
a compact initial set using ﬁnite approximation ε-nets. In , it was established that the guaranteeing
control problem in the class of program packages is equivalent to the extended program control problem
for a target set, and a solvability criterion for the problem was proved. In , a way of searching for
a guaranteeing program package was proposed, based on compressing the control domain using an
apparatus of support functions and the hyperplane separation theorem. In this method, it was required
to ﬁnd the compression coeﬃcient of the control domain from a complicated equation containing the
operation of ﬁnding maximum of a function of a large number of variables over a bounded compact set.
In this work, we produce an algorithm that is a version of the subsequent approximations method
which allows us to ﬁnd the compression coeﬃcient of the control domain and simultaneously construct
a guaranteeing program package.