MATHEMATICS
On the construction of partially non-anticipative multiselector and its application to dynamic optimization problems
D. A. Serkovab a Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, ul. S. Kovalevskoi, 16, Yekaterinburg, 620990, Russia
b Ural Federal University, ul. Mira, 19, Yekaterinburg, 620062, Russia
Abstract:
Let sets of functions
$Z$ and
$\Omega$ on the time interval
$T$ be given, let there also be a multifunction (m/f)
$\alpha$ acting from
$\Omega$ to
$Z$ and a finite set
$\Delta$ of moments from
$T$. The work deals with the following questions: the first one is the connection between the possibility of stepwise construction (specified by
$\Delta$) of a selector
$z$ of
$\alpha(\omega)$ for an unknown step-by-step implemented argument
$\omega\in\Omega$ and the existence of a multiselector (m/s)
$\beta$ of the m/f
$\alpha$ with a non-anticipatory property of special kind (we call it partially or
$\Delta$-non-anticipated); the second question is when and how non-anticipated m/s could be expressed by means of partially non-anticipated one; and the last question is how to build the above
$\Delta$-non-anticipated m/s
$\beta$ for a given pair
$(\alpha,\Delta)$.
The consideration of these questions is motivated by the presence of such step-by-step procedures in the differential game theory, for example, in the alternating integral method, in pursuit–evasion problems posed with use of counter-strategies, and in the method of guide control.
It is shown that the step-by-step construction of the value
$z\in\alpha(\omega)$ can be carried out for any steps-implemented argument
$\omega$ if and only if the above m/s
$\beta$ is non-empty-valued. The key point of the work is the description of finite-step procedure for calculation of this
$\Delta$-non-anticipated m/s
$\beta$. Conditions are given that guarantee the m/s
$\beta$ be a non-anticipative one. Illustrative examples are considered that include, in particular, control problems with disturbance.
Keywords:
non-anticipative multi-selectors, set-valued strategies, optimization of guarantee
UDC:
517.977
MSC: 49N70,
54C65 Received: 19.07.2024
Accepted: 26.08.2024
Language: English
DOI:
10.35634/vm240307
Fulltext:
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