[1,2]Dmitrii Serkov

[1]Krasovskii Institute of Mathematics and Mechanics, S. Kovalevskaya Str.,
16, Ekaterinburg, 620990, RF

2]Ural Federal University, Mira Str., 32, Ekaterinburg, 10587,RF

# On a construction of a partially non-anticipative multiselector and its
applications to dynamic optimization problems

<EMAIL_ADDRESS><EMAIL_ADDRESS>*  [

###### Abstract

Let sets of functions $\mathrm{Z}$ and $\Omega$ on the time interval
$\mathrm{T}$ be given, let there also be a multifunction (m/f) $\alpha$ acting
from $\Omega$ to $\mathrm{Z}$ and a finite set $\Delta$ of moments from
$\mathrm{T}$. The work deals with two questions: the first one is the
connection between the possibility of stepwise construction (specified by
$\Delta$) of a value $z$ of $\alpha(\omega)$ for an unknown step-by-step
implemented argument $\omega\in\Omega$ and the existence of a multiselector
$\beta$ of the m/f $\alpha$ with a non-anticipatory property of special kind
defined by $\Delta$; and the second question is how to build the above $\beta$
for a given pair $(\alpha,\Delta)$. The consideration of these questions is
motivated by the presence of similar 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 in steps implemented
argument $\omega$ if and only if the multiselector $\beta$ is non-empty-
valued. In this case, the desired value $z$ can be selected from
$\beta(\omega)$ in step-by-step procedure for any unknown in advance argument
$\omega$. The key point of the work is the procedure for calculation the
multiselector $\beta$, for which a constructive and finite-step description is
given. Illustrative examples are considered that include, in particular,
problems of a guaranteed result optimization under functional constraints on
control and/or disturbance implementations.

###### keywords:

control problem, disturbances, optimal guaranteed result, step-by-step
procedures, non-anticipative strategies, non-anticipative multi-selectors.

## Introduction

Let there be non-empty sets of functions $\mathrm{Z}$ and $\Omega$ defined on
the interval $\mathrm{T}$. Let there also be a multifunction (m/f) $S$ defined
on $\Omega$ with values in $\mathrm{Z}$ and a set $N\subset\mathrm{Z}$. We
will interpret these objects as an abstract control problem under conditions
of uncertainty (namely, an abstract game problem of retention): the
uncontrolled factor $\omega$ from the set $\Omega$, acting on the dynamics $S$
of the system, determines the bundle of possible trajectories
$S(\omega)\subset\mathrm{Z}$; the task of the control side is to select the
motion $z\in S(\omega)$ satisfying the phase constraints $N$: $z\in N$.

Without additional informational requirements, the solution of the problem is
built explicitly: we put $\alpha(\omega)\mathrel{\triangleq}N\cap S(\omega)$,
$\omega\in\Omega$; then the criterion for the solvability of the problem is
the non-emptiness of the m/f $\alpha$ values, and the solution is any of its
selectors. So, if we knew the current disturbance $\bar{\omega}\in\Omega$, the
retention problem would be solved by calling any trajectory $z$ of
$\alpha(\bar{\omega})$.

At the same time, in most control problems, information about the acting
disturbance is not available at all. In the rest cases, the best that the
control side can count on by the time $\tau\in\mathrm{T}$ is the knowledge of
the disturbance on the interval $[t_{0},\tau+\delta]$ with a small $\delta>0$
($t_{0}$ is the initial moment of the process). Such a prediction of
disturbance behavior is admissible in certain control problems, as well as in
auxiliary control structures, due to properties of the system dynamics (the
mapping $S$) and restrictions describing the set of admissible disturbances
$\Omega$.

Thus, we suppose that the control side has the m/f $\alpha$ of global
responses and the possibility of a small time-ahead disturbance prediction.
Under these conditions, the following step-by-step scheme of the desired
trajectory constructing naturally arises.

Let $\mathrm{T}\mathrel{\triangleq}[t_{0},\vartheta]$ and $\Delta$,
$\Delta\subset\mathrm{T}$, be a finite set of instants that splits
$\mathrm{T}$ into a finite set of half-intervals (control steps) of length
shorter then the above value $\delta$:
$\Delta\mathrel{\triangleq}\\{t_{0}=\tau_{0}<\tau_{1}<\ldots<\tau_{n_{\Delta}}=\vartheta\\}$.
We call $\Delta$ _the partition_ of $\mathrm{T}$ and the pair
$(\alpha,\Delta)$ will be called _step-by-step conditions_. The procedure of
step-by-step construction of the trajectory that meets the conditions
$(\alpha,\Delta)$ and the unknown disturbance $\bar{\omega}\in\Omega$ works as
follows:

— at the instant $\tau_{0}$, by the found disturbance $\omega_{1}\in\Omega$
such that
$(\omega_{1}|[\tau_{0},\tau_{1}))=(\bar{\omega}|[\tau_{0},\tau_{1}))$, the
control side chooses the trajectory $h_{1}\in\mathrm{Z}$ corresponding to the
disturbance $\omega_{1}$, i.e. such that $h_{1}\in\alpha(\omega_{1})$; here,
$(f|C)$ is the restriction of a function $f$ to the set $C$;

— at any instant $\tau_{i}\in\Delta$, $i\in{{1}..{(n_{\Delta}-1)}}$, the
control side finds a disturbance $\omega_{i+1}\in\Omega$ reconstructing the
unknown $\bar{\omega}$ up to the moment $\tau_{i+1}$:

$(\omega_{i+1}|[\tau_{0},\tau_{i+1}))=(\bar{\omega}|[\tau_{0},\tau_{i+1}));$
(1)

and, so, $\omega_{i+1}$ coincides with $\omega_{i}$ up to the moment
$\tau_{i}$

$(\omega_{i+1}|[\tau_{0},\tau_{i}))=(\omega_{i}|[\tau_{0},\tau_{i})).$ (2)

At the previous steps of partition $\Delta$, we have already supplied the
desired trajectory $h_{i}$ for the disturbance $\omega_{i}$, i.e.
$h_{i}\in\alpha(\omega_{i})$; taking this into account, the control side looks
for the trajectory $h_{i+1}\in\alpha(\omega_{i+1})$ corresponding to the
disturbance $\omega_{i+1}$, that also coincides with the choice at the
previous steps:

$(h_{i+1}|[\tau_{0},\tau_{i}))=(h_{i}|[\tau_{0},\tau_{i})).$

The procedure is repeated for all moments of the partition $\Delta$ except the
last one, $\tau_{n_{\Delta}}$. As a result, we get the desired trajectory
$h_{n_{\Delta}}$ that corresponds to the unknown in advance disturbance
$\omega_{n_{\Delta}}=\bar{\omega}$:

$h_{n_{\Delta}}\in\alpha(\omega_{n_{\Delta}})=\alpha(\bar{\omega}).$

The possibility of realization the above step-by-step procedure in response to
any admissible disturbance $\omega\in\Omega$ will be referred to as _the
feasibility of the conditions $(\alpha,\Delta)$_.

The work deals with two questions. The first one is the connection between the
feasibility of conditions $(\alpha,\Delta)$ and the existence of some special
multi-selector of the m/f $\alpha$. The second question is the construction of
this multiselector for given conditions $(\alpha,\Delta)$.

The motivation for the consideration was the above step-by-step scheme and
similar ones, which starting from convergence problem [1, sec. III] in theory
of differential games arise, for example, in the method of alternating
integral [2], in pursuit — evasion problems using counter-strategies [3, 4,
5], or in controlling with a guide under functional constraints on a
disturbance (see [6, 7] and references). Besides game-theoretical problems,
the above informational conditions for control side can be found in the field
of robotics: suppose a robot-manipulator extracts from the container and
submits for the further processing some parts poured into it. In this case the
disturbance/uncertainty is the arrangement of the parts in the container. It
changes (unpredictably as a rule) after the extraction of a part and remains
practically constant/unchanged during inactivity time. So, the disturbance can
be effectively predicted.

It is clear that the non-anticipatory and non-emptiness of values of $\alpha$
or of its multiselector implies feasibility of $(\alpha,\Delta)$ for any
$\Delta$. Thus, the conditions for the existence of non-anticipative
multiselectors (see, for example, [8, 9, 10]) are sufficient conditions for
the feasibility property. Note that the existence of a non-anticipative
selector [11, 12, 13] is also closely related to the existence of a non-
anticipative multiselector.

In this paper, we show that the feasibility of the conditions
$(\alpha,\Delta)$ is equivalent to the existence of a _partially non-
anticipative_ and non-empty-valued multiselector of the m/f $\alpha$: the
above step-by-step process can be implemented by means of this partially non-
anticipative multiselector for any disturbance. Here, the property of partial
non-anticipatory is understood as the classical non-anticipatory property that
is satisfied at moments from $\Delta$ only. This property is certainly weaker
than the classical one when it should be satisfied for all moments from
$\mathrm{T}$. Moreover, even the feasibility of the conditions
$(\alpha,\Delta)$ for any $\Delta$ does not in general ensure the existence of
a non-anticipative non-empty-valued multiselector of $\alpha$.

So, to implement the above step-by-step procedure under conditions
$(\alpha,\Delta)$, we need this partially non-anticipative and non-empty-
valued multiselector of $\alpha$. Here arises the second question: how to
build such a partially non-anticipative multiselector.

With the aim, for any instant $\tau\in\mathrm{T}$ we introduce a "projection"
operator acting on set of all m/f with values in the set of all $\tau$-non-
anticipative m/f. Then, by means of a superposition of such "projections"
(corresponding to all $\tau\in\Delta$), we get the required partially non-
anticipative multiselector. The procedure is completed in $n_{\Delta}$ steps.

It seems the idea of constructing a non-anticipative multiselector of a m/f by
an iterative method in order to obtain a direct solution of the dynamic
optimization problem under conditions of an uncertainty appeared in [14] (see
also [8, 9]). The obstacle for applications of such an iterative procedure is
that it requires infinite number of iterations in the general case. For some
classes of control problems (see, for example, [15, ch. 5]), conditions are
given that ensure the finiteness of iterations required to construct the
function of the optimal guaranteed result, a resolving set of initial
positions, or a resolving non-anticipative strategy.

The work is close to the problems considered in [7]; the constructions used
are similar to those from [10] and the results supplements the results
announced in [16]. In simple cases, the connection of partially non-
anticipative multiselectors with ordinary non-anticipative multiselectors was
noted in the course of the presentation, but is not covered in detail. The
outline of the article is as follows: Sec. 1 contains basic notation and
terms; in Sec. 2, a more detailed and meaningful description of the step-by-
step procedure for constructing an optimal trajectory, its formalization, and
the definition of the feasibility of the $(\alpha,\Delta)$ conditions are
given (see (10), (11)); in Sec. 3, the notion of a partially non-anticipative
m/f (multiselector) is defined and the feasibility criterion (21) is
formulated in its terms; in Sec. 4, the above-mentioned "projection" operator
on the set of all $\tau$-non-anticipative m/f is defined (see (28)); then, the
definition of such "projection" operator on the set of m/f that are non-
anticipative for several such "moments" (see (44)) is given and the finite
step procedure for its constructing is provided (see Theorem 2); in Sec. 5,
illustrative examples are given: in Example 5.2, it is shown that the
"projection" operators are not commutative; Example 5.3 provides the case when
the feasibility of conditions $(\alpha,\Delta)$ for arbitrary $\Delta$ do not
allow, nevertheless, a resolving non-anticipative strategy (a non-anticipative
non-empty-valued multiselector of $\alpha$); in Example 5.4, an ordinal
resolving non-anticipative strategy is constructed using the proposed technic.
in Sec. 6, open formal questions and prospects for using this approach in
applications are briefly discussed.

##  1 Definitions and notation

In the following, set-theoretic symbolism is used (quantifiers, propositional
connectives, etc.); hereinafter $\varnothing$ — empty set,
$\mathrel{\triangleq}$ — equality by definition,
$\stackrel{{\scriptstyle\mathrel{\text{def}}}}{{\mathrel{\Leftrightarrow}}}$ —
equivalency by definition; a family is a set all of whose elements are sets.

Let $\mathcal{P}({X})$ and $\mathcal{P^{\prime}}({X})$ denote respectively the
families of all (Boolean of $X$) and of all non-empty subsets of an arbitrary
set $X$. If $A$ and $B$ are non-empty sets, then we denote by $B^{A}$ the set
of all mappings from $A$ to $B$. If $g\in\mathcal{P}({B})^{A}$, then by
$(\mathrm{DOM})\left[{g}\right]$ we denote the region where the m/f $g$ takes
nonempty values: $(\mathrm{DOM})\left[{g}\right]\mathrel{\triangleq}\\{a\in
A\mid g(a)\neq\varnothing\\}$. If $f\in B^{A}$ and
$C\in\mathcal{P^{\prime}}({A})$, then $(f|C)$, where $(f|C)\in B^{C}$, is the
restriction of $f$ to the set $C$: $(f|C)(x)\mathrel{\triangleq}f(x)$ $\forall
x\in C$; in the case when $F\in\mathcal{P^{\prime}}({B^{A}})$, we set
$\llparenthesis F|C\rrparenthesis\mathrel{\triangleq}\\{(f|C)\colon f\in
F\\}$.

We call a pair $(X,\mathrel{\prec})$ _partially ordered set_ (poset) if $X$ is
a non-empty set and ${\mathrel{\prec}}\in\mathcal{P}({X\times X})$ is a non-
strict partial order relation on $X$. In particular,
$(\mathcal{P}({X}),\subset)$ is a poset on Boolean of $X$ with the inclusion
relation ${\subset}$. For any poset $(X,\mathrel{\prec})$ and arbitrary subset
$S\in\mathcal{P}({X})$ we call $S$ a _chain_ if any elements of $S$ are
comparable: $(x\mathrel{\prec}y)\lor(y\mathrel{\prec}x)$, $\forall x,y\in S$.
If $(X,\mathrel{\prec})$ is a poset and $M\subset X$, denote $\top_{M}$,
$\top_{M}\in M$, the greatest element of $M$, if it exists:
$x\mathrel{\prec}\top_{M}$ for all $x\in M$.

We fix non-empty sets $\mathrm{X}$, $\mathrm{Y}$ and $\mathrm{T}$, as well as
non-empty sets $\Omega\in\mathcal{P^{\prime}}({\mathrm{Y}^{\mathrm{T}}})$,
$\mathrm{Z}\in\mathcal{P^{\prime}}({\mathrm{X}^{\mathrm{T}}})$ and the family
$\mathcal{T}\in\mathcal{P^{\prime}}({\mathcal{P^{\prime}}({\mathrm{T}})})$; in
other words, $\mathcal{T}$ is a non-empty family of non-empty subsets of
$\mathrm{T}$. In order to reveal the connection with dynamic optimization
problems, we will assume that $\mathrm{T}$ is a "time interval":
$\mathrm{T}\mathrel{\triangleq}[t_{0},\vartheta]$,
$t_{0},\vartheta\in\mathbb{R}$ ; and $\mathcal{T}$ is a chain of the form
$\mathcal{T}\mathrel{\triangleq}\\{[t_{0},\tau]:\tau\in\mathrm{T}\\}$. Then
$\Omega$ is a set of admissible disturbances, $\mathrm{Z}$ is a set of
possible system trajectories.

Let $\mathrel{\sqsubseteq}$ be a partial order on the set mappings
$\mathcal{P}({\mathrm{Z}})^{\Omega}$:
$\forall\alpha,\beta\in\mathcal{P}({\mathrm{Z}})^{\Omega}$

$(\alpha\mathrel{\sqsubseteq}\beta)\stackrel{{\scriptstyle\mathrel{\text{def}}}}{{\mathrel{\Leftrightarrow}}}(\alpha(\omega)\subset\beta(\omega)\
\forall\omega\in\Omega).$ (3)

For $\omega\in\Omega$, $z\in\mathrm{Z}$ and
$A\in\mathcal{P^{\prime}}({\mathrm{T}})$ we denote

$\Omega(\omega|A)\mathrel{\triangleq}\\{\eta\in\Omega\mid(\eta|A)=(\omega|A)\\},\qquad\mathrm{Z}(z|A)\mathrel{\triangleq}\\{g\in\mathrm{Z}\mid(g|A)=(z|A)\\}.$

Note that the family of partitions
$(\\{\Omega(\omega|A)\mid\omega\in\Omega)_{A\in\mathcal{T}}$ forms a chain in
the set of partitions of $\Omega$ with an embedding relation (see, for
example, [17, Sec. 3.1]), namely, for any $A,A^{\prime}\in\mathcal{T}$, we
have

$(A\subset
A^{\prime})\mathrel{\Rightarrow}((\Omega(\omega|A^{\prime})\subset\Omega(\omega|A)\
\forall\omega\in\Omega).$ (4)

Here by _partition_ of a set $M$ we call a family of its nonempty mutually
disjont subsets that cover the set.

In terms of the family $\mathcal{T}$ we define the basic notion of _non-
anticipatory_. Denote by $\mathbf{N}_{(\mathcal{T})}$ the set of all _non-
anticipative_ m/f from ${\mathcal{P}({\mathrm{Z}})}^{\Omega}$:

$\mathbf{N}_{(\mathcal{T})}\mathrel{\triangleq}\left\\{{\bf
z}\in\mathcal{P}({\mathrm{Z}})^{\Omega}\mid\llparenthesis{\bf
z}(\omega)|A\rrparenthesis=\llparenthesis{\bf
z}(\omega^{\prime})|A\rrparenthesis\ \forall A\in\mathcal{T}\
\forall\omega\in\Omega\ \forall\omega^{\prime}\in\Omega(\omega|A)\right\\}.$

By $\mathbf{N}_{(\mathcal{T})}^{0}$ we denote the subset of all nonempty-
valued m/f:

$\mathbf{N}_{(\mathcal{T})}^{0}\mathrel{\triangleq}\\{{\bf
z}\in\mathbf{N}_{(\mathcal{T})}\mid(\mathrm{DOM})\left[{{\bf
z}}\right]=\Omega\\}.$

The most important are multiselectors of a given m/f: for
$\alpha\in\mathcal{P}({\mathrm{Z}})^{\Omega}$ we denote

$\mathbf{N}_{(\mathcal{T})}[{\alpha}]\mathrel{\triangleq}\\{{\bf
z}\in\mathbf{N}_{(\mathcal{T})}\mid{\bf z}\mathrel{\sqsubseteq}\alpha\\};$ (5)

then we define the subset of nonempty-valued multiselectors of m/f $\alpha$:

$\mathbf{N}_{(\mathcal{T})}^{0}[{\alpha}]\mathrel{\triangleq}\\{{\bf
z}\in\mathbf{N}_{(\mathcal{T})}^{0}\mid{\bf z}\mathrel{\sqsubseteq}\alpha\\}.$
(6)

Let us introduce (see [18, § 3, item V]) pointwise defined "union" and
"intersection" of m/f from $\mathcal{P}({\mathrm{Z}})^{\Omega}$. Let
$\mathbb{M}$ be a subset of $\mathcal{P}({\mathrm{Z}})^{\Omega}$. We assume
that the pointwise union $\bigvee_{{\bf z}\in\mathbb{M}}{{\bf z}}$ and the
pointwise intersection $\bigwedge_{{\bf z}\in\mathbb{M}}{{\bf z}}$, for each
$\omega\in\Omega$ are determined by:

$\bigvee_{{\bf z}\in\mathbb{M}}{\bf
z}(\omega)\mathrel{\triangleq}\bigcup_{{\bf z}\in\mathbb{M}}{{\bf
z}}(\omega),\qquad\bigwedge_{{\bf z}\in\mathbb{M}}{\bf
z}(\omega)\mathrel{\triangleq}\bigcap_{{\bf z}\in\mathbb{M}}{{\bf
z}}(\omega).$ (7)

Note that in the form $\bigvee_{{\bf z}\in\mathbb{M}}{\bf z}$ and
$\bigwedge_{{\bf z}\in\mathbb{M}}{\bf z}$ we have, respectively, supremum and
infimum of the set $\mathbb{M}$ in the poset
$(\mathcal{P}({\mathrm{Z}})^{\Omega},\mathrel{\sqsubseteq})$. Unlike pointwise
intersection, the result of pointwise union inherits non-anticipatory property
of operands (see lemma 1 below).

##  2 Feasibility of step-by-step procedure

Let us define the property of feasibility in formal terms. Let $\Delta$,
$\Delta\subset\mathrm{T}$, be a finite set of instants:
$\Delta\mathrel{\triangleq}\\{t_{0}=\tau_{0}<\tau_{1}<\ldots<\tau_{n_{\Delta}}=\vartheta\\}$.
We will also refer to $\Delta$ as the partition of the time interval
$\mathrm{T}$. Denote by ${\cal H}_{\Delta}$ the subset of $\mathcal{T}$ of the
form ${\cal
H}_{\Delta}\mathrel{\triangleq}\\{H_{i}\mathrel{\triangleq}[\tau_{0},\tau_{i}]:i\in{{1}..{n_{\Delta}}}\\}$.

For partition $\Delta$ and a m/f $\alpha\in\mathcal{P}({\mathrm{Z}})^{\Omega}$
denote by $\Omega_{\Delta}$ ($\Omega_{\Delta}\subset\Omega^{n_{\Delta}}$) and
$\mathrm{Z}_{\Delta}$
($\mathrm{Z}_{\Delta}\subset{\mathcal{P}({\mathrm{Z}})}^{n_{\Delta}}$) the
sets defined as follows (in expressions like $Q^{n}$, where $n\in\mathbb{N}$,
the number $n$ is not considered as a set):

$\Omega_{\Delta}\mathrel{\triangleq}\\{(\omega_{i})_{i\in{{1}..{n_{\Delta}}}}\in\Omega^{n_{\Delta}}\mid(\omega_{i}|H_{i})=(\omega_{i+1}|H_{i}),\
i\in{{1}..{n_{\Delta}-1}}\\},$ (8)
$\mathrm{Z}_{\Delta}\mathrel{\triangleq}\\{({\beta}_{i})_{i\in{{1}..{n_{\Delta}}}}\in{\mathcal{P^{\prime}}({\mathrm{Z}})}^{n_{\Delta}}\mid\llparenthesis{\beta}_{i}|H_{i}\rrparenthesis=\llparenthesis{\beta}_{i+1}|H_{i}\rrparenthesis,\
i\in{{1}..{n_{\Delta}-1}}\\}.$ (9)

Conditions $(\alpha,\Delta)$ for the step-by-step procedure we call feasible
if there exists a tuple of multiselectors
$(\phi_{i})_{i\in{{1}..{n_{\Delta}}}}$ of m/f $\alpha$,

$\phi_{i}\in\mathcal{P}({\mathrm{Z}})^{\Omega},\quad\phi_{i}\mathrel{\sqsubseteq}\alpha,\qquad
i\in{{1}..{n_{\Delta}}},$ (10)

such that the following inclusions are fulfilled:

$(\phi_{i}(\omega_{i}))_{i\in{{1}..{n_{\Delta}}}}\in\mathrm{Z}_{\Delta}\qquad\forall(\omega_{i})_{i\in{{1}..{n_{\Delta}}}}\in\Omega_{\Delta}.$
(11)

This definition simply retells the above informal description of the step-by-
step procedure in mathematical terms. Namely, let given a tuple
$(\phi_{i})_{i\in{{1}..{n_{\Delta}}}}$ satisfying (10), (11), then the
construction of a step-by-step response to an unknown disturbance
$\bar{\omega}$, due to the $(\alpha,\Delta)$ conditions, can be realised like
this:

— we find $\omega_{1}\in\Omega$ such that
$(\omega_{1}|H_{1})=(\bar{\omega}|H_{1})$ and choose $h_{1}\in\mathrm{Z}$ that
satisfies $h_{1}\in\phi_{1}(\omega_{1})$. This can always be done, because by
completing formally $\omega_{1}$ to an arbitrary tuple
$(\omega_{1},\ldots,\omega_{n_{\Delta}})$ from $\Omega_{\Delta}$ (for example,
by setting
$\omega_{n_{\Delta}}\mathrel{\triangleq}\ldots\mathrel{\triangleq}\omega_{2}\mathrel{\triangleq}\omega_{1}$),
due to (9) and (11) we get the inequality
$\phi_{1}(\omega_{1})\neq\varnothing$ which enables us to choose the desired
trajectory $h_{1}$;

— if there are $(\omega_{i})_{i\in{{1}..{k+1}}}\in\Omega^{k+1}$ and
$h_{k}\in\mathrm{Z}$ such that

$\displaystyle(\omega_{k+1}|H_{k+1})=(\bar{\omega}|H_{k+1}),$ (12)
$\displaystyle(\omega_{i+1}|H_{i})=(\omega_{i}|H_{i}),\qquad i\in{{1}..{k}}$
(13) $\displaystyle h_{i}\in\phi_{i}(\omega_{i}),\qquad i\in{{1}..{k}}$ (14)

we choose $h_{k+1}$ from the condition

$h_{k+1}\in\phi_{k+1}(\omega_{k+1})\cap\mathrm{Z}(h_{k}|H_{k}).$

Such a choice is possible: the tuple $(\omega_{1},\ldots,\omega_{k+1})$ can be
completed (see (13), (8)) to a tuple $(\omega_{1},\ldots,\omega_{n_{\Delta}})$
from $\Omega_{\Delta}$ (for example, by setting
$\omega_{n_{\Delta}}\mathrel{\triangleq}\ldots\mathrel{\triangleq}\omega_{k+2}\mathrel{\triangleq}\omega_{k+1}$).
Then, by virtue of (9), (11) we get the equality

$(\phi_{k+1}(\omega_{k+1})|H_{k})=(\phi_{k}(\omega_{k})|H_{k}),$

from which, taking into account (14), we get
$\phi(\omega_{k+1})\cap\mathrm{Z}(h_{k}|H_{k})\neq\varnothing$. Hence, the
choice is possible to undertake. So, by induction, the step-by-step procedure
can be continued up to $k+1=n_{\Delta}$.

It is clear that the non-emptiness of values and non-anticipatory property of
$\alpha$ (that is, when $\alpha\in\mathbf{N}_{(\mathcal{T})}^{0}$) or, in
general, the existence of a non-empty-valued non-anticipative multiselector
$\beta$ of m/f $\alpha$ implies the feasibility of the conditions
$(\alpha,\Delta)$ for any partition $\Delta$. Indeed, then it suffices to put
$\phi_{i}\mathrel{\triangleq}\beta$, where
$\beta\in\mathbf{N}_{(\mathcal{T})}^{0}[{\alpha}]$. At the same time, it can
be seen from the example 5.3 that the feasibility of the conditions
$(\alpha,\Delta)$ even for all partitions $\Delta$ do not imply existence of
non-anticipative and non-empty-valued multiselector of $\alpha$. In following
sections, we consider the property of partial non-anticipatory of a m/f that
is equivalent to the feasibility property.

##  3 Partially non-anticipative mappings: basic properties

For an arbitrary $A\in\mathcal{T}$, we introduce the notion of _$A$ -non-
anticipative_ m/f: a mapping
$\mathbf{z}\in{\mathcal{P}({\mathrm{Z}})}^{\Omega}$ is called $A$-non-
anticipative if the implication

$\left((\omega_{1}|A)=(\omega_{2}|A)\right)\mathrel{\Rightarrow}\left(\llparenthesis\mathbf{z}(\omega_{1})|A\rrparenthesis=\llparenthesis\mathbf{z}(\omega_{2})|A\rrparenthesis\right)$

is true for all $\omega_{1},\omega_{2}\in\Omega$. The subset of all $A$-non-
anticipative m/f we denote by $\mathbf{N}_{(\\{A\\})}$. The subset of all non-
empty-valued and $A$-non-anticipative m/f we denote by
$\mathbf{N}^{0}_{(\\{A\\})}$:
$\mathbf{N}^{0}_{(\\{A\\})}\mathrel{\triangleq}\\{{\bf
z}\in\mathbf{N}_{(\\{A\\})}\mid(\mathrm{DOM})\left[{\bf z}\right]=\Omega\\}$.

Let ${\cal H}\in\mathcal{P^{\prime}}({\mathcal{T}})$. We call a m/f from
${\mathcal{P}({\mathrm{Z}})}^{\Omega}$ _${\cal H}$ -non-anticipative_ if the
m/f is $A$-non-anticipative for all $A\in{\cal H}$. The set all ${\cal
H}$-non-anticipative (non-empty-valued and ${\cal H}$-non-anticipative) m/f we
denote by $\mathbf{N}_{(\cal H)}$ ($\mathbf{N}^{0}_{(\cal H)}$):

$\mathbf{N}_{(\cal H)}=\bigcap_{A\in{\cal
H}}\mathbf{N}_{(\\{A\\})}\qquad\bigl{(}\mathbf{N}^{0}_{(\cal
H)}=\bigcap_{A\in{\cal H}}\mathbf{N}^{0}_{(\\{A\\})}\bigr{)}.$

For any $\alpha\in\mathcal{P}({\mathrm{Z}})^{\Omega}$ and ${\cal
H}\in\mathcal{P^{\prime}}({\mathcal{T}})$ designations $\mathbf{N}_{(\cal
H)}\left[{\alpha}\right]$ and $\mathbf{N}^{0}_{(\cal H)}\left[{\alpha}\right]$
are defined similarly to those in (5) and (6), respectively.

Note the implication following from these definitions (isotonicity): for
arbitrary $\alpha\in\mathcal{P}({\mathrm{Z}})^{\Omega}$, ${\cal H}$, ${\cal
H^{\prime}}\in\mathcal{P}({\mathcal{T}})$

$\left({\cal H}\subset{\cal
H^{\prime}}\right)\mathrel{\Rightarrow}\left(\mathbf{N}^{0}_{(\cal
H^{\prime})}\left[{\alpha}\right]\subset\mathbf{N}^{0}_{(\cal
H)}\left[{\alpha}\right]\right).$ (15)

The following is definitions and some properties of point-wise operations on
non-anticipative m/f.

######  Lemma 1.

Let $\alpha\in\mathcal{P}({\mathrm{Z}})^{\Omega}$, ${\cal
H}\subset\mathcal{T}$ and
$\mathbb{M}\in\mathcal{P^{\prime}}({\mathbf{N}_{(\cal
H)}\left[{\alpha}\right]})$. Then

$\bigvee_{{\bf z}\in\mathbb{M}}{\bf z}\in\mathbf{N}_{(\cal
H)}\left[{\alpha}\right],$ (16) $\Bigl{(}\forall\omega\in\Omega\ \exists{\bf
z}\in\mathbb{M}:{\bf
z}(\omega)\neq\varnothing\Bigr{)}\mathrel{\Rightarrow}\left(\bigvee_{{\bf
z}\in\mathbb{M}}{\bf z}\in\mathbf{N}^{0}_{(\cal
H)}\left[{\alpha}\right]\right),$ (17) $\top_{\mathbf{N}_{(\cal
H)}\left[{\alpha}\right]}=\bigvee_{{\bf z}\in\mathbf{N}_{(\cal
H)}\left[{\alpha}\right]}{\bf z}.$ (18)

P r o o f. 1\. It is clear that

$\left(\bigvee_{{\bf z}\in\mathbb{M}}{\bf
z}\mathrel{\sqsubseteq}\alpha\right)\mathrel{\&}\left(\bigwedge_{{\bf
z}\in\mathbb{M}}{\bf z}\mathrel{\sqsubseteq}\alpha\right).$ (19)

We show that the mapping $\bigvee_{{\bf z}\in\mathbb{M}}{\bf z}$ inherits the
property of ${\cal H}$-non-anticipatory of elements from $\mathbb{M}$: let
$\omega,\omega^{\prime}\in\Omega$, $A\in{\cal H}$,
$(\omega|A)=(\omega^{\prime}|A)$ and $\gamma\in\llparenthesis\bigvee_{{\bf
z}\in\mathbb{M}}{\bf z}(\omega)|A\rrparenthesis$. Then there are $\bar{\bf
z}\in\mathbb{M}$ and $h\in\bar{\bf z}(\omega)$ such that $\gamma=(h|A)$.
Therefore, taking into account the relation $\bar{\bf z}\in\mathbf{N}_{(\cal
H)}\left[{\alpha}\right]$, we have the inclusion
$\gamma\in\llparenthesis\bar{\bf}z(\omega^{\prime})|A\rrparenthesis$. Hence,
there exists $h^{\prime}\in\bar{\bf z}(\omega^{\prime})$ such that
$\gamma=(h^{\prime}|A)$, whence we obtain the inclusion
$h^{\prime}\in\bigvee_{{\bf z}\in\mathbb{M}}{\bf z}(\omega^{\prime})$ and, as
a consequence, the inclusion $\gamma\in\llparenthesis\bigvee_{{\bf
z}\in\mathbb{M}}{\bf z}(\omega^{\prime})|A\rrparenthesis$. Since $\gamma$ was
chosen arbitrarily, we have the relation

$\llparenthesis\bigvee_{{\bf z}\in\mathbb{M}}{\bf
z}(\omega)|A\rrparenthesis\subset\llparenthesis\bigvee_{{\bf
z}\in\mathbb{M}}{\bf z}(\omega^{\prime})|A\rrparenthesis.$

Hence, due to symmetry of considerations and an arbitrary choice of $\omega$,
$\omega^{\prime}$, the implication follows

$((\omega|A)=(\omega^{\prime}|A))\mathrel{\Rightarrow}\llparenthesis\bigvee_{{\bf
z}\in\mathbb{M}}{\bf z}(\omega)|A\rrparenthesis=\llparenthesis\bigvee_{{\bf
z}\in\mathbb{M}}{\bf
z}(\omega^{\prime})|A\rrparenthesis\qquad\forall\omega,\omega^{\prime}\in\Omega.$

Due to the implication, taking into account the arbitrary choice of $A$, we
have

$\bigvee_{{\bf z}\in\mathbb{M}}{\bf z}\in\mathbf{N}_{(\cal H)}.$ (20)

Relations (19), (20) together give the desired inclusion (16).

2\. Premise in (17) implies the equality $(\mathrm{DOM})\left[{\bigvee_{{\bf
z}\in\mathbb{M}}{\bf z}}\right]=\Omega$, which in combination with (16) gives
the desired assertion.

3\. By construction, the expression on the right side (18)
$\mathrel{\sqsubseteq}$-majorizes each element of the set $\mathbf{N}_{(\cal
H)}\left[{\alpha}\right]$, while (16) for
$\mathbb{M}\mathrel{\triangleq}\mathbf{N}_{(\cal H)}\left[{\alpha}\right]$
implies the inclusion $\bigvee_{{\bf z}\in\mathbf{N}_{(\cal
H)}\left[{\alpha}\right]}{\bf z}\in\mathbf{N}_{(\cal
H)}\left[{\alpha}\right]$.

The lemma is proven.

######  Theorem 1.

For a m/f $\alpha\in\mathcal{P}({\mathrm{Z}})^{\Omega}$ and a partition
$\Delta$ of interval $\mathrm{T}$, the conditions $(\alpha,\Delta)$ are
feasible if and only if m/f $\alpha$ has an non-empty-valued and ${\cal
H}_{\Delta}$-non-anticipative multiselector:

$\left((\alpha,\Delta)-\text{feasible}\right)\mathrel{\Leftrightarrow}\left(\mathbf{N}^{0}_{({\cal
H}_{\Delta})}\left[{\alpha}\right]\neq\varnothing\right).$ (21)

P r o o f. Let
$\Delta\mathrel{\triangleq}\\{t_{0}=\tau_{0}<\tau_{1}<\ldots<\tau_{n_{\Delta}}=\vartheta\\}$.
Remind that ${\cal
H}_{\Delta}\mathrel{\triangleq}\\{H_{i}\mathrel{\triangleq}[\tau_{0},\tau_{i}]:i\in{{1}..{n_{\Delta}}}\\}$.
The definitions (8), (9) imply the equalities

$\Omega_{\Delta}=\\{(\omega_{i})_{i\in{{1}..{n_{\Delta}}}}\in\Omega^{n_{\Delta}}\mid(\omega_{i}|H_{\overline{ij}})=(\omega_{j}|H_{\overline{ij}}),\
i,j\in{{1}..{n_{\Delta}}}\\},$ (22) $\mathrm{Z}_{\Delta}=\\{({\bf
z}_{i})_{i\in{{1}..{n_{\Delta}}}}\in{\mathcal{P^{\prime}}({\mathrm{Z}})}^{n_{\Delta}}\mid\llparenthesis{\bf
z}_{i}|H_{\overline{ij}}\rrparenthesis=\llparenthesis{\bf
z}_{j}|H_{\overline{ij}}\rrparenthesis,\ i,j\in{{1}..{n_{\Delta}}}\\},$ (23)

where $\overline{ij}\mathrel{\triangleq}\min\\{i,j\\}$ for all
$i,j\in{{1}..{n_{\Delta}}}$.

1\. Let $\mathbf{N}^{0}_{({\cal
H}_{\Delta})}\left[{\alpha}\right]\neq\varnothing$ and
$\beta\in\mathbf{N}^{0}_{({\cal H}_{\Delta})}\left[{\alpha}\right]$. Consider
a tuple $(\phi_{i})_{i\in{{1}..{n_{\Delta}}}}$ of the form
$\phi_{i}\mathrel{\triangleq}\beta$, $i\in{{1}..{n_{\Delta}}}$. By definition,
it satisfies the equalities $(\mathrm{DOM})\left[{\phi_{i}}\right]=\Omega$,
$i\in{{1}..{n_{\Delta}}}$, and the conditions
$\phi_{i}\mathrel{\sqsubseteq}\beta\mathrel{\sqsubseteq}\alpha$,
$i\in{{1}..{n_{\Delta}}}$. This implies the fulfillment of the condition (10).

Let us check (11). Suppose a tuple $(\omega_{i})_{i\in{{1}..{n_{\Delta}}}}$ is
such that $(\omega_{i})_{i\in{{1}..{n_{\Delta}}}}\in\Omega_{\Delta}$. Then,
taking into account the ${\cal H}_{\Delta}$-non-anticipatory property of
$\beta$, from the equalities
$(\omega_{k}|H_{\overline{km}})=(\omega_{m}|H_{\overline{km}})$ we get the
equalities

$\llparenthesis\phi_{k}(\omega_{k})|H_{\overline{km}}\rrparenthesis=\llparenthesis\beta(\omega_{k})|H_{\overline{km}}\rrparenthesis=\llparenthesis\beta(\omega_{m})|H_{\overline{km}}\rrparenthesis=\llparenthesis\phi_{m}(\omega_{m})|H_{\overline{km}}\rrparenthesis\quad
k,m\in{{1}..{n_{\Delta}}}.$

Since $(\omega_{i})_{i\in{{1}..{n_{\Delta}}}}$, $k$ and $m$ were chosen
arbitrarily, for the tuple $(\phi_{i})_{i\in{{1}..{n_{\Delta}}}}$ condition
(11) is met. We have shown that the left side of (21) follows from the right
side.

2\. Let us show that the right side of (21) follows from the left side. Assume
that the conditions $(\alpha,\Delta)$ are feasible. Then (see sec. 2) there
exists a tuple $(\phi_{i})_{i\in{{1}..{n_{\Delta}}}}$ of the form (10) such
that for any tuple $(\omega_{i})_{i\in{{1}..{n_{\Delta}}}}\in\Omega_{\Delta}$
(see (22),(23)):

$(\phi_{i}(\omega_{i}))_{i\in{{1}..{n_{\Delta}}}}\in\mathrm{Z}_{\Delta}.$ (24)

Let m/f $\phi_{\alpha}$ be defined by
$\phi_{\alpha}\mathrel{\triangleq}\phi_{n_{\Delta}}$. Then we have the
comparison $\phi_{\alpha}\mathrel{\sqsubseteq}\alpha$ and equality
$(\mathrm{DOM})\left[{\phi_{\alpha}}\right]=\Omega$ (indeed, from (23), (24)
follow the inclusions
$\phi_{\alpha}(\omega)\in\mathcal{P^{\prime}}({\mathrm{Z}})$ for all
$\omega\in\Omega$). To show that $\phi_{\alpha}$ is ${\cal H}_{\Delta}$-non-
anticipative, suppose $\omega,\omega^{\prime}\in\Omega$ and
$m\in{{1}..{n_{\Delta}}}$ are such that

$(\omega|H_{m})=(\omega^{\prime}|H_{m}).$ (25)

We put

$\omega_{i}\mathrel{\triangleq}\omega,\quad
i\in{{1}..{n_{\Delta}}},\qquad\omega^{\prime}_{i}\mathrel{\triangleq}\begin{cases}\omega,&i\in{{1}..{m}},\\\
\omega^{\prime},&i\in{{(m+1)}..{n_{\Delta}}}.\end{cases}$

For the tuple $(\omega_{i})_{i\in{{1}..{n_{\Delta}}}}$, we obviously have the
inclusion $(\omega_{i})_{i\in{{1}..{n_{\Delta}}}}\in\Omega_{\Delta}$.

Show the inclusion
$(\omega^{\prime}_{i})_{i\in{{1}..{n_{\Delta}}}}\in\Omega_{\Delta}$. For the
tuple $(\omega^{\prime}_{i})_{i\in{{1}..{n_{\Delta}}}}$ and any
$i,j\in{{1}..{n_{\Delta}}}$ we have:

if $m<i\leqslant j$, then

$(\omega^{\prime}_{i}|H_{\overline{ij}})=(\omega^{\prime}|H_{\overline{ij}})=(\omega^{\prime}_{j}|H_{\overline{ij}});$

if $i\leqslant m<j$, then (see (25))

$(\omega^{\prime}_{i}|H_{\overline{ij}})=(\omega|H_{\overline{ij}})=(\omega|H_{i})=(\omega^{\prime}|H_{i})=(\omega^{\prime}_{j}|H_{i})=(\omega^{\prime}_{j}|H_{\overline{ij}});$

if $i\leqslant j\leqslant m$, then

$(\omega^{\prime}_{i}|H_{\overline{ij}})=(\omega|H_{\overline{ij}})=(\omega^{\prime}_{j}|H_{\overline{ij}}).$

Thus (see (22)), the inclusion
$(\omega^{\prime}_{i})_{i\in{{1}..{n_{\Delta}}}}\in\Omega_{\Delta}$ takes
place.

Therefore, due to (24), the inclusions

$(\phi_{i}(\omega_{i}))_{i\in{{1}..{n_{\Delta}}}}\in\mathrm{Z}_{\Delta},\qquad(\phi_{i}(\omega^{\prime}_{i}))_{i\in{{1}..{n_{\Delta}}}}\in\mathrm{Z}_{\Delta},$

considered (see (23)) under $i=m$ and $j=n_{\Delta}$, imply the equalities

$\llparenthesis\phi_{m}(\omega_{m})|H_{\overline{mn_{\Delta}}}\rrparenthesis=\llparenthesis\phi_{n_{\Delta}}(\omega_{n_{\Delta}})|H_{\overline{mn_{\Delta}}}\rrparenthesis,\qquad\llparenthesis\phi_{m}(\omega^{\prime}_{m})|H_{\overline{mn_{\Delta}}}\rrparenthesis=\llparenthesis\phi_{n_{\Delta}}(\omega^{\prime}_{n_{\Delta}})|H_{\overline{mn_{\Delta}}}\rrparenthesis.$
(26)

Moreover, from the definitions of $\omega^{\prime}_{m}$ and $\omega_{m}$ we
have $\omega^{\prime}_{m}=\omega_{m}=\omega$ and hence the equalities

$\llparenthesis\phi_{m}(\omega_{m})|H_{\overline{mn_{\Delta}}}\rrparenthesis=\llparenthesis\phi_{m}(\omega)|H_{m}\rrparenthesis=\llparenthesis\phi_{m}(\omega^{\prime}_{m})|H_{\overline{mn_{\Delta}}}\rrparenthesis.$
(27)

From the given equalities and the definition of $\phi_{\alpha}$ we obtain (the
2nd and the 4th equalities follow from (23) and (26), the 3rd — from (27)):

$\llparenthesis\phi_{\alpha}(\omega)|[\tau_{0},\tau_{m}]\rrparenthesis=\llparenthesis\phi_{n_{\Delta}}(\omega_{n_{\Delta}})|[\tau_{0},\tau_{\overline{mn_{\Delta}}}]\rrparenthesis=\llparenthesis\phi_{m}(\omega_{m})|[\tau_{0},\tau_{\overline{mn_{\Delta}}}]\rrparenthesis\\\
=\llparenthesis\phi_{m}(\omega^{\prime}_{m})|[\tau_{0},\tau_{\overline{mn_{\Delta}}}]\rrparenthesis=\llparenthesis\phi_{n_{\Delta}}(\omega^{\prime}_{n_{\Delta}})|[\tau_{0},\tau_{\overline{mn_{\Delta}}}]\rrparenthesis=\llparenthesis\phi_{\alpha}(\omega^{\prime})|[\tau_{0},\tau_{m}]\rrparenthesis.$

Since $m$, $\omega$ and $\omega^{\prime}$ were chosen arbitrarily, from the
last equalities we obtain the property of ${\cal H}_{\Delta}$-non-anticipatory
of m/f $\phi_{\alpha}$. Taking into account the indicated properties of
$\phi_{\alpha}$, we have the inclusion $\phi_{\alpha}\in\mathbf{N}^{0}_{({\cal
H}_{\Delta})}\left[{\alpha}\right]$, i.e., the right side of (21) is
fulfilled.

The proof is complete.

##  4 A construction of the partially non-anticipative multiselector

In this section, we give a description of partially non-anticipative
multiselectors in terms of explicitly defined operators that are non-
anticipative at one point of $\mathrm{T}$. In general, such description is
certainly non-constructive. Meanwhile, when applied to step-by-step
procedures, due to the finiteness of operations, the description allows to
construct and analyze corresponding partially non-anticipative multiselectors
(see examples).

For an arbitrary $A\in\mathcal{T}$, denote by $\langle{\cdot}\rangle_{A}$ the
operator that transforms the set $\mathcal{P}({\mathrm{Z}})^{\Omega}$ and is
given by:

$\langle{\alpha}\rangle_{A}(\omega)\mathrel{\triangleq}\bigl{\\{}h\in\alpha(\omega)\
\bigr{|}\
(h|A)\in\bigcap_{\omega^{\prime}\in\Omega(\omega|A)}\llparenthesis\alpha(\omega^{\prime})|A\rrparenthesis\bigr{\\}}\qquad\forall\alpha\in\mathcal{P}({\mathrm{Z}})^{\Omega},\
\forall\omega\in\Omega.$ (28)

It immediately follows that $\langle{\cdot}\rangle_{A}$ is non-expansive and
isotonic as an operator in the poset
$(\mathcal{P}({\mathrm{Z}})^{\Omega},\mathrel{\sqsubseteq})$: for arbitrary
$A\in\mathcal{T}$, $\alpha,\beta\in\mathcal{P}({\mathrm{Z}})^{\Omega}$

$\langle{\alpha}\rangle_{A}\mathrel{\sqsubseteq}\alpha,$ (29)
$(\alpha\mathrel{\sqsubseteq}\beta)\Rightarrow\left(\langle{\alpha}\rangle_{A}\mathrel{\sqsubseteq}\langle{\beta}\rangle_{A}\right).$
(30)

We also note (see subsection 5.1) that in the general case the mapping
$\mathcal{T}\ni
A\mapsto\langle{\alpha}\rangle_{A}\in\mathcal{P}({\mathrm{Z}})^{\Omega}$ is
not isotonic as a mapping from poset $(\mathcal{T},\subset)$ to poset
$(\mathcal{P}({\mathrm{Z}})^{\Omega},\mathrel{\sqsubseteq})$.

######  Lemma 2.

For $A\in\mathcal{T}$ and $\alpha\in\mathcal{P}({\mathrm{Z}})^{\Omega}$

(i) values of the operator $\langle{\cdot}\rangle_{A}$ consist the set of all
$A$-non-anticipative m/f, as well as the set of its fixed points:

$\langle{\mathcal{P}({\mathrm{Z}})^{\Omega}}\rangle_{A}=\mathbf{N}_{(\\{A\\})}={\mathbf{Fix}({\langle{\cdot}\rangle_{A}})};$
(31)

(ii) $\langle{\alpha}\rangle_{A}$ is the $\mathrel{\sqsubseteq}$-greatest
$A$-non-anticipative multiselector of $\alpha$:

$\langle{\alpha}\rangle_{A}=\top_{\mathbf{N}_{(\\{A\\})}\left[{\alpha}\right]};$
(32)

(iii) the operator $\langle{\cdot}\rangle_{A}$ is idempotent: for any
$\alpha\in\mathcal{P}({\mathrm{Z}})^{\Omega}$ the following equality takes
place

$\langle{\langle{\alpha}\rangle_{A}}\rangle_{A}=\langle{\alpha}\rangle_{A}.$
(33)

P r o o f. 1\. Let $\beta\mathrel{\triangleq}\langle{\alpha}\rangle_{A}$. We
show that $\beta\in\mathbf{N}_{(\\{A\\})}\left[{\alpha}\right]$. Due to (29),
we have $\beta\mathrel{\sqsubseteq}\alpha$. It remains to verify the $A$-non-
anticipatory property of $\beta$.

Let $\omega,\omega^{\prime}\in\Omega$ be such that
$(\omega|A)=(\omega^{\prime}|A)$ and
$\xi\in\llparenthesis\beta(\omega)|A\rrparenthesis$. Then (see (28))

$\xi\in\bigcap_{\bar{\omega}\in\Omega(\omega|A)}\llparenthesis\alpha(\bar{\omega})|A\rrparenthesis.$
(34)

By the choice of $\omega^{\prime}$, we have
$\omega^{\prime}\in\Omega(\omega|A)$ and hence (see (34))
$\xi\in\llparenthesis\alpha(\omega^{\prime})|A\rrparenthesis$. Then, there
exists $h^{\prime}\in\alpha(\omega^{\prime})$ such that $(h^{\prime}|A)=\xi$.
From the equality $(\omega|A)=(\omega^{\prime}|A)$ it also follows that
$\Omega(\omega^{\prime}|A)=\Omega(\omega|A)$ and, therefore,

$\bigcap_{\bar{\omega}\in\Omega(\omega|A)}\llparenthesis\alpha(\bar{\omega})|A\rrparenthesis=\bigcap_{\bar{\omega}\in\Omega(\omega^{\prime}|A)}\llparenthesis\alpha(\bar{\omega})|A\rrparenthesis.$

As a result, $h^{\prime}$ satisfies relations

$h^{\prime}\in\alpha(\omega^{\prime}),\qquad(h^{\prime}|A)\in\bigcap_{\bar{\omega}\in\Omega(\omega^{\prime}|A)}\llparenthesis\alpha(\bar{\omega})|A\rrparenthesis,$

i.e. (see (28)) $h^{\prime}\in\langle{\alpha}\rangle_{A}(\omega^{\prime})$.
Then $\xi\in\llparenthesis\beta(\omega^{\prime})|A\rrparenthesis$. Since $\xi$
was chosen arbitrarily, we have

$\llparenthesis\beta(\omega)|A\rrparenthesis\subset\llparenthesis\beta(\omega^{\prime})|A\rrparenthesis.$

From the inclusion, due to an arbitrary choice of $\omega$, $\omega^{\prime}$
and to symmetry of them in the considerations, we obtain the desired $A$-non-
anticipatory of $\beta$.

So, taking into account the definition of $\beta$, we have the inclusions

$\langle{\alpha}\rangle_{A}\in\mathbf{N}_{(\\{A\\})}\left[{\alpha}\right]\subset\mathbf{N}_{(\\{A\\})}.$
(35)

Due to the arbitrary choice of $\alpha$, (35) implies an embedding

$\langle{\mathcal{P}({\mathrm{Z}})^{\Omega}}\rangle_{A}\subset\mathbf{N}_{(\\{A\\})}.$
(36)

2\. Let us verify that $\langle{\alpha}\rangle_{A}$ is the
$\mathrel{\sqsubseteq}$-greatest m/f in
$\mathbf{N}_{(\\{A\\})}\left[{\alpha}\right]$. Let

$\beta\in\mathbf{N}_{(\\{A\\})}\left[{\alpha}\right],$ (37)

$\bar{\omega}\in\Omega$ and $\bar{h}\in\beta(\bar{\omega})$. Then (37) implies
that

$(\bar{h}|A)\in\llparenthesis\beta(\omega)|A\rrparenthesis\subset\llparenthesis\alpha(\omega)|A\rrparenthesis\quad\forall\omega\in\Omega(\bar{\omega}|A).$

So, $\bar{h}$ satisfies the relations

$\bar{h}\in\beta(\bar{\omega})\subset\alpha(\bar{\omega}),\qquad(\bar{h}|A)\in\bigcap_{\omega\in\Omega(\bar{\omega}|A)}\llparenthesis\beta(\omega)|A\rrparenthesis\subset\bigcap_{\omega\in\Omega(\bar{\omega}|A)}\llparenthesis\alpha(\omega)|A\rrparenthesis,$

i.e. (see (28)) $\bar{h}\in\langle{\beta}\rangle_{A}(\bar{\omega})$ and
$\bar{h}\in\langle{\alpha}\rangle_{A}(\bar{\omega})$. Since $\bar{\omega}$ and
$\bar{h}$ were chosen arbitrarily, we have comparisons

$\beta\mathrel{\sqsubseteq}\langle{\beta}\rangle_{A},$ (38)
$\beta\mathrel{\sqsubseteq}\langle{\alpha}\rangle_{A}.$ (39)

Comparing (39), due to arbitrary choice of $\beta$, gives the equality (32).

3\. Relation (38) and the non-expansion property of
$\langle{\cdot}\rangle_{A}$ (see (29)) imply the equality
$\beta=\langle{\beta}\rangle_{A}$. Hence, due to arbitrary choice of $\beta$,
we obtain for any $\alpha\in\mathcal{P}({\mathrm{Z}})^{\Omega}$ the embedding
$\mathbf{N}_{(\\{A\\})}\left[{\alpha}\right]\subset{\mathbf{Fix}({\langle{\cdot}\rangle_{A}})}$.
For $\alpha$ of the form $\alpha(\omega)\mathrel{\triangleq}\mathrm{Z}$,
$\omega\in\Omega$, we obviously have
$\mathbf{N}_{(\\{A\\})}\left[{\alpha}\right]=\mathbf{N}_{(\\{A\\})}$. So the
following inclusion is correct

$\mathbf{N}_{(\\{A\\})}\subset{\mathbf{Fix}({\langle{\cdot}\rangle_{A}})}.$
(40)

For any $\beta\in{\mathbf{Fix}({\langle{\cdot}\rangle_{A}})}$, by the
definition of a fixed point the equality $\beta=\langle{\beta}\rangle_{A}$ is
satisfied, i.e., $\beta$ lies in the image of the set
$\mathcal{P}({\mathrm{Z}})^{\Omega}$ under the mapping
$\langle{\cdot}\rangle_{A}$. Then the next inclusion is fulfilled:

${\mathbf{Fix}({\langle{\cdot}\rangle_{A}})}\subset\langle{\mathcal{P}({\mathrm{Z}})^{\Omega}}\rangle_{A}.$
(41)

From relations (36), (40) and (41), we get the equalities (31).

4\. Equalities (31) imply the equality (33).

The lemma is proven.

From the lemma, we immediately obtain a corollary that allows us to filter out
m/f that do not have an non-empty-valued and non-anticipative multiselector.

######  Corollary 1.

For $\alpha\in\mathcal{P}({\mathrm{Z}})^{\Omega}$ and ${\cal
H}\in\mathcal{P^{\prime}}({\mathcal{T}})$ the following implications hold:

$\left(\beta\in\mathbf{N}_{(\cal
H)}\left[{\alpha}\right]\right)\Rightarrow\left(\beta\mathrel{\sqsubseteq}\bigwedge_{A\in{\cal
H}}\langle{\alpha}\rangle_{A}\right),$ (42)
$\left((\mathrm{DOM})\left[{\bigwedge_{A\in{\cal
H}}\langle{\alpha}\rangle_{A}}\right]\neq\Omega\right)\Rightarrow\left(\mathbf{N}^{0}_{(\cal
H)}\left[{\alpha}\right]=\varnothing\right)\Rightarrow\left(\mathbf{N}_{(\mathcal{T})}^{0}[{\alpha}]=\varnothing\right).$
(43)

P r o o f. 1\. Let $\beta$ satisfy the premise of (42). Then from the
relations

$\beta\in\mathbf{N}_{(\\{A\\})}\left[{\alpha}\right],\qquad\forall A\in{\cal
H},$

and (32) we have

$\beta\mathrel{\sqsubseteq}\langle{\alpha}\rangle_{A},\qquad\forall A\in{\cal
H},$

which implies the conclusion of (42).

2\. From (42) and (15) the implications (43) follow.

The the corollary is proven.

Since for arbitrary ${\cal H}\in\mathcal{P^{\prime}}({\mathcal{T}})$ and
$\alpha\in\mathcal{P}({\mathrm{Z}})^{\Omega}$ there exists the greatest
(unique) element $\top_{\mathbf{N}_{(\cal H)}\left[{\alpha}\right]}$ (see the
lemma 1), then we introduce an operator $\langle{\cdot}\rangle_{{\cal
H}}:\mathcal{P}({\mathrm{Z}})^{\Omega}\mapsto\mathcal{P}({\mathrm{Z}})^{\Omega}$
of the form

$\langle{\alpha}\rangle_{{\cal H}}\mathrel{\triangleq}\top_{\mathbf{N}_{(\cal
H)}\left[{\alpha}\right]},\qquad\alpha\in\mathcal{P}({\mathrm{Z}})^{\Omega}.$
(44)

For ${\cal H}\subset\mathcal{T}$ we denote by ${\bf F}_{\cal H}$ the family of
operators defined by ${\bf F}_{\cal
H}\mathrel{\triangleq}\\{\langle{\cdot}\rangle_{A}\mid A\in{\cal H}\\}$. By
${\mathbf{Fix}({{\bf F}_{\cal H}})}$ we refer to the set of joint fixed points
of the family ${\bf F}_{\cal H}$: ${\mathbf{Fix}({{\bf F}_{\cal
H}})}\mathrel{\triangleq}\bigcap_{A\in{\cal
H}}{\mathbf{Fix}({\langle{\cdot}\rangle_{A}})}$.

######  Remark 1.

It follows from the definition of $\mathbf{N}_{(\cal H)}$ and the equality
(31) that the joint fixed points of the family ${\bf F}_{\cal H}$ are ${\cal
H}$-non-anticipative mappings: ${\mathbf{Fix}({{\bf F}_{\cal
H}})}=\mathbf{N}_{(\cal H)}$. Thus, in the case when ${\cal H}$ is a singleton
(${\cal H}=\\{A\\}$, $A\in\mathcal{T}$), we get the equality (see (32))
$\langle{\cdot}\rangle_{\\{A\\}}=\langle{\cdot}\rangle_{A}$.

We turn to the representation of the operator $\langle{\cdot}\rangle_{\cal H}$
in terms of superposition of operators from ${\bf F}_{\cal H}$. As already was
noted, the representation of non-anticipative multiselectors in the form of
fixed points of non-expansive isotonic operators in a poset and, as a
consequence, in the form of limits of their iterative sequences, was proposed
and studied in [14, 8, 9]. The iterative process in some cases turns out to be
finite (see, for example, [15, ch. 5]), which makes it possible to obtain
efficient solutions to problems.

In this paper despite the fact that the operators from ${\bf F}_{\cal H}$ are
in general non-commutative (see subsection 5.2), a finite-step construction of
the operator $\langle{\cdot}\rangle_{\cal H}$ is given in theorem 2 for all
cases when the set ${\cal H}$ is finite. To this end, we give some definitions
and auxiliary results.

######  Lemma 3.

Let $H_{1},H_{2}\in\mathcal{T}$,
$\alpha\in\mathcal{P}({\mathrm{Z}})^{\Omega}$,
$\Upsilon\in\mathcal{P^{\prime}}({\Omega})$ and
$K\in\mathcal{P}({\llparenthesis\mathrm{Z}|H_{1}\rrparenthesis})$ are such
that

$\displaystyle H_{1}\subset H_{2},$ (45)
$\displaystyle\llparenthesis\alpha(\omega)|H_{2}\rrparenthesis=\llparenthesis\alpha(\omega^{\prime})|H_{2}\rrparenthesis,\qquad\forall\omega,\omega^{\prime}\in\Upsilon.$
(46)

Then for $\beta\in\mathcal{P}({\mathrm{Z}})^{\Omega}$ of the form

$\beta(\omega)\mathrel{\triangleq}\begin{cases}\\{h\in\alpha(\omega)\mid(h|H_{1})\in
K\\},&\omega\in\Upsilon,\\\
\alpha(\omega),&\omega\in\Omega\setminus\Upsilon,\end{cases}$ (47)

the equalities (46) are also satisfied:

$\llparenthesis\beta(\omega)|H_{2}\rrparenthesis=\llparenthesis\beta(\omega^{\prime})|H_{2}\rrparenthesis,\qquad\forall\omega,\omega^{\prime}\in\Upsilon.$
(48)

P r o o f. Let $\omega,\omega^{\prime}\in\Upsilon$ and
$\gamma\in\llparenthesis\beta(\omega)|H_{2}\rrparenthesis$. Then, by the
choice of $\gamma$, there exists $h\in\alpha(\omega)$ such that

$(h|H_{2})=\gamma,$ (49)

and at the same time (see (47))

$(h|H_{1})\in K.$ (50)

From (46) and the choice of $\omega$, $\omega^{\prime}$ it follows that there
is $h^{\prime}\in\alpha(\omega^{\prime})$ satisfying the equality

$(h^{\prime}|H_{2})=(h|H_{2}).$ (51)

From (49), (51) we have

$(h^{\prime}|H_{2})=\gamma.$ (52)

In addition, (50), (51), and (45) imply

$(h^{\prime}|H_{1})\in K.$ (53)

From the inclusion of $h^{\prime}\in\alpha(\omega^{\prime})$, (53) and the
definition of $\beta$ (see (47)), we get
$h^{\prime}\in\beta(\omega^{\prime})$, whence, taking into account (52), we
have $\gamma\in\llparenthesis\beta(\omega^{\prime})|H_{2}\rrparenthesis$.
Then, due to the arbitrary choice of $\gamma$, we have
$\llparenthesis\beta(\omega)|H_{2}\rrparenthesis\subset\llparenthesis\beta(\omega^{\prime})|H_{2}\rrparenthesis$.
From here, in view of the symmetry of the occurrence of $\omega$,
$\omega^{\prime}$, the desired equality (48) is extracted. The proof is
complete.

######  Corollary 2.

In particular, if (45) is true, the application of
$\langle{\cdot}\rangle_{H_{1}}$ to $H_{2}$-non-anticipative m/f $\alpha$ does
not violate this property.

P r o o f. Indeed, let $H_{1},H_{2}\in\mathcal{T}$ satisfy (45),
$\alpha\in\mathbf{N}_{(H_{2})}$, and $\omega\in\Omega$ is fixed. Let us define
$K$ and $\Upsilon$ from lemma 3 as follows:

$K\mathrel{\triangleq}\bigcap_{\bar{\omega}\in\Omega(\omega|H_{1})}\llparenthesis\alpha(\bar{\omega})|H_{1}\rrparenthesis,\qquad\Upsilon\mathrel{\triangleq}\Omega(\omega|H_{2}).$

Then, all conditions of lemma 3 are satisfied and m/f $\beta$ specified in
(47) satisfies the equalities (see (28)) for all
$\tilde{\omega}\in\Upsilon=\Omega(\omega|H_{2})$:

$\beta(\tilde{\omega})\mathrel{\triangleq}\\{h\in\alpha(\tilde{\omega})\mid(h|H_{1})\in
K\\}\mathrel{\triangleq}\\{h\in\alpha(\tilde{\omega})\mid(h|H_{1})\in\bigcap_{\bar{\omega}\in\Omega(\omega|H_{1})}\llparenthesis\alpha(\bar{\omega})|H_{1}\rrparenthesis\\}\\\
=\\{h\in\alpha(\tilde{\omega})\mid(h|H_{1})\in\bigcap_{\bar{\omega}\in\Omega(\tilde{\omega}|H_{1})}\llparenthesis\alpha(\bar{\omega})|H_{1}\rrparenthesis\\}\mathrel{\triangleq}\langle{\alpha}\rangle_{H_{1}}(\tilde{\omega}).$

Using the last relations and (48), we obtain:

$\llparenthesis\langle{\alpha}\rangle_{H_{1}}(\omega^{\prime\prime})|H_{2}\rrparenthesis=\llparenthesis\beta(\omega^{\prime\prime})|H_{2}\rrparenthesis=\llparenthesis\beta(\omega^{\prime})|H_{2}\rrparenthesis=\llparenthesis\langle{\alpha}\rangle_{H_{1}}(\omega^{\prime})|H_{2}\rrparenthesis\qquad\forall\omega^{\prime},\omega^{\prime\prime}\in\Upsilon.$
(54)

Due to arbitrary choice of $\omega$, relations (54) are true for all
$\omega^{\prime},\omega^{\prime\prime}\in\Omega(\omega|H_{2})$ and all
$\omega\in\Omega$, i. e. we have inclusion
$\langle{\alpha}\rangle_{H_{1}}\in\mathbf{N}_{(H_{2})}$. The proof is
complete.

The following statement provides for a finite chain ${\cal H}$ and an
arbitrary m/f a representation of its greatest ${\cal H}$-non-anticipative
multiselector as a finite superposition of operators from ${\bf F}_{\cal H}$.
The provided construction inherits the features of the backward recurrent
procedures [19, 2, 20] on the one hand, and of the method of programmed
iterations of A. G. Chentsov, on the other hand.

######  Theorem 2.

Let $\alpha\in\mathcal{P}({\mathrm{Z}})^{\Omega}$ and ${\cal
H}=\\{H_{i}\in\mathcal{T}\mid i\in{{1}..{k}},k\in\mathbb{N}\\}$ be a finite
chain: $(i\leqslant j)\mathrel{\Leftrightarrow}(H_{i}\subset H_{j})$, $\forall
i,j\in{{1}..{k}}$. Then the equality is fulfilled:

$\langle{\ldots\langle{\alpha}\rangle_{H_{k}}\ldots}\rangle_{H_{1}}=\langle{\alpha}\rangle_{\cal
H}.$ (55)

That is, the expression on the left gives (see (44)) the greatest ${\cal
H}$-non-anticipative multiselector of m/f $\alpha$.

P r o o f. 1\. Denote
$\phi\mathrel{\triangleq}\langle{\ldots\langle{\alpha}\rangle_{H_{k}}\ldots}\rangle_{H_{1}}$.
Then, successively applying the operators
$\langle{\cdot}\rangle_{H_{k}}$,…,$\langle{\cdot}\rangle_{H_{1}}$ to the m/f
$\alpha$ and using the property (29) we arrive at the inequality

$\phi\mathrel{\sqsubseteq}\alpha;$ (56)

also applying successively the operators
$\langle{\cdot}\rangle_{H_{k}}$,…,$\langle{\cdot}\rangle_{H_{1}}$ to the
inequality $\langle{\alpha}\rangle_{\cal H}\mathrel{\sqsubseteq}\alpha$ taking
into account isotonicity (see (30)) we get the ratio

$\langle{\alpha}\rangle_{\cal H}\mathrel{\sqsubseteq}\phi.$ (57)

Therefore (see (ii) of lemma 2), the assertion will be proven if we establish
the ${\cal H}$-non-anticipatory property of $\phi$.

2\. Let’s show that for all $i\in{{1}..{k}}$ m/f $\phi$ holds the property
(58) of $H_{i}$-non-anticipatory:

$\llparenthesis\phi(\omega)|H_{i}\rrparenthesis=\llparenthesis\phi(\omega^{\prime})|H_{i}\rrparenthesis,\qquad\forall\omega,\omega^{\prime}\in\Omega,\
\omega^{\prime}\in\Omega(\omega|H_{i}).$ (58)

Case $i=k$. M/f $\langle{\alpha}\rangle_{H_{k}}$ is $H_{k}$-non-anticipative
by construction (see (i) of lemma 2):

$\llparenthesis\langle{\alpha}\rangle_{H_{k}}(\omega)|H_{k}\rrparenthesis=\llparenthesis\langle{\alpha}\rangle_{H_{k}}(\omega^{\prime})|H_{k}\rrparenthesis,\qquad\forall\omega,\omega^{\prime}\in\Omega,\
\omega^{\prime}\in\Omega(\omega|H_{k}).$ (59)

Since ${\cal H}$ is a chain and hence $H_{k-1}\subset H_{k}$, taking into
account the corollary 2 from (59), we get the property of $H_{k}$-non-
anticipatory of m/f
$\langle{\langle{\alpha}\rangle_{H_{k}}}\rangle_{H_{k-1}}$:

$\llparenthesis\langle{\langle{\alpha}\rangle_{H_{k}}}\rangle_{H_{k-1}}(\omega)|H_{k}\rrparenthesis=\llparenthesis\langle{\langle{\alpha}\rangle_{H_{k}}}\rangle_{H_{k-1}}(\omega^{\prime})|H_{k}\rrparenthesis,\qquad\forall\omega,\omega^{\prime}\in\Omega,\
\omega^{\prime}\in\Omega(\omega|H_{k}).$ (60)

Due to relations (60) we can apply the reasoning to m/f
$\langle{\langle{\alpha}\rangle_{H_{k}}}\rangle_{H_{k-1}}$ and operator
$\langle{\cdot}\rangle_{H_{k-2}}$. Continuing these arguments up to the
application of the operator $\langle{\cdot}\rangle_{H_{1}}$ inclusively, we
obtain the property of $H_{k}$-non-anticipatory for the m/f $\phi$, i.e., we
prove statement (58) for the case $i=k$.

Case $i\in{{1}..{(k-1)}}$. Consider m/f
$\psi\mathrel{\triangleq}\langle{\ldots\langle{\alpha}\rangle_{H_{k}}\ldots}\rangle_{H_{i}}$.
Since the operator $\langle{\cdot}\rangle_{H_{i}}$ was used last, by virtue of
item (i) of lemma 2 $\psi$ is a $H_{i}$-non-anticipative m/f. Further,
repeating the arguments from the case $i=k$ for the operators
$\langle{\cdot}\rangle_{H_{j}}$, $j\in{{1}..{(i-1)}}$, we conclude that these
operators, applied to m/f $\psi$ when constructing m/f $\phi$, preserve the
$H_{i}$-non-anticipatory property of m/f $\psi$. Namely, for all
$j\in{{1}..{(i-1)}}$ the equalities are hold true:

$\llparenthesis\langle{\ldots\langle{\psi(\omega)}\rangle_{H_{i-1}}\ldots}\rangle_{H_{j}}|H_{i}\rrparenthesis=\llparenthesis\langle{\ldots\langle{\psi(\omega^{\prime})}\rangle_{H_{i-1}}\ldots}\rangle_{H_{j}}|H_{i}\rrparenthesis,\qquad\forall\omega,\omega^{\prime}\in\Omega,\
\omega^{\prime}\in\Omega(\omega|H_{i}).$

In particular, for $j=1$ we have (58) for $i\in{{1}..{(k-1)}}$. So, m/f $\phi$
is ${H_{i}}$-non-anticipative for all $i\in{{1}..{k}}$:

$\phi\in\mathbf{N}_{(\cal H)}\left[{\alpha}\right].$ (61)

From the relations (56), (61) and item (ii) of lemma 2, the relation
$\phi\mathrel{\sqsubseteq}\langle{\alpha}\rangle_{\cal H}$ follows. Together
with (57) the relation gives us equality $\phi=\langle{\alpha}\rangle_{\cal
H}$, i.e. the required equality (55) is true. The proof is complete.

######  Remark 2.

Lemma 3 and theorem 2 remain true in general case of sets $\mathrm{T}$ and
$\mathcal{T}$, i.e. when $\mathrm{T}$ is a set and $\mathcal{T}$ is a chain in
poset $(\mathcal{P^{\prime}}({\mathrm{T}}),\subset)$.

Denote by ${{\bf s}(\omega,\omega^{\prime})}\in\mathcal{P}({\mathcal{T}})$ the
set of the form (see [7, (3.1),(3.6)])

${{\bf
s}(\omega,\omega^{\prime})}\mathrel{\triangleq}\\{H\in\mathcal{T}\mid(\omega|H)=(\omega^{\prime}|H)\\},\qquad\omega,\omega^{\prime}\in\Omega,$

and define the mapping
${{\bf\overline{s}}(\cdot,\cdot)}\in\mathcal{T}^{\Omega\times\Omega}$ given by

${{\bf\overline{s}}(\omega,\omega^{\prime})}\mathrel{\triangleq}\bigcup_{H\in{{\bf
s}(\omega,\omega^{\prime})}}H,\qquad\omega,\omega^{\prime}\in\Omega.$

The result of Corollary 3 is close to the idea of a monotonicity of non-
anticipatory m/fs (see [12, Remark 2.8]).

######  Corollary 3.

Let $\alpha\in\mathcal{P}({\mathrm{Z}})^{\Omega}$, $\Omega$ be a finite set,
and

${{\bf\overline{s}}(\omega,\omega^{\prime})}\in{{\bf
s}(\omega,\omega^{\prime})},\qquad\forall\omega,\omega^{\prime}\in\Omega.$
(62)

Then the set $\mathcal{T}_{\Omega}$,
$\mathcal{T}_{\Omega}\in\mathcal{P}({\mathcal{T}})$, defined by
$\mathcal{T}_{\Omega}\mathrel{\triangleq}\\{{{\bf\overline{s}}(\omega,\omega^{\prime})}\,:\,\omega,\omega^{\prime}\in\Omega\\}$
is a finite chain,
$\mathcal{T}_{\Omega}=\\{\bar{H}_{1}\subset\ldots\subset\bar{H}_{k_{\mathcal{T}_{\Omega}}}\\}$,
and

$\top_{\mathbf{N}_{(\mathcal{T})}[{\alpha}]}=\langle{\alpha}\rangle_{\mathcal{T}_{\Omega}}=\langle{\ldots\langle{\alpha}\rangle_{\bar{H}_{k_{\mathcal{T}_{\Omega}}}}\ldots}\rangle_{\bar{H}_{1}}.$
(63)

P r o o f. By definitions the inclusion
$\mathbf{N}_{(\mathcal{T})}[{\alpha}]\subset\mathbf{N}_{(\mathcal{T}_{\Omega})}\left[{\alpha}\right]$
is fulfilled (see (5), sec. 3, (15)). On the other hand, suppose
$\beta\in\mathbf{N}_{(\mathcal{T}_{\Omega})}\left[{\alpha}\right]$ and
$(\omega,\omega^{\prime},H)\in\Omega\times\Omega\times\mathcal{T}$. If
$(\omega|H)=(\omega^{\prime}|H)$, then $H\in{{\bf
s}(\omega,\omega^{\prime})}$,
${{\bf\overline{s}}(\omega,\omega^{\prime})}\in\mathcal{T}_{\Omega}$, and
$H\subset{{\bf\overline{s}}(\omega,\omega^{\prime})}$. Then due to inclusion
(62), we have

$(\omega|{{\bf\overline{s}}(\omega,\omega^{\prime})})=(\omega^{\prime}|{{\bf\overline{s}}(\omega,\omega^{\prime})})$

and by the property of $\mathcal{T}_{\Omega}$-non-anticipatory of $\beta$, the
relation is fulfilled:

$\llparenthesis\beta(\omega)|{{\bf\overline{s}}(\omega,\omega^{\prime})}\rrparenthesis=\llparenthesis\beta(\omega^{\prime})|{{\bf\overline{s}}(\omega,\omega^{\prime})}\rrparenthesis.$

Wherefrom the equality

$\llparenthesis\beta(\omega)|H\rrparenthesis=\llparenthesis\beta(\omega^{\prime})|H\rrparenthesis$

immediately follows. Since the choice of $(\omega,\omega^{\prime},H)$ was
arbitrary, we have the inclusion
$\beta\in\mathbf{N}_{(\mathcal{T})}[{\alpha}]$, i.e.
$\mathbf{N}_{(\mathcal{T}_{\Omega})}\left[{\alpha}\right]\subset\mathbf{N}_{(\mathcal{T})}[{\alpha}]$.
Then, the equality
$\mathbf{N}_{(\mathcal{T})}[{\alpha}]=\mathbf{N}_{(\mathcal{T}_{\Omega})}\left[{\alpha}\right]$
is true. By definition, this implies the first equality in (63). The second
equality in (63) follows from finiteness of $\mathcal{T}_{\Omega}$ and theorem
2.

The proof is complete.

Let us return to step-by-step finding of the selector under the conditions
$(\alpha,\Delta)$. Theorem 2 specifies a way to construct m/f
$\langle{\alpha}\rangle_{{\cal H}_{\Delta}}$ — the
$\mathrel{\sqsubseteq}$-greatest ${\cal H}_{\Delta}$-non-anticipative
multiselector of $\alpha$. If this multiselector turns out to be non-empty-
valued, then, by virtue of theorem 1, the step-by-step procedure can be
implemented by means of $\langle{\alpha}\rangle_{{\cal H}_{\Delta}}$ for any
disturbance $\omega\in\Omega$ (see the examples 5.3, 5.4). At the same time,
as in example 5.3, an ordinary non-anticipative multiselector of m/f $\alpha$
may be absent ($\mathbf{N}_{(\mathcal{T})}^{0}[{\alpha}]=\varnothing$).

In the case, when for some $\omega\in\Omega$ the value of the multiselector is
empty ($\langle{\alpha}\rangle_{{\cal H}_{\Delta}}(\omega)=\varnothing$) due
to $\mathrel{\sqsubseteq}$-majority of $\langle{\alpha}\rangle_{{\cal
H}_{\Delta}}$ in $\mathbf{N}_{({\cal H}_{\Delta})}\left[{\alpha}\right]$, we
obtain the fact of unrealizability (theorem 1) of step by step procedure under
the conditions $(\alpha,\Delta)$ and, as a consequence (see (15)), under any
other conditions $(\alpha,\Delta^{\prime})$ where
$\Delta\subset\Delta^{\prime}$.

##  5 Examples

###  5.1 Example 1

$0$$1$$2$$3$$\,\omega_{1}\mapsto\\{h_{1},h_{2}\\}$$\,\omega_{2}\mapsto\\{h_{1},h_{2},h_{3}\\}$$\,\omega_{3}\mapsto\\{h_{2},h_{3}\\}$$0$$1$$2$$3$${\,h_{1}}$${\,h_{2}}$${\,h_{3}}$$\Omega$$\mathrm{Z}$
Figure 1: Example 1

Let $\mathrm{T}\mathrel{\triangleq}[0,3]$,
$\mathcal{T}\mathrel{\triangleq}\\{[0,\tau]\mid\tau\in\mathrm{T}\\}$ and
$\mathrm{Y}=\mathrm{X}=\mathbb{R}$. The sets $\Omega$ and $\mathrm{Z}$ are
shown in figure 1. Let m/f $\beta$ be of the form

$\beta(\omega)=\begin{cases}\\{h_{1},h_{2}\\},&\omega=\omega_{1},\\\
\\{h_{1},h_{2},h_{3}\\},&\omega=\omega_{2},\\\
\\{h_{2},h_{3}\\},&\omega=\omega_{3}.\end{cases}$

For this m/f, using definition (28), we get:

$\langle{\beta}\rangle_{[0,1]}(\omega)=\begin{cases}\\{h_{1},h_{2}\\},&\omega=\omega_{1},\\\
\\{h_{1},h_{2}\\},&\omega=\omega_{2},\\\
\\{h_{2}\\},&\omega=\omega_{3},\end{cases}\qquad\langle{\beta}\rangle_{[0,2]}(\omega)=\begin{cases}\\{h_{1},h_{2}\\},&\omega=\omega_{1},\\\
\\{h_{2},h_{3}\\},&\omega=\omega_{2},\\\
\\{h_{2},h_{3}\\},&\omega=\omega_{3}.\end{cases}$

It is clear that the inequalities
$\langle{\beta}\rangle_{[0,1]}\not\mathrel{\sqsubseteq}\langle{\beta}\rangle_{[0,2]}$,
$\langle{\beta}\rangle_{[0,2]}\not\mathrel{\sqsubseteq}\langle{\beta}\rangle_{[0,1]}$
are fulfilled. That is, in general, the mapping $\mathcal{T}\ni
H\mapsto\langle{\beta}\rangle_{H}\in{\mathcal{P}({\mathrm{Z}})}^{\Omega}$,
considered as a mapping from poset $(\mathcal{T},\subset)$ in poset
$({\mathcal{P}({\mathrm{Z}})}^{\Omega},\mathrel{\sqsubseteq})$ does not posses
the isotonic property.

###  5.2 Example 2

$0$$1$$2$$3$$\,\omega_{11}\mapsto\\{h_{10},h_{11},h_{12},h_{21},h_{32},h_{41}\\}$$\,\omega_{12}\mapsto\\{h_{20},h_{21},h_{22},h_{11},h_{32},h_{42}\\}$$\,\omega_{21}\mapsto\\{h_{30},h_{31},h_{32},h_{12},h_{21},h_{41}\\}$$\,\omega_{22}\mapsto\\{h_{40},h_{41},h_{42},h_{12},h_{22},h_{31}\\}$$0$$1$$2$$3$${\,h_{10},\,h_{11}\,h_{12},\,h_{30}\,h_{31},\,h_{32}}$${\\!\\!h_{20}}$${\\!\\!h_{40}}$${\,h_{21}}$${\,h_{41}}$${\,h_{22}}$${\,h_{42}}$$\Omega$$\mathrm{Z}$
Figure 2: Example 2

The example shows that the operators $\langle{\cdot}\rangle_{H}$,
$H\in\mathcal{T}$ can be non-commutative. Let
$\mathrm{T}\mathrel{\triangleq}[0,3]$,
$\mathcal{T}\mathrel{\triangleq}\\{[0,\tau]\mid\tau\in\mathrm{T}\\}$ and
$\mathrm{Y}=\mathbb{R}$, $\mathrm{X}=X_{1}\times X_{2}=\mathbb{R}^{2}$. Denote

$\omega_{ij}(t)\mathrel{\triangleq}(-1)^{i}\max\\{0,t-j\\},\quad
h_{ij}(t)\mathrel{\triangleq}a_{i}\left(1+\max\\{0,t-j\\}\right),\quad
i\in{{1}..{4}},\ j\in{{0}..{2}},\ t\in[0,3],$

where $a_{1}\mathrel{\triangleq}(1,0)$, $a_{2}\mathrel{\triangleq}(0,1)$,
$a_{3}\mathrel{\triangleq}(-1,0)$, $a_{4}\mathrel{\triangleq}(0,-1)$. Put (see
figure 2; for reasons of symmetry of $\mathrm{Z}$, the projection onto the
plane $\mathrm{T}\times X_{2}$ is only shown)

$\Omega\mathrel{\triangleq}\\{\omega_{ij}\mid
i,j\in{{1}..{2}}\\},\qquad\mathrm{Z}\mathrel{\triangleq}\\{h_{ij}\mid
i\in{{1}..{4}},\ j\in{{0}..{2}}\\}.$

Thus, all elements in $\Omega$ and $\mathrm{Z}$ are distinct. Moreover, if
$k=i$, then

$(\omega_{kl}|[0,\min\\{l,j\\}])=(\omega_{ij}|[0,\min\\{l,j\\}]),\quad(h_{kl}|[0,\min\\{l,j\\}])=(h_{ij}|[0,\min\\{l,j\\}]).$

Let m/f $\alpha$ be given by

$\displaystyle\alpha(\omega_{11})\mathrel{\triangleq}\\{h_{10},h_{11},h_{12},h_{21},h_{32},h_{41}\\},\quad\alpha(\omega_{21})\mathrel{\triangleq}\\{h_{30},h_{31},h_{32},h_{12},h_{21},h_{41}\\},$
$\displaystyle\alpha(\omega_{12})\mathrel{\triangleq}\\{h_{20},h_{21},h_{22},h_{11},h_{32},h_{42}\\},\quad\alpha(\omega_{22})\mathrel{\triangleq}\\{h_{40},h_{41},h_{42},h_{12},h_{22},h_{31}\\}.$

Let $H_{1}\mathrel{\triangleq}[0,1]$. Then $\Omega=\Omega(\omega|H_{1})$ for
all $\omega\in\Omega$; hence, by definition (28), we have the equalities

$\displaystyle\langle{\alpha}\rangle_{H_{1}}(\omega_{11})=\\{h_{11},h_{12},h_{21},h_{32},h_{41}\\},$
$\displaystyle\langle{\alpha}\rangle_{H_{1}}(\omega_{12})=\\{h_{21},h_{22},h_{11},h_{32},h_{42}\\},$
$\displaystyle\langle{\alpha}\rangle_{H_{1}}(\omega_{21})=\\{h_{31},h_{32},h_{12},h_{21},h_{41}\\},$
$\displaystyle\langle{\alpha}\rangle_{H_{1}}(\omega_{22})=\\{h_{41},h_{42},h_{12},h_{22},h_{31}\\}.$

Let $H_{2}\mathrel{\triangleq}[0,2]$. Then
$\Omega(\omega_{11}|H_{2})=\\{\omega_{11}\\}$,
$\Omega(\omega_{21}|H_{2})=\\{\omega_{21}\\}$,
$\Omega(\omega_{12}|H_{2})=\Omega(\omega_{22}|H_{2})=\\{\omega_{12},\omega_{22}\\}$;
and by virtue of (28) we get

$\displaystyle\langle{\alpha}\rangle_{H_{2}}(\omega_{11})$ $\displaystyle=$
$\displaystyle\\{h_{10},h_{11},h_{12},h_{21},h_{32},h_{41}\\},$
$\displaystyle\langle{\alpha}\rangle_{H_{2}}(\omega_{12})$ $\displaystyle=$
$\displaystyle\\{h_{22},h_{42}\\},$
$\displaystyle\langle{\alpha}\rangle_{H_{2}}(\omega_{21})$ $\displaystyle=$
$\displaystyle\\{h_{30},h_{31},h_{32},h_{12},h_{21},h_{41}\\},$
$\displaystyle\langle{\alpha}\rangle_{H_{2}}(\omega_{22})$ $\displaystyle=$
$\displaystyle\\{h_{42},h_{22}\\}.$

Then for m/f $\langle{\langle{\alpha}\rangle_{H_{1}}}\rangle_{H_{2}}$ we have

$\displaystyle\langle{\langle{\alpha}\rangle_{H_{1}}}\rangle_{H_{2}}(\omega_{11})$
$\displaystyle=$ $\displaystyle\\{h_{11},h_{12},h_{21},h_{32},h_{41}\\},$
$\displaystyle\langle{\langle{\alpha}\rangle_{H_{1}}}\rangle_{H_{2}}(\omega_{12})$
$\displaystyle=$ $\displaystyle\\{h_{22},h_{42}\\},$
$\displaystyle\langle{\langle{\alpha}\rangle_{H_{1}}}\rangle_{H_{2}}(\omega_{21})$
$\displaystyle=$ $\displaystyle\\{h_{31},h_{32},h_{12},h_{21},h_{41}\\},$
$\displaystyle\langle{\langle{\alpha}\rangle_{H_{1}}}\rangle_{H_{2}}(\omega_{22})$
$\displaystyle=$ $\displaystyle\\{h_{42},h_{22}\\}.$

Finally, for the m/f $\langle{\langle{\alpha}\rangle_{H_{2}}}\rangle_{H_{1}}$
we have the relations:

$\displaystyle\langle{\langle{\alpha}\rangle_{H_{2}}}\rangle_{H_{1}}(\omega_{11})$
$\displaystyle=$ $\displaystyle\\{h_{21},h_{41}\\},$
$\displaystyle\langle{\langle{\alpha}\rangle_{H_{2}}}\rangle_{H_{1}}(\omega_{12})$
$\displaystyle=$ $\displaystyle\\{h_{22},h_{42}\\},$
$\displaystyle\langle{\langle{\alpha}\rangle_{H_{2}}}\rangle_{H_{1}}(\omega_{21})$
$\displaystyle=$ $\displaystyle\\{h_{21},h_{41}\\},$
$\displaystyle\langle{\langle{\alpha}\rangle_{H_{2}}}\rangle_{H_{1}}(\omega_{22})$
$\displaystyle=$ $\displaystyle\\{h_{42},h_{22}\\}.$

It is easy to see that there is an inequality
$\langle{\langle{\alpha}\rangle_{H_{2}}}\rangle_{H_{1}}\neq\langle{\langle{\alpha}\rangle_{H_{1}}}\rangle_{H_{2}}$,
indicating that the operators $\langle{\cdot}\rangle_{H_{1}}$ and
$\langle{\cdot}\rangle_{H_{2}}$ are not commutative.

Since the composition $\langle{\langle{\cdot}\rangle_{H_{2}}}\rangle_{H_{1}}$
corresponds to the order specified in theorem 2, the result —
$\langle{\langle{\alpha}\rangle_{H_{2}}}\rangle_{H_{1}}$ — presents an
$\\{H_{1},H_{2}\\}$-non-anticipative multiselector of $\alpha$.

###  5.3 Example 3

In this example, we consider an approaching game problem in which the m/f of
optimal trajectories does not have any non-empty-valued and non-anticipative
multiselector, that is, the problem is not solvable in the class of non-
anticipative strategies built on usual (not relaxed) controls. At the same
time, the m/f of optimal trajectories has the property of feasibility for all
partitions of the control interval; i. e. the step-by-step procedure can be
fulfilled for any partition and any disturbance.

$0$$1$$\tau$$2$${\,{\bf v}_{1}}$${\,{\bf v}_{2}}$${\,{\bf v}_{3}}$${\,{\bf
v}_{4}}$${\,\bf\ \cdot}$${\,\bf\ \cdot}$${\,\bf\ \cdot}$${\,\bf\
\cdot}$${\,\bf\ \cdot}$${\,\bf\ \cdot}$${\,\bf\ \cdot}$$0$$1$$\tau$$2$${\,{\bf
u}_{1}}$${\,{\bf u}_{2}}$${\,{\bf u}_{3}}$${\,{\bf u}_{4}}$${\,\bf\
\cdot}$${\,\bf\ \cdot}$${\,\bf\ \cdot}$${\,\bf\ \cdot}$${\,\bf\
\cdot}$${\,\bf\ \cdot}$${\,\bf\
\cdot}$${\bar{\mathbf{V}}}$${\bar{\mathbf{U}}}$ Figure 3: Example 3

Let the trajectories of the controlled system be given by solutions of the
following Cauchy problem:

$\begin{cases}\dot{x}(t)=u(t)-v(t),&t\in\mathrm{T}\mathrel{\triangleq}[0,2],\\\
x(0)=0\in\mathbb{R},&u\in{\mathbf{U}},v\in{\mathbf{V}},\end{cases}$ (64)
${\mathbf{U}}\mathrel{\triangleq}\\{u_{i}\mid
i\in\mathbb{N}\\},\qquad{\mathbf{V}}\mathrel{\triangleq}\\{v_{i}\mid
i\in\mathbb{N}\\},$ (65)
$u_{i}(t)\mathrel{\triangleq}\begin{cases}0,&t\in[0,1],\\\
1-1/i,&t\in(1,2],\end{cases}\qquad
v_{i}(t)\mathrel{\triangleq}\begin{cases}0,&t\in[0,1+1/i],\\\
1,&t\in(1+1/i,2],\end{cases}\qquad i\in\mathbb{N}.$ (66)

For system (64)–(66), consider the problem of meeting its trajectories
$x(\cdot)$ with the set
$M\mathrel{\triangleq}\\{(t,x)\in\mathrm{T}\times\mathbb{R}\mid(t=2)\mathrel{\&}(x\geqslant
0)\\}$ by choosing of control $u(\cdot)\in{\mathbf{U}}$ for all possible
disturbances $v(\cdot)\in{\mathbf{V}}$. Denote by $x(\cdot,u,v)$, where
$x(\cdot,u,v)\in C(\mathrm{T},\mathbb{R})$, the solution of the Cauchy problem
(64)–(66) where a control $u\in{\mathbf{U}}$ and a disturbance
$v\in{\mathbf{V}}$ are given. Denote
${\bar{\mathbf{U}}}\mathrel{\triangleq}\\{{\bf u}_{i}\mid i\in\mathbb{N}\\}$,
${\bar{\mathbf{V}}}\mathrel{\triangleq}\\{{\bf v}_{i}\mid i\in\mathbb{N}\\}$
(see figure 3)

${\bf u}_{i}(t)\mathrel{\triangleq}\int_{0}^{t}u_{i}(s)ds,\quad{\bf
v}_{i}(t)\mathrel{\triangleq}\int_{0}^{t}v_{i}(s)ds,\qquad i\in\mathbb{N},\
t\in\mathrm{T}.$

It is easy to verify that for
$x_{ij}(\cdot)\mathrel{\triangleq}x(\cdot,u_{i},v_{j})$ the equalities (see
(65), (66))

$x_{ij}(t)={\bf u}_{i}(t)-{\bf v}_{j}(t),\quad{\bf u}_{i}(2)={\bf
v}_{i}(2)=1-\frac{1}{i},\qquad i,j\in\mathbb{N},t\in\mathrm{T},$

and, consequently, the equalities

$x_{ij}(2)=\frac{1}{j}-\frac{1}{i},\qquad i,j\in\mathbb{N}$ (67)

are fulfilled. Hence, given a disturbance $v_{j}(\cdot)\in{\mathbf{V}}$ and a
control $u_{i}(\cdot)$, the meeting criterion can be written as the inequality

$i\geqslant j.$

Therefore, for any disturbance $v_{j}(\cdot)\in{\mathbf{V}}$, denoting by
$\alpha(v_{j}(\cdot))$ the set of all controls in ${\mathbf{U}}$ that solve
the meeting problem, we can write

$\alpha(v_{j})=\\{u_{j},u_{j+1},\ldots\\}=\\{u_{i}\in{\mathbf{U}}\mid
i\geqslant j\\}.$ (68)

That is, in (68) we have the m/f of optimal answers. The family of subsets
$\mathcal{T}$, implementing "flow of time", has its usual form:
$\mathcal{T}\mathrel{\triangleq}\\{[0,\tau]\mid\tau\in\mathrm{T}\\}$.

We choose a finite partition $\Delta\subset\mathrm{T}$ and show that
conditions $(\alpha,\Delta)$ are feasible. To this end, for ${\cal
H}_{\Delta}\in\mathcal{T}$ we construct using (55) ${\cal H}_{\Delta}$-non-
anticipative multiselector of $\alpha$ and verify that its values be non-
empty. Thus, by virtue of (21), the feasibility of step-by-step procedure
under conditions $(\alpha,\Delta)$ will be proven for arbitrary finite
partition $\Delta$.

1\. Denote $H_{\tau}\mathrel{\triangleq}[0,\tau]$. Then

$\Omega(v|H_{\tau})={\mathbf{V}},\qquad\tau\in[0,1],\ v\in{\mathbf{V}};$
$\Omega(v_{j}|H_{\tau})=\begin{cases}\\{v_{i},\ldots,v_{1}\\},&j\leqslant
i,\\\
\\{v_{j}\\},&j>i,\end{cases}\qquad\tau\in\left(1+\frac{1}{i+1},1+\frac{1}{i}\right],\quad
i,j\in\mathbb{N}.$

Hence, by direct calculation (see (28), (68)), we obtain the representation of
m/f $\langle{\alpha}\rangle_{H_{\tau}}$:

$\displaystyle\langle{\alpha}\rangle_{H_{\tau}}=\alpha,\qquad\tau\in[0,1],$
(69)
$\displaystyle\langle{\alpha}\rangle_{H_{\tau}}(v_{j})=\begin{cases}\alpha(v_{i}),&j\leqslant
i,\\\
\alpha(v_{j}),&j>i,\end{cases}\qquad\tau\in\left(1+\frac{1}{i+1},1+\frac{1}{i}\right],\quad
i,j\in\mathbb{N}.$ (70)

From the equalities (69) and (70) the relations follow

$(\mathrm{DOM})\left[{\langle{\alpha}\rangle_{H_{\tau}}}\right]=\Omega,\qquad\forall\tau\in\mathrm{T}.$
(71)

Moreover, it is not difficult to check that the representations (69) and (70)
hold for a wider set of m/f, namely, for any
$\beta\mathrel{\sqsubseteq}\alpha$ take place formulas:

$\langle{\beta}\rangle_{H_{\tau}}=\beta,\qquad\tau\in[0,1],$ (72)

and for any $\beta$ of the form
$\beta\mathrel{\triangleq}\langle{\alpha}\rangle_{H_{\xi}}$,
$\xi\in\mathrm{T}$, the equalities are true:

$\langle{\beta}\rangle_{H_{\tau}}(v_{j})=\begin{cases}\beta(v_{i}),&i\geqslant
j,\\\ \beta(v_{j}),&i<j,\end{cases}\quad\
\tau\in\left(1+\frac{1}{i+1},1+\frac{1}{i}\right],\quad i,j\in\mathbb{N}.$
(73)

2\. Using presentations (69)–(73) in accordance with the (55), we finally
obtain the equality

$\langle{\alpha}\rangle_{{\cal
H}_{\Delta}}=\langle{\alpha}\rangle_{[0,\min\\{\Delta\cap(1,2]\\}]}.$ (74)

Sines $\Delta$ is finite and $\Delta\cap(1,2]\neq\varnothing$ we have
$\langle{\alpha}\rangle_{{\cal
H}_{\Delta}}=\langle{\alpha}\rangle_{H_{\bar{\tau}}}$ for same
$H_{\bar{\tau}}\in{\cal H}_{\Delta}$. Then, in view of (71), we receive non-
emptiness of values of m/f $\langle{\alpha}\rangle_{{\cal H}_{\Delta}}$. That
is, non-emptiness of the set $\mathbf{N}^{0}_{({\cal
H}_{\Delta})}\left[{\alpha}\right]$ of all non-empty-valued and ${{\cal
H}_{\Delta}}$-non-anticipative multiselectors of $\alpha$. Hence, taking into
account theorem 1, we conclude that conditions $(\alpha,\Delta)$ are feasible.

3\. Let us show that the problem has no solution in the class of non-
anticipative strategies (quasi-strategies), that is, m/f $\alpha$ (68) does
not have non-empty-valued non-anticipative multiselectors:
$\mathbf{N}_{(\mathcal{T})}^{0}[{\alpha}]=\varnothing$.

Let’s say the contrary — m/f
$\beta\in\mathbf{N}_{(\mathcal{T})}^{0}[{\alpha}]$ was found. Then (15)
implies that
$\mathbf{N}^{0}_{(\mathcal{T})}\left[{\alpha}\right]\subset\mathbf{N}^{0}_{(\cal
H)}\left[{\alpha}\right]$, where ${\cal H}\mathrel{\triangleq}\\{H_{k}\mid
k\in\mathbb{N}\\}$ and $H_{k}\mathrel{\triangleq}[0,1+1/k]$. Let’s use the
implication (42); by assumption we have
$\beta\in\mathbf{N}^{0}_{(\mathcal{T})}\left[{\alpha}\right]\subset\mathbf{N}^{0}_{(\cal
H)}\left[{\alpha}\right]$, hence
$\beta\in\bigwedge_{k\in\mathbb{N}}\langle{\alpha}\rangle_{H_{k}}$. Then,
taking into account (70) and (68), for each $j\in\mathbb{N}$ we obtain

$\beta(v_{j}(\cdot))\subset\bigcap_{k\in\mathbb{N}}\langle{\alpha}\rangle_{H_{k}}(v_{j}(\cdot))=\bigcap_{k>j}\alpha(v_{k}(\cdot))=\varnothing.$

So, $(\mathrm{DOM})\left[{\beta}\right]=\varnothing$ and relation
$\beta\in\mathbf{N}_{(\mathcal{T})}^{0}[{\alpha}]$ is impossible. Then, we
have $\mathbf{N}_{(\mathcal{T})}^{0}[{\alpha}]=\varnothing$.

###  5.4 Example 4

Consider an example from [7, §5]. Here, we have the case where the
construction of partially non-anticipative multiselector provides a classic
non-anticipative multiselector.

As in previous example, the controlled system is given by solutions of the
following Cauchy problem:

$\dot{x}(t)=u(t)+v(t),\quad t\in\mathrm{T}\mathrel{\triangleq}[0,3],\quad
x(0)=0,$

where $x(t)\in\mathbb{R}$, the control $u$ and disturbance $v$ are Borel
measurable functions subject to the instantaneous constraints

$u(t)\in P\mathrel{\triangleq}[-1,1],\quad v(t)\in
Q\mathrel{\triangleq}\\{-1,0,1\\},\quad t\in\mathrm{T}.$

Denote by $x(\cdot,u,v)$, where $x(\cdot,u,v)\in C(\mathrm{T},\mathbb{R})$,
the solution of the Cauchy problem where a control $u\in{\mathbf{U}}$ and a
disturbance $v\in{\mathbf{V}}$ are given.

Suppose that the set ${\mathbf{V}}$ of admissible disturbances consists of two
functions, ${\mathbf{V}}\mathrel{\triangleq}\\{v_{1},v_{2}\\}$, where:

$\displaystyle v_{1}(t)\mathrel{\triangleq}\begin{cases}0,&\mbox{if
}t\in[0,1]\cup(2,3],\\\ 1,&\mbox{if }t\in(1,2],\end{cases}\quad
v_{2}(t)\mathrel{\triangleq}\begin{cases}0,&\mbox{if }t\in[0,1],\\\
-1,&\mbox{if }t\in(1,3].\end{cases}$

The example meet the conditions of Corollary 3, in particular, the set of
disturbances is finite. So, due to the corollary, the class of non-
anticipative strategies in this problem is limited by the set of all
$[0,1]$-non-anticipative m/f (from
$\mathcal{P^{\prime}}({{\mathbf{U}}})^{\mathbf{V}}$); here ${\mathbf{U}}$ is
the set of all possible realizations of control.

The control minimizes the following cost functional:

$J(u,v)\mathrel{\triangleq}-|x(3;u,v)|,\quad u\in{\mathbf{U}},\quad
v\in{\mathbf{V}}.$

Namely, the goal of control side is to minimize the guaranteed result in the
class of non-anticipative strategies from
$\mathcal{P^{\prime}}({{\mathbf{U}}})^{\mathbf{V}}$.

Since we are considering a guaranteed state of the problem, denote by $\rho$
(yet unknown) optimal (minimal) guaranteed result. Let us compose the m/f
$\alpha_{\rho}\in\mathcal{P}({{\mathbf{U}}})^{\mathbf{V}}$, which describes
the $\rho$-optimal control responses to the realized disturbance.

Taking into account the obvious inequality $\rho\leqslant 0$, we write the
values of $\alpha_{\rho}$ depending on $\rho$:

$\alpha_{\rho}(v)=\left\\{u\in{\mathbf{U}}\mid\left|\int_{0}^{3}u(s)+v(s)\mathrm{d}s\right|\geqslant-\rho\right\\},\qquad
v\in{\mathbf{V}}.$ (75)

Then we get the equalities

$\alpha_{\rho}(v_{1})=\left\\{u\in{\mathbf{U}}\mid\left(\int_{0}^{3}u(s)\mathrm{d}s\leqslant\rho-1\right)\vee\left(-\rho-1\leqslant\int_{0}^{3}u(s)\mathrm{d}s\right)\right\\},$
(76)
$\alpha_{\rho}(v_{2})=\left\\{u\in{\mathbf{U}}\mid\left(\int_{0}^{3}u(s)\mathrm{d}s\leqslant\rho+2\right)\vee\left(-\rho+2\leqslant\int_{0}^{3}u(s)\mathrm{d}s\right)\right\\}.$
(77)

It is clear that the set of values of the parameter $\rho$ for which
$\alpha_{\rho}$ has a non-empty-valued and non-anticipative multiselector is
bounded. That is, the set has infimum $\bar{\rho}$, which is the optimal
result of the control side in the class of non-anticipative strategies. So,
keeping in mind Corollary 3, we can write:

$\bar{\rho}=\min\left\\{\rho\in\mathbb{R}\mid(\mathrm{DOM})\left[{\top_{\mathbf{N}_{(\mathcal{T})}[{\alpha_{\rho}}]}}\right]={\mathbf{V}}\right\\}=\min\left\\{\rho\in\mathbb{R}\mid(\mathrm{DOM})\left[{\langle{\alpha_{\rho}}\rangle_{\\{[0,1]\\}}}\right]={\mathbf{V}}\right\\}.$

Let us calculate the m/f $\langle{\alpha_{\rho}}\rangle_{\\{[0,1]\\}}$:

$\langle{\alpha_{\rho}}\rangle_{\\{[0,1]\\}}(v)\mathrel{\triangleq}\left\\{u\in\alpha_{\rho}(v)\mid(u|[0,1])\in\bigcap_{v=v_{1},v_{2}}\llparenthesis\alpha_{\rho}(v)|[0,1]\rrparenthesis\right\\},\qquad
v\in\\{v_{1},v_{2}\\}.$

Note that the condition
$(\mathrm{DOM})\left[{\langle{\alpha_{\rho}}\rangle_{\\{[0,1]\\}}}\right]={\mathbf{V}}$
is equivalent to the inequality
$\bigcap_{v=v_{1},v_{2}}\llparenthesis\alpha_{\rho}(v)|[0,1]\rrparenthesis\neq\varnothing$.
The last inequality is true if and only if (see (76), (77)) the following two
relations are fulfilled:

$\displaystyle\left(\rho-1-\min_{u\in{\mathbf{U}}}\int_{1}^{3}u(s)\mathrm{d}s\geqslant\int_{0}^{1}u(s)\mathrm{d}s\right)\vee\left(\int_{0}^{1}u(s)\mathrm{d}s\geqslant-\rho-1-\max_{u\in{\mathbf{U}}}\int_{1}^{3}u(s)\mathrm{d}s\right),$
$\displaystyle\left(\rho+2-\min_{u\in{\mathbf{U}}}\int_{1}^{3}u(s)\mathrm{d}s\geqslant\int_{0}^{1}u(s)\mathrm{d}s\right)\vee\left(\int_{0}^{1}u(s)\mathrm{d}s\geqslant-\rho+2-\max_{u\in{\mathbf{U}}}\int_{1}^{3}u(s)\mathrm{d}s\right),$

that is, the relations below are true:

$\displaystyle\left(\rho+1\geqslant\int_{0}^{1}u(s)\mathrm{d}s\right)\vee\left(\int_{0}^{1}u(s)\mathrm{d}s\geqslant-\rho-3\right),$
$\displaystyle\left(\rho+4\geqslant\int_{0}^{1}u(s)\mathrm{d}s\right)\vee\left(\int_{0}^{1}u(s)\mathrm{d}s\geqslant-\rho\right).$

Hence, we come to the equality

$\bar{\rho}=\min\left\\{\rho\in\mathbb{R}\mid\left\\{u\in{\mathbf{U}}\mid\rho+4\geqslant\int_{0}^{1}u(s)\mathrm{d}s\geqslant-\rho-3\right\\}\neq\varnothing\right\\}.$

The latter condition implies that $\rho+4\geqslant-\rho-3$ or
$\rho\geqslant-3.5$.

So, for the value $\bar{\rho}$ of optimal guarantee in the class of non-
anticipative strategies, we have the equality $\bar{\rho}=-3.5$. In this case,
the greatest $[0,1]$-non-anticipative multiselector of $\alpha_{\bar{\rho}}$
is non-empty-valued and takes the form

$\displaystyle\langle{\alpha_{\bar{\rho}}}\rangle_{[0,1]}(v_{1})(t)\mathrel{\triangleq}\begin{cases}u(t),\
u\in{\mathbf{U}}_{([0,1],0.5)},&t\in[0,1],\\\
1=\operatornamewithlimits{\mathrm{argmax}}_{w\in P}w,&t\in(1,3],\end{cases}$
$\displaystyle\langle{\alpha_{\bar{\rho}}}\rangle_{[0,1]}(v_{2})(t)\mathrel{\triangleq}\begin{cases}u(t),\
u\in{\mathbf{U}}_{([0,1],0.5)},&t\in[0,1],\\\
-1=\operatornamewithlimits{\mathrm{argmin}}_{w\in P}w,&t\in(1,3],\end{cases}$

where

${\mathbf{U}}_{([0,1],0.5)}\mathrel{\triangleq}\Big{\\{}u\in{\mathbf{U}}\mid(u|[0,1])\in\bigcap_{v\in\\{v_{1},v_{2}\\}}\llparenthesis\alpha_{\bar{\rho}}(v)|[0,1]\rrparenthesis\Big{\\}}\\\
=\Big{\\{}u\in{\mathbf{U}}\mid\int_{0}^{1}u(s)\mathrm{d}s=\bar{\rho}+4=-\bar{\rho}-3=0.5\Big{\\}}.$

Unlike Example 5.3, the optimal result here is achieved in the class of non-
anticipative strategies, and can be explicitly described due to finiteness of
the disturbances set.

##  6 Conclusion

Since the topic of the paper is at the initial stage of development, many
details and even essential issues remain unexplored. Here, we make some
remarks on them. Expression (55), in view of the maximality properties of (32)
and (44), gives some hope for new results on existence of a m/f non-
anticipative multiselector. Concerning applications, an interesting question
arises about the convergence of guaranteed results for ${\cal
H}_{\Delta}$-non-anticipatory multiselectors of a m/f $\alpha$ to the
guaranteed result of the non-anticipative multiselector of $\alpha$ as the
step of the partition $\Delta$ tends to zero. Another question is the
implementation of the proposed constructions in a solution of dynamic
optimization problem. Namely, we need a systematic approach to resolving of
the mathematical programming and the parametric optimization problems that
arise when "calculating" partially non-anticipatory multiselectors (see the
last example).

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