ON THE EXISTENCE OF A SOLUTION FOR SOME DISTRIBUTED OPTIMAL CONTROL HYPERBOLIC SYSTEM

We consider a Bolza problem governed by a linear time-varying DarbouxGoursat system and a nonlinear cost functional, without the assumption of the convexity of an integrand with respect to the state variable. We prove a theorem on the existence of an optimal process in the classes of absolutely continuous trajectories of two variables and measurable controls with values in a fixed compact and convex set.


Introduction.
Let us consider a control system described by a system of ordinary differential equations of the form ẋ = f (t, x, u), x(0) = x 0 , x(1) = x 1 , ( with a cost functional where f : One of the fundamental problems of optimization theory is the question of the existence of optimal processes for the system of (1.1) and (1.2).This problem was the topic of investigations in many papers and monographs (cf.[1,5] and the references therein).The natural spaces in which one studies the existence of solutions for the system (1.1) and (1.2) are the space of absolutely continuous trajectories AC([0, 1], R n ) and the space of essentially bounded controls with values in the set M. Under some assumptions about the functions f , f 0 , and the set M (the growth conditions of the function f 0 , the convexity of f 0 with respect to u as well as the convexity and the compactness of M), it is possible to prove that the system (1.1) and (1.2) possesses a solution in the space AC([0, 1], R n ) × L ∞ ([0, 1], R r ) (cf. [1,5]).
In the present paper, we consider the problem of the existence of solutions for a system with distributed parameters of the form ∂ 2 z ∂x ∂y = A 0 (x, y)z + A 1 (x, y) ∂z ∂x + A 2 (x, y) ∂z ∂y + B(x, y)u a.e. on K, (1.3) z(•, 0) ≡ 0 on [0, 1], z(0, •) ≡ 0 on [0, 1] (1.4) with a cost functional I(z, u) = K f 0 x, y, z(x, y), u(x, y) dx dy, u ∈ M, (1.5) where z = (z 1 ,...,z n ), u = (u 1 ,...,u r ), (x, y) / / R, M ⊂ R r is a convex and compact set.Control system (1.3) and (1.4) is con- sidered in the space of trajectories which are absolutely continuous on K(z ∈ AC) (cf.[11]) and in the space ᐁ M of controls u essentially bounded and such that u(x, y) ∈ M for (x, y) ∈ K a.e.The basic result of our paper is a theorem on the existence of solutions, stating that if the function f 0 is convex with respect to u, continuous with respect to (z, u), measurable with respect to (x, y), and satisfies some growth condition, then the system (1.3), (1.4), and (1.5) possesses an optimal solution.This theorem has a form quite analogous to existence theorems for ordinary systems.
Systems of the form (1.3) were the objects of investigations in many papers.Essential results concerning the existence of smooth solutions can be found in [2].The problem of the existence of solutions in Sobolev spaces is considered in [9].In [3,8], the existence and uniqueness of a solution in the class of continuous functions is assumed.Under the above assumptions, the maximum principle for piecewise continuous controls is proved.In [13], the system (1.3) and (1.4), with a cost functional of the form is considered in the spaces of absolutely continuous trajectories and measurable controls with values in a fixed compact and convex subset of R r .Using Dubovitskii-Milyutin method, the author gives necessary conditions for optimality that are analogous to the Pontryagin maximum principle for ordinary systems.
In our paper, we introduce the notion of equiabsolute continuity of a family of absolutely continuous functions of two variables and give necessary and sufficient conditions for such a family to be equiabsolutely continuous (the analogue of [1, 10.2(i)]).Next, we prove the Ascoli-Arzela theorem for absolutely continuous functions of two variables.Making use of this theorem, we prove an analogue of [1, 10.8(iv)] for system (1.3).Finally, on the basis of the lower semicontinuity theorem (cf.[1, 10.8(i)]), we obtain a theorem on the existence of an optimal solution of problem (1.3), (1.4), and (1.5).
Systems of the form (1.3), (1.4), and (1.5) have a natural physical interpretation which is given at the end of this paper.The associated function F z of an interval is defined by the formula Let us recall that a function F of an interval Q ⊂ K is called absolutely continuous if, for any ε > 0, there exists δ > 0 such that where µ 2 denotes Lebesgue measure in K (cf.[6]).
In [11], it was shown that z : K / / R is absolutely continuous if and only if there for all (x, y) ∈ K. Making use of the above integral representation, we can demonstrate that the absolutely continuous function z possesses (in the classical sense) the partial derivatives defined for (x, y) ∈ K a.e.These derivatives are, of course, integrable on K.
The space of all absolutely continuous vector functions z denoted by AC.The norm in this space is defined by the formula It is easy to see that the space AC with this norm is a Banach space.

Families of equiabsolutely continuous functions of two variables
; the Ascoli-Arzela theorem.First, we recall some definitions.
A family {ϕ s (•), s ∈ S} of functions defined on for all finite systems of nonoverlapping intervals ) and for all s ∈ S, where µ 1 denotes Lebesgue measure in [0, 1].
We have the following.

Lemma 3.1. If {ϕ s , s ∈ S} is a family of absolutely continuous functions on [0, 1], then this family is equiabsolutely continuous if and only if the family of derivatives {ϕ s , s ∈ S} is equiabsolutely integrable.
The above definitions and the proof of Lemma 3.1 can be found in [1, 10.2].Now, let us introduce the notion of equiabsolute continuity of a family of absolutely continuous functions of an interval that are defined on the collection of all closed intervals contained in K.
So, a family {F s : s ∈ S} of functions of an interval, which are absolutely continuous on K, is called equiabsolutely continuous if, for any ε > 0, there exists δ = δ(ε) > 0 such that for all finite systems of nonoverlapping closed intervals P i , i = 1,...,N, in K with N i=1 µ 2 (P i ) ≤ δ and for all s ∈ S. Before we prove an analogue of Lemma 3.1 for functions of an interval, we recall (cf.[6]) that an absolutely continuous function F on K of an interval possesses a derivative DF (x) for x ∈ K a.e.This derivatives is integrable on K and for any interval P ⊂ K.
Lemma 3.2.If {F s : s ∈ S} is a family of functions of an interval, which are absolutely continuous on K, then this family is equiabsolutely continuous if and only if the family of derivatives {DF s : s ∈ S} is equiabsolutely integrable on K. Proof Sufficiency.Let us fix ε > 0 and let δ > 0 be the number in the definition of equiabsolute integrability of the family of derivatives {DF s , s ∈ S}.
for all s ∈ S because µ 2 N i=1 P i ≤ δ.Necessity.Let us fix ε > 0 and let δ > 0 be the number in the definition of equiabsolute continuity on K of the family {F s , s ∈ S} for ε/6.Now, let us fix s ∈ S and the set where From the integrability of DF s it follows that there exists σ > 0 (depending on ε and s) such that for any measurable set F ⊂ K with µ(F ) ≤ σ .Without loss of generality, we may assume that σ ≤ δ/2.
Let G be an open set such that From [6, Lemma V. 4.1], it follows that there exists at most countable family Consequently, If we denote for N ∈ N, then we get for sufficiently large N. Thus, (3.16) In an analogous way, we can show that for any measurable set E ⊂ K with µ 2 (E) ≤ δ/2 and for any s ∈ S.
Now, let us introduce the notion of equiabsolute continuity of a family of absolutely continuous functions of two variables.
Using equalities (2.1) and (2.2), we easily notice that, for an absolutely continuous function z, We end the considerations of this section with Ascoli-Arzela theorem for absolutely continuous functions of two variables.Theorem 3.4.Let (z n ) n∈N be a sequence of absolutely continuous functions on K.If it is equibounded and equiabsolutely continuous on K, then we can choose a subsequence (z n k ) k∈N that is uniformly convergent on K to some function z 0 , which is absolutely continuous on K.
Proof.It is easy to see that the equiabsolute continuity of the sequence (z n ) n∈N carries its equicontinuity.Indeed, let ε > 0 and δ = min{δ 1 ,δ 2 ,δ 3 }, where δ 1 , δ 2 , δ 3 are the numbers in the definition of equiabsolute continuity of the sequences (z n (•, 0)) n∈N , (z n (0, •)) n∈N , (F zn ) n∈N , respectively, for ε/4.Then, for any points ( x, ȳ), ( x, ȳ) ∈ K, with | x − x|+| ȳ − ȳ| < δ, we have for any n ∈ N. Applying Ascoli-Arzela theorem for continuous functions (cf.[4, 1.5.4]),we assert that we can choose a subsequence (z n k ) k∈N that converges uniformly on K to some function z 0 continuous on K. Now, we show that the function z 0 is absolutely continuous on K. Indeed, from the equiabsolute continuity of the sequences F zn k k∈N , we have for any ε > 0, there exists δ > 0 such that for all finite systems of nonoverlapping closed intervals P i ⊂ K, i = 1,...,N, with N i=1 µ 2 (P i ) < δ and for all k ∈ N. If we denote , i = 1,...,N, then inequality (3.21) can be written in the form Using the pointwise convergence of the sequence (z n k ) k∈N to z 0 , we obtain from (3.22) This means that the function F z 0 of an interval is absolutely continuous on K.
In an analogous way, we can show that the equiabsolute continuity of the sequence (z n k (•, 0)) k∈N implies the absolute continuity of the function z 0 (•, 0), and the equiabsolute continuity of the sequence (z n k (0, •)) k∈N implies the absolute continuity of the function z 0 (0, •).
So, the function z 0 is absolutely continuous on K and the proof is completed.

4.
On the existence of an optimal solution.Let us consider system (1.3) and (1.4).In the sequel, we assume that the functions are measurable and essentially bounded.
The class of admissible controls is defined as follows: where M ⊂ R r is a fixed compact and convex set.
In [11], the author proved the following.
Since, in the sequel, we use some facts from the proof of Theorem 4.1, we reproduce the proof here.
Proof of Theorem 4.1.Let us define the following operator: It is easy to see that this operator is continuous.Consider a sequence (l k ) k∈N defined by the recurrence relation Of course, l k can be represented in the form where By definition, where and, consequently, It can be easily noticed that Ᏺ 2 (Bu) is the sum of 3 2 components, and that each component may be estimated by C 2 N. Thus, and, consequently, On the basis of the induction principle, it can be shown that Ᏺ s (Bu) is the sum of 3 s components.Each component of that sum is the product of s coefficients A i , i = 0, 1, 2, and a k-fold, k ≥ s, multiple integral.In this integral, there are at least [(s + 1)/2] integrations with respect to x or y.This implies that each component of the sum may be estimated by Consequently, Since the series of numbers is convergent, there exists a limit (in From the continuity of Ᏺ and from (4.4), we obtain we obtain a solution of system (1.3) in the space AC, satisfying the boundary conditions (1.4).
Proof.Let (u n ) n∈N be a sequence of controls from ᐁ M and let us choose from it, on the basis of Lemma 4.3, a subsequence (u n k ) k∈N such that u n k k→∞ / u 0 weakly in L 1 (K, R r ), where u 0 is some function belonging to ᐁ M .From Lemmas 4.5 and 4.6, it follows that the sequence (z n k ) k∈N of the corresponding solutions of system (1.3) and (1.4) satisfies the assumptions of Theorem 3.4.So, we may choose a subsequence, say still (n k ), such that z n k / / / / k→∞ z 0 uniformly on K, where z 0 is some function from AC. From Lemma 4.6, it follows that the sequence (∂ 2 z n k /∂x ∂y k∈N ) is equiabsolutely integrable on K. Thus, making use of Dunford-Pettis theorem (cf.[1, 10.3(i)]), we may choose a subsequence, say still (n k ) k∈N , such that ∂ 2 z n k /∂x ∂y k→∞ / σ weakly in L 1 (K, R n ), where σ is some function from L 1 (K, R n ).In view of the above, let us observe that, for any (x, y) ∈ K, for (x, y) ∈ K a.e.So, from the linearity and the continuity of the operator To complete the proof, it is sufficient to show that the pair (z 0 ,u 0 ) satisfies system (1.3) and (1.4).
Indeed, the fact that z 0 satisfies the boundary conditions (1.4) follows immediately from the uniform convergence of the sequence (z n k ) k∈N to z 0 .
The fact that (z 0 ,u 0 ) satisfies (1.3) follows from the convergences weakly in L 1 (K, R n ) and from the fact that each pair (z n k ,u n k ), k ∈ N, satisfies (1.3).Now, let us consider Bolza problem (1.3), (1.4), and (1.5) in the spaces AC of trajectories and U M of controls.Consider the function We assume that process plays no essential role in the motion of the gas.In this case, the process of the absorption of the poison gas by the filter, filled up with the substance S, is described by a differential equation of the form where y 0 is the gas concentration at the inlet to the filter (y 0 -const.),v(x, t) denotes the speed of the flow of the mixture of air and gas through the filter at the moment t and the distance x from the inlet of the filter, v 0 = v(0, 0), β and γ are physical quantities characterizing the given gas (for details, see [ x = 0, z(x, 0) = 0, z(0,t) = 0. (5.4) Let us suppose that we have some influence on the process of the filtering of the gas, and that our control has a linear character.In this situation, we can assume that the system describing this process is of the form Assume that f 0 satisfies conditions (1), ( 2), (3), and (4).By Theorem 4.7, control system (5.5),(5.6), and (5.7) possesses an optimal process (z 0 ,u 0 ) in the space of absolutely continuous trajectories z ∈ AC and in the set of admissible controls u ∈ L ∞ ([0, 1], [c, d]).

2 .Definition 2 . 1 .
Preliminaries.First, we recall the definition of an absolutely continuous function on K, introduced in[11].A function z : K / / R is called an absolutely continuous func- tion on K (shortly, an AC function) if the associated function F z of an interval is an absolutely continuous function of an interval and the functions z(•, 0), z(0, •) are absolutely continuous functions of one variable on [0, 1].

6 )
The function u :K / / [c, d],where −∞ < c < d < ∞ are fixed numbers, is treated as a control.Suppose that the cost functional has the form I(z, u) , t, z(x, t), u(x, t) dx dt.(5.7) .24) χ A denotes the characteristic function of the set A.On the other hand, since the sequence (z n k ) k∈N converges uniformly on K to z 0 , we have 10, Chapter II]).Without loss of generality, we may assume that x ∈ [0, 1] and t ∈ [0, 1].Put It is easy to demonstrate that the system (5.1) and (5.2) is equivalent to a system of the form