Relaxation methods for optimal control problems

We consider a nonlinear optimal control problem with dynamics described by a differential inclusion involving a maximal monotone map [Formula: see text]. We do not assume that [Formula: see text], incorporating in this way systems with unilateral constraints in our framework. We present two relaxation methods. The first one is an outgrowth of the reduction method from the existence theory, while the second method uses Young measures. We show that the two relaxation methods are equivalent and admissible.


Introduction
We consider the following optimal control problem: (1) In this problem, the dynamics of the system are described by a differential inclusion involving a maximal monotone map A : R N → 2 R N . We do not assume that D(A) = R N (recall that D(A) = {x ∈ R N : A(x) = ∅} is the domain of A(·)). This way we incorporate in our framework systems with unilateral constraints (differential variational inequalities). In addition, the control constraint set U (t, x) is state-dependent, that is, the system has a priori feedback, a setting which is of interest in engineering and economic problems. The existence theory of such problems is based on the so-called "reduction technique", which was developed in the pioneering works of Cesari [10,11] and Berkovitz [3] (see also the books of Berkovitz [4] and Cesari [12]). According to this method, the original optimal control problem is reduced to a calculus of variations problem with multivalued dynamics. This problem is obtained by elimination of the control variable u ∈ R m . For this approach to work, we need to have enough convex structure in the problem, usually expressed in terms of the "property Q" of Cesari. In the absence of such a convex structure, a minimizing sequence of state-control pairs need not converge to an admissible pair. To rectify this, we need to augment the system and pass to a "convexified" version known as the "relaxed problem", which captures the asymptotic behaviour of the minimizing sequences. The process of relaxation is a delicate one since we have to strike a sensitive balance between competing requirements. We want that the relaxed problem exhibits the following three fundamental properties: (a) Every original state is also a relaxed state (that is, the original problem is embedded in the relaxed one). (b) Every relaxed state can be approximated by original ones (that is, we want to make sure that we did not augment the system too much). (c) The values of the relaxed and original problems are equal and the relaxed problem has a solution (that is, there exists an optimal state-control pair).
Note that the first two requirements concern the dynamics of the system, while the third one concerns the cost functional. Any relaxation method which meets these three requirements, is said to be "admissible".
Key words and phrases. Admissible relaxation, maximal monotone map, Young measure, convex conjugate, weak norm.
In this paper, under general conditions on the data of problem (1), we present two such admissible relaxation methods. The first one is an outgrowth of the reduction method from the existence theory, while the second method uses Young measures.

Mathematical background and hypotheses
and the graph of A(·) is the set then we say that A(·) is "strictly monotone". We say that A(·) is "maximal monotone", if Gr A is maximal with respect to inclusion among the graphs of all monotone maps. This is equivalent to saying that Suppose that Y, Z are Banach spaces and V : Y → Z. We say that V (·) is "compact", if it is continuous and maps bounded sets in Y onto relatively compact subsets of Z. Also, we say that V (·) is "completely continuous", if y n w − → y in Y , implies that V (x n ) → V (x) in Z. In general, these two notions are distinct. However, if Y is reflexive, then complete continuity implies compactness. Moreover, if in addition, V (·) is linear, then the two notions coincide.
From fixed point theory, we will need the so-called "Leray-Schauder alternative theorem", which we recall here.
Theorem 2.1. (See e.g. [17,28]) If Y is a Banach space, V : Y → Y is a compact map, and S = {y ∈ Y : y = λV (y) for some 0 < λ < 1}, then one of the following two statements is true: (a) S is bounded; (b) V has a fixed point. Now let (Ω, Σ, µ) be a finite measure space and X a separable Banach space. We introduce the following families of subsets of X: and the converse is true if Σ is µ-complete. In general, a multifunction F : Ω → 2 X \{∅} is said to be "graph measurable", if Gr F ∈ Σ ⊗ B(X). Given 1 p ∞ and a multifunction F : Ω → 2 X \{∅}, we introduce the set e. in Ω}. This set can be empty. For a graph measurable multifunction F (·), S p F = ∅ if and only if w → inf {||v|| X : v ∈ F (ω)} belongs in L p (Ω). The set S p F is "decomposable", in the sense that for every Here, for every C ∈ Σ, χ C is the characteristic function of C and C c is the complement of C (that is, C c = X\C). Since χ A c = 1 − χ A , the notion of decomposability formally looks like that of convexity. Only now the coefficients are functions. Nevertheless, decomposability is a good substitute of convexity and several results valid for convex sets have their counterparts for decomposable sets (see Fryszkowski [16]).
Let Y be Hausdorff topological space and G : Y → 2 X \{∅} a multifunction. We say that G(·) is "lower semicontinuous" (lsc for short), resp. "upper semicontinuous" (usc for short), if for every Recall that on P f (X) we can define a generalized metric, known as the "Hausdorff metric", by The next theorem, due to Bressan & Colombo [6] and Fryszkowski [15], is an illustration of how decomposability can serve as a substitute of convexity. It extends the celebrated Michael selection theorem.  Let (Ω, Σ, µ) be a complete finite measure space and Y a Polish space. Recall that this means that Y is a separable Hausdorff topological space and there is a metric d on Y compatible with the topology of Y such that (Y, d) is complete. By P (Y ) we denote the set of all probability measures on Y endowed with the narrow topology τ n (see Papageorgiou & Winkert [29,p. 375]). Let B(Y ) be a Borel σ-field on Y , ca(Σ ⊗ B(Y ) the space of all R-valued signed measures on Σ ⊗ B(Y ) and p Ω : Ω × Y → Ω the projection map. Given λ ∈ ca(Σ ⊗ B(Y )) and if µ = λ • p −1 Ω , the disintegration theorem says that there exists a Σ-measurable mapλ : Ω → P (Y ) such that By ca + (Σ ⊗ B(Y )) we denote the R + -valued elements of ca(Σ × B(Y )). A "Young measure" on Ω × Y is a λ ∈ ca + (Σ ⊗ B(Y )) such that . On account of the disintegration theorem mentioned above, we can identify λ ∈ Y(Ω × Y ) with its disintegrationλ(·). So, we say that a Young measure is a measurable mapλ : Ω → P (Y ). Such maps are also known as "transition measures". The space of transition measures is denoted by R(Ω, Y ). We know that the following statements are equivalent: . Given a Σ-measurable function u : Ω → Y , the "Young measure associated with u", is the transition probability defined bŷ with δ u(ω) (·) being the Dirac measure defined by Let ϕ : Ω × Y → R be a "Carathéodory function", that is, for all y ∈ Y the mapping ω → ϕ(ω, y) is Σ-measurable and for µ-a.e. ω ∈ Ω the mapping y → ϕ(ω, y) is continuous. We know that such The "Young narrow topology" on Y(Ω × Y ), is the weakest topology on Y(Ω × Y ) for which the maps withλ(·) being the disentegration of λ and ϕ ∈ Car b (Ω × Y ), are all continuous. This topology on Y(Ω × Y ) is denoted by τ y n . Now we introduce the hypotheses on the data of problem (1).
Remark 2.1. We do not require that D(A) = R N . In this way our framework incorporates systems with unilateral constraints (differential variational inequalities).
for almost all t ∈ T and all |x|, |y| r; for almost all t ∈ T and all x ∈ R N , with a 0 ∈ L ∞ (T ).
(ii) for every r > 0, there exists ϑ r ∈ L 1 (T ) such that for almost all t ∈ T and all |x|, |y|, |u|, |v| r; (iii) for every r > 0, there exists a r ∈ L 1 (T ) such that |L(t, x, u)| a r (t) for almost all t ∈ T and all |x|, |u| r.
We introduce the "convexified" dynamics of problem (1), namely the following control system Then we define the following two sets: is an admissible state-control pair for (2)}.
These are the sets of admissible trajectories for the original system (the set S ) and for the convexified system (the set S c ).

First relaxation method
The first relaxation method is motivated by the "reduction method" of the existence theory and it uses the convexified control system (2).
Proof. Consider the following orientor field We fix x ∈ R N and consider the multifunction t → U (t, x). The measurability of this multifunction (see hypothesis H(U )(i)) implies that we can find a sequence {u x n } n 1 ⊆ S U(·,x) = {u : T → R m measurable and u(t) ∈ U (t, x) for almost all t ∈ T } such that [20,Theorem 24,p.156]). Then we have This proves Claim 3.1.
Let r > 0 and consider x, y ∈ R N such that |x|, |y| r. If ξ ∈ F (t, x), then Given ǫ > 0, we choose v ∈ U (t, y) such that for almost all t ∈ T, where k r ∈ L 1 (T ).
Let ǫ → 0 + , and conclude that for almost all t ∈ T , the multifunction We consider the following multivalued Cauchy problem: We will prove the existence of solutions for this problem. To this end, let h ∈ L 1 (T, R N ) and consider the following Cauchy problem: By virtue of the Bénilan-Brezis theorem (see Brezis [7, Proposition 3.8, p.82]), we know that problem (5) has a unique solution x ∈ W 1,1 ((0, b), R N ) = AC 1 (T, R N ). So, we can define the solution map K : So, up to a subsequence, we may assume that (6) x n → x * in C(T, R N ).
Exploiting the monotonicity of A(·), we obtain (6) and recall that h n We deduce that for the original sequence we have This proves Claim 3.3.
Let N F : C(T, R N ) → 2 L 1 (T,R N ) be the multivalued Nemitsky operator corresponding to the multifunction F (t, x), that is, F (·,x(·))) for all x ∈ C(T, R N ). The measurability of t → F (t, x(t)) and hypotheses H(f )(iii) and H(U )(iii) imply that N F (x) ∈ P wk (L 1 (T, R N )) for all x ∈ C(T, R N ).  [20, p.61]). So, we can apply Theorem 2.2 and produce a continuous map g : C(T, R N ) → L 1 (T, R N ) such that (7) g(x) ∈ N F (x) for all x ∈ C(T, R N ).
Evidently, the map Hypotheses H(f )(iii) and H(U )(iii) and the complete continuity of K(·), imply that (K • g)(·) maps bounded sets in C(T, R N ) to relatively compact sets in C(T, R N ). Therefore (9) x → (K • g)(x) is compact.
Therefore C ⊆ C(T, R N ) is bounded. This proves Claim 3.4. Then (9) and Claim 3.4 permit the use of Theorem 2.1 (the Leray-Schauder alternative theorem). So, we can findx ∈ C(T, R N ) such that

Consider the multifunction
with L T being the Lebesgue σ-field of T and B(R m ) the Borel σ-field of R m . Applying the Yankovvon Neumann-Aumann selection theorem (see Hu & Papageorgiou [20, Theorem 2.14, p.158]), we obtain a measurable mapû : T → R m such that u(t) ∈ E(t) for almost all t ∈ T, ⇒ g(x)(t) = f (t,x(t))û(t) for almost all t ∈ T, ⇒ (x,û) ∈ P.
Moreover, as in the proof of Claim 3.4, we can show that there existsĉ > 0 such that ||x|| C(T,R N ) ĉ for all x ∈ S C .

The proof of Proposition 3.1 is now complete.
In what follows, we denote by L 1 (T, R m ) w the Lebesgue space L 1 (T, R N ) equipped with the weak topology and by L 1 w (T, R N ) the same space furnished with the weak norm. Proof. Let {(x n , u n )} n 1 ⊆ P c . Hypothesis H(U )(iii) implies that by passing to a subsequence if necessary, we may assume that Recall that ||x n || C(T,R N ) ĉ for all n ∈ N. Hence by hypothesis H(f )(iii) and the Dunford-Pettis theorem, we see that {g n (·) = f (·, x n (·))u n (·)} n 1 ⊆ L 1 (T, R n ) is relatively w-compact. So, we may assume that for some g ∈ L 1 (T, R N ) we have By Claim 3.4 in the proof of Proposition 3.1, we have Note that ⇒ ||f (·, x n (·)) * h(·) − f (·, x(·)) * h(·)|| ∞ → 0 as n → ∞ (see (12)).
If in (13) we pass to the limit as n → ∞ and use (10), (11), (14), From Proposition 3.6 of Brezis [7, p. 70], we have for all 0 s t b, (y, y * ) ∈ Gr A, n ∈ N, Invoking once again Proposition 3.6 of Brezis [7, p. 70], we have for almost all t ∈ T, (see (15) and hypothesis H(U )(ii)). Therefore (x, u) ∈ P c and so we can conclude that P c is sequentially compact in C(T, R N ) × L 1 (T, R m ) w . The proof of Proposition 3.2 is now complete.

The proof of Proposition 3.3 is now complete.
We introduce the integrandL : ByL * * (t, x, u) we denote the second convex conjugate of the function u →L(t, x, u) (see, for example, Gasinski & Papageorgiou [17, p. 512]).
(b) Similarly, from (19) and (16), we see that for almost all t ∈ T and all (x, u) ∈ R N × R m .
The proof of Proposition 3.4 is now complete.
We will also need the following result about the integrable selectors of a graph measurable multifunction, which is actually of independent interest.
is graph measurable and S 1 G = ∅ and is uniformly integrable, then S 1 On the other hand, Proposition 7.16 of Hu & Papageorgiou [20, p. 232], says that given any ǫ > 0, we can find T ǫ ⊆ T closed such that |T \T ǫ | 1 < ǫ and G| Tǫ is h-continuous (here by | · |, we denote the Lebesgue measurable on R). Then we have G(T ǫ ) ∈ P k (R d ). Moreover, since S 1 G is uniformly integrable, so is conv S 1 G = S 1 conv G . Hence S 1 conv G has property U of Bourgain [5] and so applying the theorem of Gutman [19] (see also Hu & Papageorgiou [20,Proposition 4.14,p.195]), we infer that on S 1 conv G the w-topology and the || · || w -topology coincide. So, from (20), we have
The proof of Proposition 3.6 is now complete. Now we can establish the density of P in P c in the space C(T, R N ) × L 1 (T, R m ) w .
Since {û} n 1 ⊆ L 1 (T, R m ) is bounded (see hypothesis H(U )(iii)), it follows by Proposition 2.3 that We consider the following control system: Reasoning as in the proof of Proposition 3.1, we can show that for every n ∈ N, problem (24 n ) has admissible state-control pairs. So, let (x n ,v n ) be such a pair for (24 n ). Then (x n ,v n ) ∈ P for all n ∈ N.
As in the proof of Proposition 3.1, using the theorem of Baras [2], we see that So, we may assume that (25)x n →x * in C(T, R N ).
We have Exploiting the monotonicity of A(·), we obtain for almost all t ∈ T, We estimate the integral on the left-hand side of (26). Then We examine each summand on the right-hand side of (27). We have for some a 1 ∈ L 1 (T ), all n ∈ N (see (23), (25) and hypothesis H(f )(iii)) for some a 2 ∈ L 1 (T ), c 2 > 0 and all n ∈ N.
So, if in (26) we pass to the limit as n → ∞, and use (25) and (34), we get Therefore for the original sequence we have From (23) and (35), we see that (21) and (22)).
The proof of Proposition 3.7 is now complete.
Now we are ready to show that our first relaxation method which produces problem (18), is admisible. Proof. (a) Let {(ŷ n ,v n )} n 1 ⊆ P c be a minimizing sequence for the relaxed problem (18). So, we have J r (ŷ n ,v n ) ↓ m r . We know that However, from (38) and Proposition 3.2, we have (x,û) ∈ P c , ⇒ J r (x,û) = m r (see (39)).
The proof of Theorem 3.8 is now complete.
Recall that on P (B M ) we consider the narrow topology τ n . The orientor field for this new convexified control system iŝ So, if we denote byP c the set of admissible state-control pairs for (41), thenP c is compact in C(T, R N ) × Y(T ×B M ) τ y n . We show that this new relaxed optimal control problem is equivalent to (18) and hence also admissible. Letx n ∈ W 1,1 ((0, b), R N ) = AC 1 (T, R N ) be the relaxed state generated by the relaxed control λ n ∈ S Σ(·,xn(·)) . We know that {x n } n 1 ⊆ C(T, R N ) is relatively compact (see the proof of Proposition 3.1).
We use this as control constraint multifunction and consider the control system (24 n ). We can find admissible state-control pairs (x n ,v n ), n ∈ N for this system such that The proof of Theorem 4.1 is now complete.