Duality of the alternating integral for quasi-linear differential games

. The concept of alternated integrals proved to be useful in differential games and in control theory under the conditions of uncertainty. In this article the connection between upper and lower alternating integrals for quasi-linear differential games is established and its applications to the problem of pursuit is studied.


Introduction
To solve the problem of pursuit in linear differential games, L.S. Pontryagin suggested two direct methods [1,2]. Pontryagin's second direct method, based on concept of the alternating integral, which has no analogs in integration of real function. In definition of alternating integral participate operations of integration of set-valued mappings and geometric difference (Minkovski difference) of sets. These operations make difficulties for computation of alternating integral. A simplified schemes for constructing of alternating integral were proposed in [3,4]. An evaluation error of the numerical constructing of the alternating integral has been described in [5]. This elegant construction admits a generalization, which makes possible to define a differential and an integral of a setvalued mapping in such way that these two operations become mutually reverse. Such a generalization based on quasi-affine mapping has been proposed in [6,7]. Numerical algorithms for evaluation of a generalized alternating integral were proposed in [8,9]. An important facts has been established in [10,11]: A generalized alternating integral describes the epigraph of a function, which is the viscosity solution to a Hamilton-Jacobi equation.
Another set-valued integration procedure unifying Rieman-type integral, Auman's integral and Pontryagin's alternating integral were proposed in [12,13]. In these works the following important observation has been made: an alternating integral may serve cross-cuts of "bridge of Krasovski" and a level set of value functions for related Hamilton-Jacobi-Bellman-Isaacs equations. Some aspects of computation alternating integral basic on ellipsoidal calculation is studied. This procedure allowed to apply an alternating integral to the control synthesis problem.
Pontryagin's second direct method has played the great role in development of the differential games and control theory. Therefore, many works are devoted to investigation of this method (see [14][15][16][17][18][19][20][21][22]). In particular, a lower analogue of the Pontryagin alternating integral for linear differential games was introduced in [14] (to complete the symmetry the alternating integral defined in [2] was called the upper Pontryagin alternating integral). The lower alternating integral has been found useful in solving problems of pursuit under certain information discrimination of the pursuer compared to the evader. In [14], a connection was also established between these concepts, and it was applied to the problem of informality [4]. In this article a connection between the upper and lower alternating integrals for quasi-linear differential games and its applications to the problem of pursuit are established.
A differential game described by the equatioṅ is considered, where z ∈ R d , t ∈ I, u and v are control parameters, u is the pursuer parameter, v is the evader parameter, u ∈ P ∈ Ccm(R d ), υ ∈ Q ∈ Ccm(R d ), and f : I × P × Q → R d is a continuous function. The terminal set is M , M ⊂ R d . An arbitrary measurable function u(·) ∈ P (I) (v(·) ∈ Q(I)) is called admissible control parameter of the pursuer (evader). A given initial point z 0 and a pair of controlling parameters u(·) ∈ P (I), v(·) ∈ Q(I) give rise to a unique trajectory z(t) = z(t, z 0 , u(·), v(·)), t ≥ 0, of the system (1) (precisely definition of a trajectory is given in Section 4). Pursuit starts from a point z 0 ∈ R d \ M and it is considered to be ended, when the phase point hits the set M . In other words, the pursuer aims to realize the inclusion z(τ ) ∈ M . Then, we say that pursuit from a point z 0 is completed at the time τ in the game (1).
Naturally, there is a question: From which initial points z 0 pursuit can be completed at the time τ in the game (1)?
To solve this problem L.S. Pontryagin has introduced the second method of pursuit in a linear differential game. The second method of pursuit is formulated in terms of an alternating integral.
To each partition ω n of the interval I we assign upper and lower alternating sums S τ (M, ω n ) and S τ (M, ω n ) as described below: [14,15].
Further, if it will be necessary, we shall indicate in notations, the dependence of sums and integrals not only of ω or τ , but also of other initial data.
A concepts of the upper and lower alternating integrals have the following role in a quasi-linear differential games: For points z 0 with z 0 ∈ W τ (M ) (z 0 ∈ W τ (M )) the pursuit can be completed at the time τ with (without) discriminating against the evader controls [1, 2, [14][15][16].
Now we formulate the basic results. The following theorems establish connection between the upper and lower alternating integrals. holds.
Theorem 2. Let the set M ⊂ R d has the convex closed complement, and f (t, u, Q) is convex for any t ∈ I and u ∈ P . Then the equality [14,19].
Remark 2. It should be noted (see proof of the Theorem 1 and Theorem 2), for any set M ⊂ R d , the following relations are valid: where A c is a complement of a set A.

Preliminary lemmas
In this section we give some preliminary facts which are necessary for proving basic theorems.
Lemma 1. (See [14].) Let a sequence X k ∈ cl(R d ) decreases monotonically by inclusion and Y ∈ cm(R d ). Then the equality is valid.
It should be noted, for any family X α and a set Y ⊂ R d , the following relations hold.
Further, we assume that the set f (t, P, v) is convex for any t ∈ I and v ∈ Q.
One can easily verify the validity of the following Proof. It is obvious, www.mii.lt/NA Applying the inequality (6) to the right side of (8), we obtain By virtue of (8), we get from here (7).
Let ω n ∈ Ω, ξ i ∈ ∆ i , and Lemma 5. For any ε > 0, there exists a positive integer N such that for all n ≥ N , the inclusion Proof. Let the partition ω n satisfies condition Γ (δ) < ε/(3τ ). By definition, Applying Lemma 4 to the right side of the last equality, we obtain By definition, Using Lemma 4, we have where ξ n−k is arbitrary point from the segment ∆ n−k .
By virtue of assumption, Now, using (5), we obtain
Then by virtue of (5) we have H .
Proof. Obviously, the left side of (9) is contained in its right side. Prove the opposite.
Let v(·) be arbitrary function from Q(∆). Then there exists a sequence of piecewise constant functions v k (·) ∈ Q(∆) convergent almost everywhere to v(·) on ∆. Then By given ε > 0, we find η > 0 such that t ∈ ∆, u ∈ P , v 1 , v 2 ∈ Q, |v 1 − v 2 | < η. By the Egorov theorem, there exists an open subset ∆ 1 ⊂ ∆ with the measure µ(∆ 1 ) < ε such that v k (·) → v(·) uniformly on the closed set ∆ \ ∆ 1 . Then |v k (t) − v(t)| < η for all t ∈ ∆ \ ∆ 1 and k > N if N is sufficiently large. Hence, by virtue of (11) we have for all t ∈ ∆ \ ∆ 1 and k > N . Let λ = max{|f (t, u, v)|, t ∈ ∆, u ∈ P , v ∈ Q}, and |∆| is the length of the segment ∆. Then by the inclusion (12) we have Moreover, Note, ∆1 f (t, u, v(t)) dt ⊂ λεH. Therefore Adding (13) and (14), we obtain Thus, by virtue of (10), we have Since the set L is closed and ε is arbitrary, inclusion (15) implies Proof. By Lemma 7, it is sufficient to prove the following inclusion Let x be arbitrary element from the left side of (16). Then for some u ∈ P . Choose arbitrary piecewise constant functionv(t) from the collection Q(∆). We shall considerv we choose point ξ j and apply twice the following inequalities at first to the segment ∆, then to the segment [t j−1 , t j ]. Then we get follows. Now we multiply these inclusions by δ j /δ and add term by term. Since, the set L is convex and by virtue of (17) we have Let ω n ∈ Ω and Applying Lemma 8 sequentially to the partial sums B i , i = n, n − 1, . . . , 1, we get the following Lemma 9. If M ∈ Ccl(R d ), then for any ε > 0, there exists a positive integer N such that the inclusion is valid at all n ≥ N.
is convex.
Proof of Lemma 10 is obvious. By virtue of Lemma 10, Lemma 9 implies the following Lemma 11. Let M ∈ Ccl(R d ). Then for any ε > 0, there exists a positive integer N such that the relation takes place for all n ≥ N .

Proof of basic theorems
Proof of Theorem 1. It follows from Lemma 12, On the other hand, we have on the base of Lemmas 2 and 1 Theorem 1 is proved.
Let L c be a notation of the complement for the set L ⊂ R d . From the duality of operations of intersection and join it follows that v(·)∈Q(∆) u(·)∈P (∆) Applying formulas (18) to the upper alternating sum, we obtain Analogously on can verify Now, relations (19) and (20) take place.
Proof of Theorem 2. Theorem 1 implies Applying the operation of complement to the both parts of this equality, using relations (21) and (22), we obtain Theorem 2 is proved.
4 Application of duality of the alternating integral to differential games of pursuit Applications of the upper and lower alternating integral to quasi-linear differential games are similarly to the linear case [3]. Therefore, in this section we turn our attention to definitions of basic concepts and restrict ourself to state basic results in connection with the system (1). Let θ > 0, X ⊂ R d . We denote by X(θ) the family of all measurable functions x(·) : [0, θ] → X.
is said to be δ-strategy of the evader in the upper game. The mapping U * δ : R d × Q(δ) → P (δ) is said to be δ-strategy of the pursuer in the upper game.
Definition 2. The mapping U δ * : R d → P (δ) is said to be δ-strategy of the pursuer in the lower game. The mapping V δ * : R d × P (δ) → Q(δ) is said to be δ-strategy of the evader in the lower game.
A given starting point z 0 and a pair of strategies U * δ , V * δ correspond to the unique absolutely continuous trajectory z(t) = z(t, z 0 , U * δ , V * δ ), t ≥ 0, defined in a following way.
The trajectory z(t) = z(t, z 0 , U δ * , V δ * ) is defined similarly. It corresponds to the given starting point z 0 and the pair of strategies U δ * , V δ * . Definition 3. Pursuit from a point z 0 can be completed at the time τ in the upper game if for any δ = τ /n, there exists δ-strategy of the pursuer U * δ such that z(τ, z 0 , U * δ , V * δ ) ∈ M for any δ-strategy of the evader V * δ . Concept of possibility to complete pursuit at the time τ in the lower game can be introduced similarly.
On the base of these definitions, Theorem 1 can be interpreted as follows. If M ∈ Ccl(R d ), then pursuit from a point z 0 can be completed at the time τ in the upper game if and only if the pursuer in the lower game can transfer the phase point from the starting state z 0 into any neighborhood of the terminal set at the time τ (see [2,14,18,[23][24][25]).