On Dependence of Sets of Functions on the Mean Value of their Elements

The paper considers, for a given closed bounded set M ⊂ R and K = (0, 1) ⊂ R, the set M = {h ∈ L2(K; R ) | h(x) ∈ M a.e. x ∈ K} and its subsets M(ĥ) = n


Introduction
Most sets of admissible control functions in the theory of optimal control are given as sets of measurable functions with values from a given set: for a given reference domain Q ⊂ R n and a given set M ⊂ R m the set of admissible controls is defined as M = h is measurable in Q | h(x) ∈ M a.e. x ∈ Q .
Here n and m are arbitrary fixed positive integers.
Provided that Q is a bounded domain and M is a bounded and closed set, the set M can be split as M = ĥ ∈coM M(ĥ), where Here by coA we denote the convex hull of the set A and by | Q | we denote the Lebesgue measure of the set Q ⊂ R n .
Such a representation of M is useful when weak limits of sequences of control functions are involved, especially in procedures of relaxation via convexification, see, for instance, Warga [4]. Analogous splitting is used in the homogenization theory defining the so-called G θ -closures, see, for instance, Milton [2]. The corresponding relaxation procedures often involve the evaluation of integrals (over the periodicity cell K = (0, 1) n ) of the kind and the investigation of continuity properties of the functionĥ → I(ĥ). To do that, obviously, one needs to know certain properties of the dependence of sets M(ĥ) onĥ. In Sections 2 and 3 we shall show the following results.
Theorem 1. Let Q ⊂ R n be bounded Lipschitz domain and let the set M ⊂ R m is bounded and closed. Then for every given sequences {ĥ k } and {h k } such that Theorem 2. Let Q ⊂ R n be bounded Lipschitz domain and let the set M ⊂ R m is finite or M is the closed convex hull of a finite set of elements. Then for every fixed sequence {ĥ k } ⊂ coM that converges in R m to an elementĥ 0 and for every given element h 0 ∈ M(ĥ 0 ) there exists a sequence {h k }, h k ∈ M(ĥ k ), k = 1, 2, . . . , such that Remark 1. Obviously, from Theorem 2 it follows immediately that the multivalued mappingĥ → M(ĥ) is lower semicontinuous on coM (for the definition and properties of multivalued mappings we refer to Kuratowski [1]).

Remark 2.
It is easy to see that under hypotheses of Theorem 1 the functionĥ → I(ĥ) defined by (1.1) is lower semicontinuous provided that f is Caratheodory function and that f has a majorant f 0 ∈ L 1 (Q) (we recall that the set M is bounded). If, in addition, the hypotheses of Theorem 2 are satisfied, then the functionĥ → I(ĥ) is continuous on coM .

Proof of Theorem 1
In this Section, we give the proof of Theorem 1. Since all reasonings below do not depend on concrete properties of the reference domain Q, then, without loosing a generality, all proofs are given for the standard case Q = K := (0, 1) n . Since the set M is bounded and closed, then the convex hull coM of M is closed too and all sets M(ĥ) withĥ ∈ coM are nonempty closed sets. In what follows, we shall use the notion of the relative interior riA for convex sets A from Euclidean spaces, for instance ricoM stands for the relative interior of the convex hull of M . For the definition of riA and other notations and properties for convex sets we refer to Rockafellar [3]. Let r 0 be dimension of coM .
Step 2. Letĥ 0 does not belong to ricoM . Because ricoM is not empty (provided that M consists of more than one element), then there exist a vector a ∈ R m and a constant c such that |a| = 1, a,ĥ 0 = c < a,ĥ for allĥ ∈ ricoM .
Without loosing generality, we can assume that c = 0, otherwise we can use the transformĥ →ĥ −ĥ 0 .
Because the sets M and M 1 are compact, then there exists a continuous function γ, Without loosing generality, we can assume that the function γ is convex, otherwise we can pass to the bipolar γ * * , which has the desired properties. By construction, for nonnegative τ there exists the inverse function τ → γ −1 (τ ), γ −1 (γ(t)) = t for t ≥ 0, which is continuous and strictly increasing on {τ ∈ R| τ ≥ 0}. Now, from (2.1) and convexity of γ it follows that for every chosen h ∈ M there exists an element h * , where c(m, M ) depends only on m and M . This way, for our situation with a fixedĥ 0 ∈ coM 1 , for everyĥ ∈ coM and arbitrary chosen h ∈ M(ĥ) there exists a corresponding h * ∈ M 1 such that By construction, M(ĥ 0 ) ⊂ M 1 and the dimension of coM 1 is less than r 0 . From now on, we have to approximate the element h * ∈ M(ĥ * ) ⊂ M 1 by elements from M(ĥ 0 ) ⊂ M 1 , i.e. we have reduced the dimension r 0 of our problem to the problem with dimension less than or equal to r 0 − 1.
Step 3. To conclude our reasoning by induction over the dimension r 0 we have to prove our assertion for the case r 0 = 1. Ifĥ 0 ∈ ricoM , then we apply reasoning from Step 1. Ifĥ 0 does not belong to ricoM , then the set M 1 from the Step 2 consists of only one elementĥ 0 and the set M 1 consists of one constant function h 0 (y) =ĥ 0 a.e. y ∈ K. For this case we can apply the same reasoning as in Step 2, what gives the statement of Theorem for r 0 = 1.
To a given vector-function h ∈ M (it has only N admissible values from M ) we can appoint an element λ whose components λ j represent the volume fractions in K of the sets where the vector-function h has the value h j , j = 1, . . . , N, respectively. Let E := {z ∈ R N | Hz = 0}.
In these notations the statement of Theorem 2 is a straight consequence of: Here the subspace E 1 can be equal to {0} if 1 is orthogonal to E. From assumptions on a k we have the existence of z 0k ∈ E 0 and z 1k ∈ E 1 such that So, if necessary, using the transform a k → a k + z 1k and replacing E by E 0 , without loosing generality, we can assume that (i) the vector 1 is orthogonal to E; (ii) a k , 1 = 0, k = 1, 2, . . ..
For the case (b), without loosing generality, we can assume that Λ s is the façade with the minimal dimension s compared to all façades, which contain λ 0 . Hence, after relabeling indexes we obtain . . , then from a k → 0 as k → ∞, and z k , 1 = 0 , k = 1, 2, . . . , it follows immediately that the sequence {z k } is bounded.
(obviously, λ k →λ 0 as k → ∞). For those indexes {j ′ }, for which entries of z 0 are equal to zero, (by the initial assumptions on the sequence {a k }), but for the rest of indexes The general case of an arbitrary sequence {z k } is treated by standard reasoning by contradiction, i.e., we assume the contrary that there exist d > 0 and a sequence of indexes {k ′ } such that the distance fromλ 0 to {λ 0 + a k ′ + E} Λ is greater than d. After that we take an arbitrary subsequence of {a k ′ }, for which the corresponding sequence {z k ′ } converges. The proof of the first part of Theorem 2 is completed. Now, let M be closed convex hull of a finite number of elements {h 1 , . . . , h N } and let Since the function is a normal integrand on Λ × K ( for every fixed h ∈ M ), then every h ∈ M has the representation with some σ ∈ S. In turn, a subset of piecewise constant elements is dense in S and sets M(ĥ) have the same property. This way, by using Cantor's diagonal process, we have that it is sufficient to show the existence of the approximating sequence {h k } for the case of a piecewise element h 0 ∈ M(ĥ 0 ). Let If {ĥ k } ⊂ M andĥ k →ĥ 0 as k → ∞ then also {ĥ k } ⊂ coM ,ĥ 0 ∈ coM and h 0 ∈M(ĥ 0 ). This way, the existence of the desired approximating sequence {h k } follows immediately from the proof of the first part of Theorem 2. The proof of Theorem 2 is completed.