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RELAXATION OF OPTIMAL CONTROL PROBLEMS DRIVEN BY NONLINEAR EVOLUTION EQUATIONS

. We consider a nonlinear optimal control problem with dynamics described by a nonlinear evolution equation deﬁned on an evolution triple of spaces. Both the dynamics and the cost functional are not convex and so an optimal pair need not exist. For this reason using tools from multivalued anal- ysis and from convex analysis, we introduce a relaxed version of the problem. No Young measures are involved in our relaxation method. We show that the relaxed problem is admissible.

1. Introduction. In this paper we study the following infinite dimensional optimal control problem governed by a nonlinear evolution equation defined on an evolution triple of spaces (see Section 2): subject to x (t) + A(t, x(t)) = G(t, x(t))u(t) for a.a. t ∈ T = [0, b], x(0) = x 0 , u(t) ∈ U (t, x(t)) for a.a. t ∈ T. (1.1) It is well known that in order to guarantee existence of optimal state-control pairs for problem (1.1), we need to have some convex structure in the dynamics and the cost functional of problem (1.1). We infer to Ahmed-Teo [1], Cesari [4], Fattorini [7,8,9], Hou [16], Liu-Liu-Fu [21], Papageorgiou [22,23] for such existence results. When this convex structure is not available, a minimizing sequence of admissible state-control pairs need not converge to an admissible pair. To overcome this difficulty, we need to enlarge the system in such a way, that the cost functional and the dynamics of the augmented system capture the asymptotic behaviour of the minimizing sequence of the original system. The new enlarged problem is known as the "relaxed problem" and it has to satisfy the following basic three requirements: (a) Every admissible state-control pair of the original system is admissible for the relaxed system, that is, the original system is embedded in the relaxed one. (b) Every relaxed state can be obtained as the uniform limit of a sequence of original states. This requirement guarantees that we did not augment the original dynamics too much. (c) The value of the two problems are equal and the relaxed problem admits optimal pairs. If these basic requirements are satisfied, then we say that the relaxation method is "admissible". The relaxation method is not unique. For finite dimensional systems various admissible relaxations and their relations can be found in Papageorgiou-Papalini [26] and Papageorgiou-Vetro-Vetro [33]. For infinite dimensional systems (distributed parameter systems), we refer to Buttazzo [2], Buttazzo-Dal Maso [3], Fattorini [10,11], Papageorgiou [24,25], Roubíček [34]. Here we develop a relaxation method which does not use transition measures (Young measures) and accommodates a large class of infinite dimensional optimal control problems.
Our work here can be extended to systems monitored by different types of evolution equations. We refer to the works of Kreulich [19] (equations driven by dissipative operators using the theory of nonlinear semigroups) and of Papageorgiou-Rȃdulescu [27], Papageorgiou-Rȃdulescu-Repovš [28,29,30,31,32] (evolutions driven by subdifferential operators, second order evolutions with viscosity, implicit evolutions).
2. Mathematical background -hypotheses. We start with some basic definitions and facts from multivalued analysis which we will need in the sequel. For details we refer to Hu-Papageorgiou [17] and Kuttler-Li-Shillor [20].
Let (Ω, Σ) be a measurable space and X a separable Banach space. We introduce the following families of subsets of X: P f (c) (X) = {E ⊆ X : E is nonempty, closed (and convex)}, , with B(X) being the Borel σ-field of X. For multifunctions with values in P f (X), measurability implies graph measurability and the converse is true if there is a σ-finite measure µ on Σ and Σ is µ-complete. Now suppose that (Ω, Σ, µ) is a finite measure space, F : Ω −→ 2 X \ {∅} is a multifunction and 1 p +∞. We introduce the following set If Y is regular and G is upper semicontinuous, then G is closed. For the converse to be true we need for G to have closed values and to be locally compact, that is, every y ∈ Y has a neighbourhood U such that Equivalently we can define the weak norm as p. 234]). This norm is equivalent to the Pettis norm (see Egghe [5]). From Lemma 2.8 of Hu-Papageorgiou [18, p. 24], we have the following result.
Now let H be a separable Hilbert space and X a separable, reflexive Banach space which is embedded continuously and densely into H. We identify H with its dual (that is, H = H * by the Riesz-Fréchet theorem). Then H is embedded continuously and densely into X * (see e.g., Gasiński-Papageorgiou [13, Lemma 2.2.27, p. 141]). Then the triple of spaces (X, H, X * ) is known in the literature as an "evolution triple". In this work we will also assume that X ⊆ H compactly. Then by Schauder's theorem (see e.g., Gasiński-Papageorgiou [13, Theorem 3.1.22, p. 275]) we also have that X ⊆ X * compactly. By | · | (respectively · , · * ) we denote the norm of H (respectively of X, X * ). Also by (·, ·) we denote the inner product of H and by ·, · the duality brackets for the dual pair (X, X * ). We know that ·, · | X×H = (·, ·). We define The time derivative involved in this definition is understood in the sense of vector valued distributions (see [13]). The space Furnished with this norm, W p (0, b) becomes a separable, reflexive Banach space. We know (see e.g., Gasiński-Papageorgiou [13]), that continuously and densely, Also we have the integration by parts formula (see e.g., Gasiński-Papageorgiou [13, Proposition 2.2.34, p. 148]).
We recall that for every r ∈ [1, +∞) and a separable reflexive Banach space V , with ·, · V being the duality brackets for the pair (V, V * ). By V * w (respectively V w ) we denote the reflexive Banach space V * (respectively V ) furnished with the weak topology. A map A : We know that a hemicontinuous, monotone map is "maximal monotone", that is A maximal monotone map A has a graph which is closed in V × V * w and in V w × V * . Moreover, a maximal monotone map which is also coercive, that is, A(u) V * −→ +∞ as u V −→ +∞, is surjective. For details on these and related issues, we refer to Gasiński-Papageorgiou [13] and Hu-Papageorgiou [17]. Now we introduce our conditions on the data of problem (1.1). So we are working on an evolution triple (X, H, X * ) in which the embedding X ⊆ H is compact. In particular, there exists β > 0 such that Also, Y is a separable reflexive Banach space and represents the control space.
H(A): A : T × X −→ X * is a map such that: there exist a 1 ∈ L p (T ), c 1 > 0 and 2 p < +∞ such that (iv): there exists c 0 > 0 such that is a map such that: (iii): there exist a ∈ L p (T ) and c > 0 such that If p = 2 we assume additionally that 2cβ < c 0 (see (2.1) and hypothesis H(A)(iv)). For the control constraint multifunction we assume: is a multifunction such that: Remark 2.4. So the control constraint multifunction is state dependent and thus the system has a priori feedback.
Above we have introduced the hypotheses on the items which determine the dynamics of the system. Next we introduce the conditions on the cost integrand L(t, x, u). H(L): L : T × H × Y −→ R is a function such that: We consider also the "convexified" control system, that is, the control system we get by convexifying the dynamics of (1.1).
x (t) + A(t, x(t)) = G(t, x(t))u(t) for a.a. t ∈ T, We introduce the following two sets: Evidently Y is the set of states of the original system, while Y c is the set of states of the relaxed (convexified) system. Clearly Y ⊆ Y c .
3. Admissible relaxation. In this section we construct an admissible relaxation using notions and techniques from multivalued analysis and from convex analysis, avoiding the use of Young measures. We start by proving the nonemptiness of Y ⊆ W p (0, b) and also by establishing useful topological properties for Y c . Proof. We introduce the multifunction F : On account of hypotheses H(U )(i), (ii), the multifunction (t, x) −→ U (t, x) is measurable (see Hu-Papageorgiou [17, Proposition 7.9, p. 229]). So, we can find measurable functions u n : (see Hu-Papageorgiou [17, Proposition 2.3, p. 155]). Therefore, we have Hypotheses H(G)(i), (ii) imply that the map (t, x) −→ G(t, x)u n (t, x) is jointly measurable (for all n ∈ N) and so we infer that Let η > 0 and consider x, y ∈ H with |x|, |y| η. Let h ∈ F (t, x). Then h = G(t, x)u with some u ∈ U (t, x).
Since ε > 0 is arbitrary, we let ε → 0 + to conclude that h(F (t, x), F (t, y)) ξ η (t)|x − y| for a.a. t ∈ T , all x, y ∈ H, |x|, |y| η, ξ η ∈ L p (T ), so We consider the following evolution inclusion We show that (2.2) and (3.4) are equivalent. Evidently, if x ∈ Y, then x also solves (3.4). On the other hand, let S(x 0 ) be the solution set of the multivalued Cauchy problem (3.4). If x ∈ S(x 0 ), then Clearly (t, u) −→ G(t, x(t))u is a Carathéodory function (that is, measurable in t and continuous in u). Therefore it is jointly measurable (see Hu-Papageorgiou [ We consider the multifunction Γ : Proposition 3.2. If hypotheses H(U ) and H(L) hold, then Γ is graph measurable and for all t ∈ T , Γ(t, ·) is closed.
We set On account of Proposition 3.2, we have We introduce the following modification of the cost integrand L: Evidently L is measurable and for all t ∈ T , L(t, ·, ·) is lower semicontinuous.
Using this proposition, we can generate a sequence of admissible original statecontrol pairs which approximate ( x, u) ∈ P c and the same is also true for the corresponding costs in L 1 w (T ).
Proof. According to Proposition 3.3 we can find v n ∈ S p U (·, x(·)) for all n ∈ N such that v n · w −→ u and v n w −→ u in L p (T ; Y ) as n → +∞. For every n ∈ N we introduce the multifunction E n : Using this multifunction as control set, we consider the following nonlinear control system x (t) + A(t, x(t)) = G(t, x(t))u(t) for a.a. t ∈ T, x(0) = x 0 , u ∈ S p En(·,x(·)) .

(3.6)
As in the proof of Proposition 3.1, we show that we can find an admissible statecontrol pair ( x n , u n ), n ∈ N, for problem (3.6). Evidently ( x n , u n ) ∈ P ∀n ∈ N. H)) is relatively compact. So, by passing to a subsequence if necessary, we may assume that x n (t) + A(t, x n (t)) = G(t, x n (t)) u n (t) for a.a. t ∈ T, x n (0) = x 0 , n ∈ N, x (t) + A(t, x(t)) = G(t, x(t)) u(t) for a.a. t ∈ T, We estimate the right hand side of (3.9): In the right hand side of (3.10), we examine the first summand: (by ·, · Y we denote the duality brackets for the pair (Y, Y * )). From the definition of the multifunction E n (t, Moreover, we have G(s, x n (s)) * L = G(s, x n (s)) L c 2 ∀n ∈ N, for some c 2 > 0. Using these estimates in (3.11), we obtain for all n ∈ N and some c 3 > 0 (see (3.7)). Now we examine the second summand in the right hand side of (3.10). We have (3.14) We return to (3.10), use (3.12), (3.13), pass to the limit as n → +∞ and then exploit (3.7) and (3.14). We obtain lim sup for some k ∈ L 1 (T ).
So, we introduce the following relaxation of the optimal control problem (1.1): subject to x (t) + A(t, x(t)) = G(t, x(t))u(t) for a.a. t ∈ T, x(0) = x 0 , u(t) ∈ conv U (t, x(t)) for a.a. t ∈ T. Proof. Let {(y n , v n )} n 1 ⊆ P c be a minimizing sequence for the relaxed problem (3.17), that is b 0 L * * (t, y n (t), v n (t)) dt m r .
From the last convergence, it follows that b 0 ( L * * (t, x, u) − L(t, x n , u n ) dt −→ 0, so m m r . Since we always have m r m, we conclude that m = m r .
Remark 3.6. The above theorem proves that the relaxation (3.17) of problem (1.1), that we have proposed, is admissible.