On a dynamic game problem with an indecomposable set of disturbances

For an abstract dynamic system, a game problem of retention of the motions in a given set of the motion histories is considered. The case of an indecomposable set of disturbances is studied. The set of successful solvability and a construction of a resolving quasistrategy based on the method of programmed iterations is proposed.


Introduction
The theory and methods of solving control problems are widely developed in the case when the instantaneous (geometric) restrictions in couple with measurability are the only claim describing the set of admissible disturbances. For these important problems, such a fundamental results as the theorem on the alternative and the extreme aiming method are established (see [5] and the references therein). This kind of restrictions imply the decomposability property [7] of the set of admissible disturbances. In control problems, this property is known as the possibility to "glue" any admissible disturbances at any time for composing a new admissible disturbance. On the other hand, a notable family of control problems is characterized by the absence of this property: typical cases are given by the systems with continuous or constant disturbances.
In the present paper we consider a dynamic control problem with an indecomposable set of disturbances. The consideration is carried out on the example of a retention problem -a simple case of the positional differential game. We search a solution of the problem in a set of quasistrategies. The description of controlled system is given in an abstract form and, in general, follows the scheme [9]. The proposed solution is based on the method of programmed iterations (see [1] and the references therein; see also [9]). The formalization studied lies in direction of the problems that use an additional information on the functional properties of the set of disturbances (see, e.g., [4,6]). The absence of topological requirements on the elements of the retention problem is compensated by an increasing of the iterations "number" [8]. As usual in the abstract setting, the control time "interval" is not assumed to be bounded or connected.

Dynamic system
Denote by P(T ) (by P ′ (T )) all (all non-empty) subsets of the set T . For non-empty sets A and B, let B A be the set of all mappings defined on the set A with values in the set B. If, in addition, f ∈ B A and C ∈ P ′ (A), then (f | C) ∈ B C denotes the restriction of the mapping f to the set C: Choose and fix a non-empty subset I of real numbers R as an analogue of a time interval. Non-empty sets X and Y specify the ranges of the spatial variables and the disturbance values respectively. If t ∈ I, then we denote I t {ξ ∈ I | ξ ≤ t} and I t {ξ ∈ I | ξ ≥ t}. Let C ∈ P ′ (X I ) and Ω ∈ P ′ (Y I ) be the sets of admissible trajectories and disturbances respectively. Denote by D I × C × Ω the state space of the controlled process. For any t ∈ I, x ∈ C, we denote As an analogue of the dynamic system, we fix a mapping S : D → P ′ (C) such that ∀t ∈ I, ∀τ ∈ I t ∀x, x ′ ∈ C and ∀ω, ω ′ ∈ Ω Thus, for every (t, x, ω) ∈ D, S(t, x, ω) denotes the set of all trajectories of the system (1.1)-(1.4) corresponding to the history x up to the "moment" t and to the disturbance realization ω after t. Choose and fix some initial history (t 0 , x 0 ) ∈ I × C. All further constructions are carried out in order to formulate and solve the retention problem for this initial history. Let us define the set SP (t 0 ,x 0 ) ∈ P ′ (D) of all the states of the controlled process arising in the system from the initial history (t 0 , x 0 ) when all admissible disturbances are implemented: For a state (t, x, ω) ∈ SP (t 0 ,x 0 ) , we determine the set Ω(t, x, ω) of all disturbances that are compatible with this state: (1.6) So (see (1.2)), we have Ω(t 0 , x 0 , ω) = Ω for all ω ∈ Ω.

Control procedures and the retention problem
We assume that for the formation of the trajectories the controlling side uses non-empty-valued and non-anticipatory multifunctions from P(C) Ω . So, for a state (t, x, ω) ∈ SP (t 0 ,x 0 ) , we determine the set M (t,x,ω) of all admissible control procedures -of quasistrategies -as follows: (1.7) Let the set D ∈ P ′ (I × C), which describes the phase constraints, satisfies the conditions On the basis of D, we define the set N in the following way: and consider the retention of states of the control process in the set N as the aim of control. Namely, we say that the aim is attainable for the initial state (t, x, ω) if the inclusions hold for some quasistrategy α ∈ M (t,x,ω) . For the initial history (t 0 , x 0 ), this definition means that projections of current states of the control process on the set I × C remain in D for any disturbance ν ∈ Ω.

The programmed absorption operator and its iterations
For In terms of (2.1), we introduce the operator A, A ∈ P(SP (t 0 ,x 0 ) ) P(SP (t 0 ,x 0 ) ) , (the programmed absorption operator) by setting for any H ∈ P(SP (t 0 ,x 0 ) ). The definition of A immediately implies that Then, following the transfinite induction method (see, for example, [3, sec. I.3]), let us introduce α-iteration A α , A α ∈ P(SP (t 0 ,x 0 ) ) P(SP (t 0 ,x 0 ) ) , of the operator A for every ordinal α. For α = 0, we assume A 0 (H) H, ∀H ∈ P(SP (t 0 ,x 0 ) ); if α has a predecessor (let it be an ordinal γ), we write Here, by ≺, the strict order relation on the class of ordinals is denoted. Then, according to the transfinite induction principle, α-iteration A α of the operator A is correctly defined for every ordinal α. As a consequence of the definitions and (2.3) (see [8, (4.4)]), we get the following embedding for any ordinal α:

Quasistrategies solving the retention problem
Let us study the issue of solvability of the retention problem in the chosen class of quasistrategies. In the following, let the ordinal σ be strictly greater than the cardinality of the set N: |N| ≺ σ.
Thus, for any (t, x, ω) ∈ A σ (N), we have obtained an explicit form of a quasistrategy solving the retention problem in N.
Theorem 1. The following equality holds: Theorem 1 states that the set A σ (N) is the greatest of the subsets of initial positions from N that admit a solution of the retention problem in N in the class of quasistrategies. By Theorem 1, the original retention problem is solvable if and only if (t 0 , x 0 , ω 0 ) ∈ A σ (N) for some ω 0 ∈ Ω; as already mentioned, when it is solvable, the quasi-strategy Π(· | (t 0 , x 0 , ω 0 ), A σ (N)) implements this solution (see Lemma 1).

Preliminaries
We begin with some auxiliary results. Lemma 2 is based on the properties (1.2)-(1.4).

Proof of Theorem 1
From the relation A σ (N) ⊂ N (see (2.4)), in view of Lemma 1, we find that, under the inequality |N| ≺ σ, the embedding below holds: 1. Denote the set from the right-hand side of (2.6) by Λ. In view of (3.21), to prove the theorem, it is sufficient to establish the relation Λ ⊂ A σ (N).