On the construction of partially non-anticipative multiselector and its application to dynamic optimization problems

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.