2 Mathematical background
Suppose that V and Z are Banach spaces and assume that V is embedded continuously and densely into Z (denoted by V↪Z). Then it is easy to check that:
Z* is embedded continuously into V*,
if V is reflexive, then Z*↪V*.
Having this observation in mind, we can introduce the notion of evolution triple of spaces, which is central in the class of evolution equations considered here.
Definition 2.1A triple (X,H,X*) of spaces is said to be an “evolution triple” (or “Gelfand triple” or “spaces in normal position”) if the following hold:
X is a separable reflexive Banach space and X* is its topological dual.
H is a separable Hilbert space identified with its dual H*=H (pivot space).
According to the remark made in the beginning of this section, we also have H*=H↪X*.
In this paper we also assume that the embedding of X into H is compact.
Hence, by Schauder’s theorem (see, for example, [20, Theorem 3.1.22]), so is the embedding of H*=H into X*.
In what follows, by ∥⋅∥ (resp. |⋅|, ∥⋅∥*) we denote the norm of the space X (resp. H,X*). By 〈⋅,⋅〉 we denote the duality brackets for the pair (X*,X) and by (⋅,⋅) the inner product of the Hilbert space H.
We know that
〈⋅,⋅〉|H×X=(⋅,⋅).
Also, let β>0 be such that
(2.1)|⋅|≤β∥⋅∥.
We introduce the following space which has a central role in the study of the evolution inclusions. So, let 1<p<∞ and set
Wp(0,b)={x∈Lp(T,X):x′∈Lp′(T,X*)} (1p+1p′=1).
In this definition the derivative of x is understood in the sense of vectorial distributions (weak derivative). In fact, if we view x as an X*-valued function, then x(⋅) is absolutely continuous, hence strongly differentiable almost everywhere. Therefore,
Wp(0,b)⊆AC1,p′(T,X*)=W1,p′((0,b),X*).
The space Wp(0,b), equipped with the norm
∥x∥Wp=∥x∥Lp(T,X)+∥x′∥Lp′(T,X*) for all x∈Wp(0,b),
becomes a separable reflexive Banach space. We know that
(2.2)Wp(0,b)↪C(T,H),
(2.3)Wp(0,b)↪Lp(T,H) compactly.
The following integration by parts formula is very helpful.
Proposition 2.2If x,y∈Wp(0,b), then t↦(x(t),y(t)) is absolutely continuous and
ddt(x(t),y(t))=〈x′(t),y(t)〉+〈x(t),y′(t)〉 for almost all t∈T.
We know that for all 1≤p<∞,
Lp(T,X)*=Lp′(T,X*)
with p′=+∞ if p=1 (see [20, Theorem 2.2.9]).
Now, let (Ω,Σ) be a measurable space and V a separable Banach space. We introduce the following hyperspaces:
Pf(c)(V)={C⊆V:C is nonempty, closed, (convex)},
P(w)k(c)(V)={C⊆V:C is nonempty, (weakly-)compact, (convex)}.
Given a multifunction F:Ω→2V∖{∅}, the “graph” of F is the set
GrF={(ω,v)∈Ω×V:v∈F(ω)}.
We say that F(⋅) is “graph measurable” if GrF∈Σ×B(V) with B(V) being the Borel σ-field of V.
If μ(⋅) is a σ-finite measure on Σ and F:Ω→2V∖{∅} is graph measurable, then the Yankov–von Neumann–Aumann selection theorem (see [22, Theorem 2.14, p. 158]) implies that F(⋅) admits a measurable selection, that is, there exists a Σ-measurable function f:Ω→V such that f(ω)∈F(ω) μ-almost everywhere.
In fact, there is a whole sequence {fn}n≥1 of such measurable selections such that F(ω)⊆{fn(ω)}¯ μ-almost everywhere (see [22, Proposition 2.17, p. 159]).
Moreover, the above results are valid if V is only a Souslin space.
Recall that a Souslin space need not be metrizable (see [21, p. 232]).
A multifunction F:Ω→Pf(V) is said to be “measurable” if for all y∈V, the function
ω↦d(y,F(ω))=infv∈F(ω)∥y-v∥V
is Σ-measurable.
A multifunction F:Ω→Pf(V) which is measurable is also graph measurable.
The converse is true if (Ω,Σ) admits a complete σ-finite measure μ.
If (Ω,Σ,μ) is a σ-finite measure space and
F:Ω→2V∖{∅} is a multifunction, then for 1≤p≤∞ we introduce the set
SFP={f∈Lp(Ω,Y):f(ω)∈F(ω)μ-a.e.}.
Evidently, SFP≠∅ if and only if ω↦infv∈F(ω)∥v∥V belongs to Lp(Ω). Moreover, the set SFP is “decomposable”, that is, if (A,f1,f2)∈Σ×SFP×SFP, then
χAf1+χΩ∖Af2∈SFP.
Here, for C∈Σ, by χC we denote the characteristic function of the set C∈Σ.
For every D⊆Σ,D≠∅, we define
|D|=supv∈D∥v∥V and σ(v*,D)=supv∈D〈v*,v〉V for all v*∈V*.
Here, 〈⋅,⋅〉V denotes the duality brackets of the pair (V*,V). The function σ(⋅,D):V*→ℝ¯=ℝ∪{+∞} is known as the “support function” of D.
Let Z,W be Hausdorff topological spaces. We say that a multifunction G:Z→2W∖{∅} is “upper semicontinuous” (USC for short), respectively, “lower semicontinuous” (LSC for short), if for all U⊆W open, the set
G+(U)={z∈Z:G(z)⊆U},respectively, G-(U)={z∈Z:G(z)∩U≠∅}
is open in Z.
If G(⋅) is both USC and LSC, then we say that G(⋅) is continuous.
On a Hausdorff topological space (W,τ) (τ being the Hausdorff topology), we can define a new topology τseq whose closed sets are the sequentially τ-closed sets.
Then topological properties with respect to this topology have the prefix “sequential”.
Note that τ⊆τseq and the two are equal if τ is first countable (see [7, p. 9] and [21, p. 808]).
We say that G:Z→2W∖{∅} is “closed” if the graph GrG⊆Z×W is closed.
For any Banach space V, on Pf(V) we can define a generalized metric, known as the “Hausdorff metric”, by setting
h(E,M)=max[supe∈Ed(e,M),supm∈Md(m,E)].
Recall that (Pf(V),h) is a complete metric space (see [22, p. 6]).
If Z is a Hausdorff topological space, a multifunction G:Z→Pf(V) is said to be “h-continuous”, if it is continuous from Z into (Pf(V),h).
Also, if E,M⊆V are nonempty, bounded, closed and convex subsets, then (Hörmander’s formula)
h(E,M)=supv*∈V*,∥v*∥V*≤1|σ(v*,E)-σ(v*,M)|.
Let (W,τ) be a Hausdorff topological space with topology τ and let {En}n≥1⊆2W∖{∅}. We define
Kseq(τ)-lim infn→∞En={y∈W:y=τ-limn→∞yn,yn∈En for all n∈ℕ},
Kseq(τ)-lim supn→∞En={y∈W:y=τ-limn→∞ynk,ynk∈Enk,n1<n2<⋯<nk<⋯}.
Sometimes we drop the Kseq-symbol and simply write τ-lim supn→∞En and τ-lim infn→∞En.
Returning to the setting of an evolution triple, we consider a sequence of multivalued maps
an,a:Lp(T,X)→2Lp′(T,X*)∖{∅} (n∈ℕ)
such that for every h*∈Lp′(T,X*) the inclusions
y′+an(y)∋h* (n∈ℕ) and y′+a(y)∋h
have unique solutions en(h*),e(h*)∈Wp(0,b).
We say that ddt+an “PG-converges” to ddt+a (denoted by ddt+an→PGddt+a as n→∞) if for every h*∈Lp′(T,X*), we have
en(h*)→𝑤e(h*) in Wp(0,b).
In what follows, by Xw (respectively, Hw,Xw*) we denote the space X (respectively, H,X*) furnished with the weak topology.
Also, by |⋅|1 we denote the Lebesgue measure on ℝ and by ((⋅,⋅)) we denote the duality brackets for the pair (Lp′(T,X*),Lp(T,X)).
So, we have
((h*,f))=∫0b〈h*(t),f(t)〉𝑑t for all h*∈Lp′(T,X*) and all f∈Lp(T,X).
Next, let us recall some useful facts from the theory of nonlinear operators of monotone type.
So, let V be a reflexive Banach space, L:D(L)⊆V→V* a linear maximal monotone operator and a:V→2V*.
We say that a(⋅) is “L-pseudomonotone” if the following conditions hold:
For every v∈V,a(v)∈Pwkc(V*).
a(⋅) is bounded (that is, maps bounded sets to bounded sets).
If {vn}n≥1⊆D(L), vn→𝑤v∈D(L) in V, L(vn)→𝑤L(v) in V*, vn*∈a(vn) for all n∈ℕ, vn*→𝑤v* in X* and lim supn→∞〈vn*,vn-v〉V≤0, then v*∈a(v) and 〈vn*,vn〉V→〈v*,v〉V.
Such maps have nice surjectivity properties.
The next result is due to Papageorgiou, Papalini and Renzacci [35], and it extends an earlier single-valued result of Lions [27, Theorem 1.2, p. 319].
Proposition 2.3Assume that V is a reflexive Banach space which is strictly convex, L:D(L)⊆V→V* is a linear maximal monotone operator and A:V→2V* is L-pseudomonotone and strongly coercive, that is,
infv*∈A(v)〈v*,v〉V∥v∥V→+∞ as ∥v∥V→+∞.
Then R(L+V)=V* (that is, L+V is surjective).
In the next section we obtain some results about a general class of evolution inclusions, which will help us study problem (1.1) (see Section 4).
3 Nonlinear evolution inclusions
Let T=[0,b] and let (X,H,X*) be an evolution triple with X↪H compactly (see Definition 2.1). In this section we deal with the following evolution inclusion:
(3.1)-x′(t)∈A(t,x(t))+E(t,x(t)) for almost all t∈T,x(0)=ξ.
The hypotheses on the data of (3.1) are as follows.
A:T×X→2X* is a map such that the following hold:
t↦A(t,x) is graph measurable for all x∈X.
GrA(t,⋅) is sequentially closed in Xw×Xw* and x↦A(t,x) is pseudomonotone for almost all t∈T.
For almost all t∈T, all x∈X and all h*∈A(t,x), we have
∥h∥*≤a1(t)+c1∥x∥p-1
with 2≤p,a1∈Lp′(T) and c1>0.
For almost all t∈T, all x∈X and all h*∈A(t,x), we have
〈h*,x〉≥c2∥x∥p-a2(t)
with c2>0,a2∈L1(T).
Remark 3.1If A(⋅,⋅) is single-valued, then in hypothesis (H$A$1) (ii) we can drop the condition on the graph of GrA(t,⋅) and only assume that x↦A(t,x) is pseudomonotone for almost all t∈T. The same applies if A(t,⋅) is maximal monotone for almost all t∈T.
An example of where the condition on the graph of A(t,⋅) is satisfied is the following (for simplicity we drop the t-dependence):
A(x)=-div∂φ(Dx)-divξ(Dx),
where φ:Lp(Ω,ℝN)→ℝ is continuous and convex, and ξ:Lp(Ω,ℝN)→ℝ is continuous and satisfies |ξ(y)|≤c^(1+|y|τ-1) for all y∈ℝN, c^>0 and 1≤τ<p. Then recalling that W1,p(Ω)↪W1,τ(Ω) compactly (see Zeidler [41, p. 1026]), we easily see that GrA is sequentially closed in W1,p(Ω)w×W1,p(Ω)w*.
F:T×H→Pfc(H) is a multifunction such that the following hold:
t↦F(t,x) is graph measurable for all x∈H.
GrF(t,⋅) is sequentially closed in H×Hw for almost all t∈T.
For almost all t∈T, all x∈H and all h∈F(t,x), we have
|h|≤a3(t)+c3|x|
with a3∈L2(T),c3>0 and if p=2, then β2c3<c2 (see (2.1)).
By a solution of problem (3.1) we understand a function x∈Wp(0,b) such that
-x′(t)=h*(t)+f(t) for almost all t∈T,
where h*∈Lp′(T,X*) and f∈L2(T,H) are such that
h*(t)∈A(t,x(t)) and f(t)∈F(t,x(t)) for almost all t∈T.
By S(ξ) we denote the set of solutions of problem (3.1). In the sequel we investigate the structure of S(ξ).
Consider the multivalued map a:Lp(T,X)→2Lp′(T,X*) defined by
(3.2)a(x)={h*∈Lp′(T,X*):h*(t)∈A(t,x(t)) for almost all t∈T} for all x∈Lp(T,X).
Note that a(⋅) has values in Pwkc(Lp′(T,X*)) (see hypotheses (H$A$1) (i) and (iii), and use the Yankov–von Neumann–Aumann selection theorem, see [22, Theorem 2.14, p. 158]).
Lemma 3.2If hypotheses (H$A$1) hold, xn→𝑤x in Wp(0,b), xn(t)→𝑤x(t) in X for almost all t∈T, hn*→𝑤h* in Lp′(T,X*) and hn*∈a(xn) for all n∈N, then h*∈a(x).
Proof.
Let v∈X and consider the function x↦σ(v,A(t,x)) (see Section 2).
We will show that it is sequentially upper semicontinuous.
To this end, we need to show that given λ∈ℝ, the superlevel set
Eλ={x∈X:λ≤σ(v,A(t,x))}
is sequentially closed in Xw. So, we consider a sequence {xn}n≥1⊆Eλ and assume that
xn→𝑤x in X.
Let hn*∈A(t,xn) (n∈ℕ) be such that
(3.3)〈hn*,v〉=σ(v,A(t,xn)) for all n∈ℕ
(recall A(t,xn)∈Pwkc(X*)).
Evidently, {hn*}n≥1⊆X* is bounded (see hypothesis (H$A$1) (iii)) and so, by passing to a subsequence if necessary, we may assume that hn*→𝑤h* in X*. Therefore, by hypothesis (H$A$1) (ii),
(3.4)h*∈A(t,x)
Then, from (3.3) and (3.4), we have
λ≤〈h*,v〉≤σ(v,A(t,x)),
and thus x∈Eλ.
This proves the upper semicontinuity of the map x↦σ(v,A(t,x)).
Now let v∈Lp(T,X). Then we have (see [22, Theorem 3.24, p. 183])
((hn*,v))≤σ(v,a(xn))=∫0bσ(v(t),A(t,xn(t)))𝑑t for all n∈ℕ,
and by Fatou’s lemma,
((h*,v))≤lim supn→∞σ(v,a(xn))≤∫0blim supn→∞σ(v(t),A(t,xn(t)))dt.
By the first part of the proof and since, by hypothesis, xn(t)→𝑤x(t) in X,
((h*,v))≤∫0bσ(v(t),A(t,x(t)))𝑑t=σ(v,a(x)).
Thus, h*∈a(x).
∎
Lemma 3.3If hypotheses (H$A$1) hold, then the multivalued map a:Lp(T,X)→2Lp′(T,X*), defined by (3.2), is L-pseudomonotone.
Proof.
Suppose xn→𝑤x in Wp(0,b),hn*∈a(xn) for all n∈ℕ, hn*→𝑤h* in Lp′(T,X*) and
(3.5)lim supn→∞((hn*,xn-x))≤0.
From (2.2), we infer that
(3.6)xn(t)→𝑤x(t) in H for all t∈T as n→∞.
We set ϑn(t)=〈hn*(t),xn(t)-x(t)〉.
Let N be the Lebesgue-null set in T=[0,b] outside of which hypotheses (H$A$1) (ii), (iii) and (iv) hold.
Using hypotheses (H$A$1) (iii) and (iv), we have
(3.7)ϑn(t)≥c2∥xn(t)∥2-a2(t)-(a1(t)+c1∥xn(t)∥p-1)∥x(t)∥ for all t∈T∖N.
We introduce the Lebesgue measurable set D⊆T defined by
D={t∈T:lim infn→∞ϑn(t)<0}.
Suppose that |D|1>0. If t∈D∩(T∖N), then from (3.7) we see that
{xn(t)}n≥1⊆X is bounded.
Then, from (3.6), it follows that
(3.8)xn(t)→𝑤x(t) in X for all t∈D∩(T∖N).
We fix t∈D∩(T∖N) and choose a subsequence {nk} of {n} (in general this subsequence depends on t) such that
limk→∞ϑnk(t)=lim infn→∞ϑn(t).
By hypothesis (H$A$1) (ii), A(t,⋅) is pseudomonotone, and since t∈D, we infer that
limk→∞〈hnk*(t),xnk(t)-x(t)〉=0,
a contradiction.
So, |D|1=0 and we have
(3.9)0≤lim infn→∞ϑn(t) for almost all t∈T.
Invoking the extended Fatou’s lemma (see [15, Theorem 2.2.33]), we have
0≤∫0blim infn→∞ϑn(t)dt(see (3.9))
≤lim infn→∞∫0bϑn(t)𝑑t
≤lim supn→∞∫0bϑn(t)𝑑t
=lim supn→∞∫0b〈hn*(t),xn(t)-x(t)〉𝑑t
=lim supn→∞((hn*,xn-x))≤0(see (3.5)).
Therefore,
(3.10)∫0bϑn(t)𝑑t→0.
We write
(3.11)|ϑn(t)|=ϑn+(t)+ϑn-(t)=ϑn(t)+2ϑn-(t).
Note that, from (3.9),
(3.12)ϑn-(t)→0 for almost all t∈T.
Moreover, from (3.7) we have
ϑn(t)≥ηn(t) for almost all t∈T and all n∈ℕ
with {ηn}n≥1⊆L1(T) uniformly integrable. Then
ϑn-(t)≤ηn-(t) for almost all t∈T and all n∈ℕ
with {ηn-}n≥1⊆L1(T) uniformly integrable.
Using (3.12) and invoking Vitali’s theorem we infer that ϑn-→0 in L1(T). Hence, from (3.10) and (3.11),
(3.13)ϑn→0 in L1(T).
Then, from (3.13) and the fact that hn*→𝑤h* in Lp′(T,X*)), we have
|((hn*,xn))-((h*,x))|≤|((hn*,xn-x))|+|((hn*-h*,x))|→0,
which implies
((hn*,xn))→((h*,x)).
In addition, from (3.8) and Lemma 3.2, we have that
h*∈a(x).
This proves the L-pseudomonotonicity of a(⋅).
∎
Remark 3.4From the above proof it is clear why in the case of a single-valued map A(t,x), in hypothesis (H$A$1) (ii) we can drop the condition on the graph of A(t,⋅) and only assume that x↦A(t,x) is pseudomonotone for almost all t∈T. Indeed, in this case, from (3.13) we have (at least for a subsequence) that
ϑn(t)→0 for almost all t∈T,
which, since A(t,⋅) is pseudomonotone, implies
A(t,xn(t))→𝑤A(t,x(t)) for almost all t∈T in X*.
In the multivalued case, there is no canonical way to identify the pointwise limit of the sequence {hn*(t)}n≥1⊆X*.
If for almost all t∈T,A(t,⋅) is maximal monotone, then again, we do not need the graph hypothesis on A(t,⋅).
In this case a(⋅) is also maximal monotone and then the lemma is a consequence of (3.5) and [4, Lemma 1.3].
It is worth mentioning that a similar strengthening of the topology in the range space was used by Defranceschi [12], while studying G-convergence of multivalued operators.
Without loss of generality, invoking the Troyanski renorming theorem (see [21, Remark 2.115]), we may assume that both X and X* are locally uniformly convex, hence Lp(T,X) and Lp′(T,X*) are strictly convex.
We are now ready for the first result concerning the solution set S(ξ).
Theorem 3.5If hypotheses (H$A$1), (H$F$1) hold and ξ∈H, then the solution set S(ξ) is nonempty, weakly compact in Wp(0,b) and compact in C(T,H).
Proof.
First suppose that ξ∈X. We define
A1(t,x)=A(t,x+ξ) and F1(t,x)=F(t,x+ξ).
Evidently, A1(t,x) and F1(t,x) have the same measurability, continuity and growth properties as the multivalued maps A(t,x) and F(t,x). So, we may equivalently consider the following Cauchy problem:
(3.14)-x′(t)∈A1(t,x(t))+F1(t,x(t)) for almost all t∈T,x(0)=0.
Note that if x∈Wp(0,b) is a solution of (3.14), then x^=x-ξ is a solution of (3.1) (when ξ∈X, that is, the initial condition is regular). Consider the linear densely defined operator L:D(L)⊆Lp(T,X)→Lp′(T,X*) defined by
L(x)=x′ for all x∈Wp0(0,b)={y∈Wp(0,b):y(0)=0}
(the evaluation y(0)=0 makes sense by virtue of (2.2)).
Consider the multivalued maps a1,G1:Lp(T,X)→2Lp′(T,X*)∖{∅} defined by
a1(x)={h*∈Lp′(T,X*):h*(t)∈A1(t,x(t)) for almost all t∈T},
G1(x)={f∈Lp′(T,X*):f(t)∈F1(t,x(t)) for almost all t∈T}.
We set K(x)=a1(x)+G1(x) for all x∈Lp(T,X). Then
K:Lp(T,X)→2Lp′(T,X*)∖{∅}.
Claim 1K is L-pseudomonotone.
Clearly, K has values in Pwkc(Lp′(T,X*)) and it is bounded (see hypotheses (H$A$1) (iii), (H$F$1) (iii)).
Next, we consider a sequence {xn}n≥1⊆D(L) such that
(3.15){xn→𝑤x∈D(L) in Lp(T,X),L(xn)→L(x) in Lp′(T,X*),kn*∈K(xn),kn*→𝑤k* in Lp′(T,X*) and lim supn→∞((kn*,xn-x))≤0.
Then we have
kn*=hn*+fn with hn*∈a1(xn),fn∈G1(xn) for all n∈ℕ.
Hypotheses (H$A$1) (iii) and (H$F$1) (iii) imply that
{hn*}n≥1⊆Lp′(T,X*) and {fn}n≥1⊆Lp′(T,H) are bounded.
So, we may assume (at least for a subsequence) that
hn*→𝑤h* in Lp′(T,X*) and fn→𝑤f in Lp′(T,H).
By (3.15) we have xn→𝑤x in Wp(0,b), and due to (2.3),
(3.16)xn→x in Lp(T,H).
Hence,
((fn,xn-x))=∫0b(fn(t),xn(t)-x(t))𝑑t→0,
and in view of (3.15),
lim supn→∞((hn*,xn-x))≤0.
Therefore, from Lemma 3.3,
h*∈a1(x) and ((hn*,xn))→((h*,x)).
Recall that
(3.17)fn(t)∈F(t,xn(t)) for almost all t∈T and all n∈ℕ.
By (3.15), (3.16), (3.17) and [34, Proposition 6.6.33], we have
f(t)∈conv¯w-lim supn→∞F(t,xn(t))⊆F(t,x(t)) for almost all t∈T
(see hypothesis (H$F$1) (ii)), and thus
f∈G1(x).
Since ((fn,xn-x))=∫0b(fn(t),xn(t)-x(t))𝑑t→0, we conclude that K is L-pseudomonotone. This proves Claim 1.
Let x∈Lp(T,X) and k*∈K(x). Then
k*=h*+f with h*∈a1(x),f∈G1(x).
We have (see hypothesis (H$A$1) (iv))
(3.18)((k*,x))=((h*,x))+∫0b(f(t),x(t))dt≥c2∥x∥Lp(T,X)p-∥a2∥1-∫0b|f(t)||x(t)|dt.
Note that from hypothesis (H$F$1) (iii) and using Young’s inequality with ϵ>0, we have
(3.19)∫0b|f(t)||x(t)|dt≤∫0b(a3(t)|x(t)|+c3|x(t)|2)dt≤∫0b(c(ϵ)a3(t)2+(c3+ϵ)|x(t)|2)dt
Returning to (3.18) and using (3.19), we see that
(3.20)((k*,x))≥c2∥x∥Lp(T,X)p-c4∥x∥Lp(T,X)2-c5 for c4,c5>0
(recall that 2≤p and in case p=2, choose ϵ>0 small so that c4<c2,
see hypothesis (H$F$1) (iii)).
From (3.20) it follows that K is coercive. This proves Claim 2.
Now Claims 1 and 2 permit the use of Proposition 2.3 to find x∈Wp(0,b) solving problem (3.1) when ξ∈X.
Next, we remove the restriction ξ∈X. So, suppose ξ∈H. We can find {ξn}n≥1⊆X such that ξn→ξ in H (recall that X is dense in H). From the first part of the proof, we know that we can find xn∈S(ξn)⊆Wp(0,b) for all n∈ℕ. We have
{-xn′(t)∈A(t,xn(t))+F(t,xn(t)) for almost all t∈T,xn(0)=ξn,n∈ℕ.
It follows that
(3.21)-xn′=hn*+fn with hn*(t)∈A(t,xn(t)),fn(t)∈F(t,xn(t))
for almost all t∈T and all n∈ℕ.
We have
((xn′,xn))+((hn*,xn))≤∫0b|fn(t)||xn(t)|dt,
and thus
(3.22)12|xn(b)|2+c2∥xn∥Lp(T,X)p≤c6+c7∥xn∥Lp(T,X)2 for c6,c7>0
(see hypotheses (H$A$1) (iv), (H$F$1) (iii) and if p=2 ,then, as before, we have c7<c2).
From (3.21), (3.22) and hypotheses (H$A$1) (iii), (H$F$1) (iii), it follows that
{xn}n≥1⊆Wp(0,b) is bounded.
So, we may assume that as n→∞ (see (2.3)),
(3.23)xn→𝑤x in Wp(0,b) and xn→x in Lp(T,H).
By (3.21), for all n∈ℕ, we have
(3.24)((xn′,xn-x))+((hn*,xn-x))=-((fn,xn-x))=-∫0b(fn(t),xn(t)-x(t))𝑑t.
By Proposition 2.2 we know that
((xn′-x′,xn-x))=12|xn(b)-x(b)|2-12|ξn-ξ|2.
Hence,
((xn′,x-xn))≤12|ξn-ξ|2+((x′,x-xn)).
By (3.23) and since ξn→ξ in H,
(3.25)lim supn→∞((xn′,x-xn))≤0.
Hypothesis (H$F$1) (iii) implies that {fn}n≥1⊆L2(T,H) is bounded.
Hence, from (3.23),
∫0b(fn(t),x(t)-xn(t))𝑑t→0,
and so, by (3.25),
lim supn→∞[((xn′,x-xn))+((fn,x-xn))]≤0.
Therefore, from (3.24),
(3.26)lim supn→∞((hn*,xn-x))≤0.
By hypothesis (H$A$1) (iii) we see that
{hn*}n≥1⊆Lp′(T,X*)
is bounded.
So, we may assume that
(3.27)hn*→𝑤h* in Lp′(T,X*) as n→∞.
From (3.23), (3.26) and (3.27), we see that we can use Lemma 3.3 and infer that
(3.28)h*(t)∈A(t,x(t)) for almost all t∈T.
As we have already mentioned {fn}n≥1⊆L2(T,H) is bounded and so we may assume that
(3.29)fn→𝑤f in L2(T,H).
Using [34, Proposition 6.6.33], we have (see hypothesis (H$F$1) (ii))
(3.30)f(t)∈conv¯w-lim supn→∞F(t,xn(t))⊆F(t,x(t)) for almost all t∈T.
In (3.21), we pass to the limit as n→∞ and use (3.23), (3.27) and (3.29) to obtain
-x′=h*+f with h*∈a(x) (see (3.28)),f∈G(x) (see (3.30)),x(0)=ξ.
Hence, x∈S(ξ).
So, we have proved that when ξ∈H, the solution set S(ξ) is a nonempty subset of Wp(0,b).
Next, we will prove the compactness of S(ξ) in Wp(0,b)w and in C(T,H). Let x∈S(ξ). For every t∈T we have
∫0t〈x′(s),x(s)〉ds+∫0t〈h*(s),x(s)〉ds≤∫0t|f(s)||x(s)|ds with h*∈a(x),
which implies
12|x(t)|2≤12c82+c9∫0t|x(s)|2ds for c8,c9>0,
and hence, by Gronwall’s inequality,
(3.31)|x(t)|≤M for some M>0, all t∈T and all x∈S(ξ).
Then let rM:H→H be the M-radial retraction defined by
rM(x)={xif |x|≤M,Mx|x|if |x|>M.
Because of the a priori bound (3.31), we can replace F(t,x) by
F^(t,x)=F(t,rM(x)).
Note that for all x∈H, t↦F^(t,x) is graph measurable (hence also measurable, see Section 2) and for almost all t∈T, x↦F^(t,x) has a graph which is sequentially closed in H×Hw. Moreover, we see that
|F^(t,x)|≤a4(t) for almost all t∈T and all x∈H with a4∈L2(T).
We introduce the set
C={f∈L2(T,H):|f(t)|≤a4(t) for almost all t∈T}.
We consider the following Cauchy problem:
(3.32){-x′(t)∈A(t,x(t))+f(t) for almost all t∈T=[0,b],x(0)=ξ∈H,f∈C.
Let H:C→2C(T,H) be the map (in general, multivalued) that assigns to each f∈C the set of solutions of problem (3.32).
It is a consequence of Proposition 2.3 and Lemma 3.3 that H(⋅) has nonempty values.
Claim 3H(C)⊆C(T,H) is compact.
Let {xn}n≥1⊆H(C). Then
(3.33)-xn′=hn*+fn with hn*∈a(xn),fn∈C for all n∈ℕ.
Hence, for all t∈T, we have
12|xn(t)|2≤12c102+∫0ta4(s)|xn(s)|𝑑s for some c10>0 and all n∈ℕ,
which implies (see [6, Lemme A.5])
(3.34)|xn(t)|≤M1 for M1>0, all t∈T and all n∈ℕ.
Also, using hypothesis (H$A$1) (iv), we have (see (3.34))
(3.35)c2∥xn∥Lp(T,X)p≤c11+∫0ba4(t)|xn(t)|𝑑t≤c12 for all n∈ℕ.
From (3.33) and (3.35) it follows that
{xn}n≥1⊆Wp(0,b) is bounded.
So, we may assume that
(3.36)xn→𝑤x in Wp(0,b),hn*→𝑤h* in Lp′(T,X*),fn→𝑤f in L2(T,H).
Passing to the limit as n→∞ in (3.33) and using (3.36), we obtain
-x′=h*+f.
Also, from (3.33) we have
(3.37)((hn*,xn-x))+((xn′,xn-x))=-∫0b(fn(t),xn(t)-x(t))𝑑t.
Note that, by (3.36) and (2.3),
(3.38)∫0b(fn(t),xn(t)-x(t))𝑑t→0.
Also using Proposition 2.2, we have (recall that xn(0)=x(0)=ξ for all n∈ℕ)
((xn′-x′,xn-x))=12|xn(b)-x(b)|2≥0 for all n∈ℕ,
and so
(3.39)((x′,xn-x))≤((xn′,xn-x)) for all n∈ℕ.
From (3.36), it follows that
(3.40)((x′,xn-x))→0.
Returning to (3.37), passing to the limit as n→∞ and using (3.38), (3.39) and (3.40), we obtain
lim supn→∞((hn*,xn-x))≤0,
and thus h*∈a(x) (see Lemma 3.3 and (3.36)). Therefore,
H(C) is w-compact in Wp(0,b).
From the proof of Lemma 3.3 (see (3.13)), we know that
(3.41)∫0b|〈hn*(t),xn(t)-x(t)〉|dt→0 as n→∞.
In a similar fashion, we also have
(3.42)∫0b|〈h*(t),xn(t)-x(t)〉|dt→0 as n→∞.
Also, by (2.3), (3.34), (3.36) and Vitali’s theorem, we have
(3.43)∫0b|(fn(t)-f(t),xn(t)-x(t))|dt→0 as n→∞.
For every t∈T and every n∈ℕ, using Proposition 2.2, we have
12|xn(t)-x(t)|2≤∫0b|〈hn*(t)-h*(t),xn(t)-x(t)〉|dt∫0b|(fn(t)-f(t),xn(t)-x(t))|dt,
and so, from (3.41), (3.42) and (3.43),
∥xn-x∥C(T,H)→0.
Thus, H(C)⊆C(T,H) is compact.
However, from the previous parts of the proof it is clear that S(ξ)⊆H(C) is weakly closed in Wp(0,b) and closed in C(T,H).
Therefore, we conclude that S(ξ) is weakly compact in Wp(0,b) and compact in C(T,H).
∎
Next, we want to produce a continuous selection of the multifunction ξ↦S(ξ) (we refer to [36] for more details about continuous selections of multivalued mappings).
Note that S(⋅) is in general not convex-valued, and so the Michael selection theorem (see [22, Theorem 4.6, p. 92]) cannot be used.
To produce a continuous selection of the solution multifunction ξ↦S(ξ), we need to strengthen the conditions on the multimap A(t,⋅) in order to guarantee that certain Cauchy problems admit a unique solution.
The new hypotheses on the map A(t,x) are as follows.
A:T×X→2X*∖{∅} is a multivalued map such that the following hold:
t↦A(t,x) is graph measurable for every x∈X.
x↦A(t,x) is maximal monotone for almost all t∈T.
For almost all t∈T, all x∈X and all h*∈A(t,x), we have
∥h*∥*≤a1(t)+c1∥x∥p-1
with a1∈Lp′(T),c1>0,2≤p.
For almost all t∈T, all x∈X and h*∈A(t,x), we have
〈h*,x〉≥c2∥x∥p-a2(t)
with c2>0,a2∈L1(T)+.
Remark 3.6As we have already mentioned in an earlier remark, since now A(t,⋅) is maximal monotone for almost all t∈T, we do not need the condition on the graph of A(t,⋅) (see hypothesis (H$A$1) (ii) and [4, Lemma 1.3]).
Also, we strengthen the condition on the multifunction F(t,⋅).
F:T×H→Pfc(H) is a multifunction such that the following hold:
t↦F(t,x) is graph measurable for every x∈H.
For almost all t∈T and all x,y∈H, we have
h(F(t,x),F(t,y))≤k(t)|x-y| with k∈L1(T)+.
For almost all t∈T, all x∈H and all h∈F(t,x), we have
|h|≤a3(t)+c3|x|
with a3∈L2(T)+, c3>0, and if p=2, then β2c3<c2 (see (2.1)).
Remark 3.7Hypothesis (H$F$2) (ii) is stronger than condition (H$F$1) (ii).
Indeed, suppose that (H$F$2) (ii) holds and we have
(3.44)xn→x,hn→𝑤h in H and hn∈F(t,xn) for all n∈ℕ.
By the definition of the Hausdorff metric (see Section 2), we have
d(hn,F(t,x))≤d(hn,F(t,xn))+h(F(t,xn),F(t,x))=h(F(t,xn),F(t,x)),
and therefore (see (3.44) and hypothesis (H$F$2) (ii))
d(hn,F(t,x))→0 as n→∞.
The function y↦d(y,F(t,x)) is continuous and convex, hence weakly lower semicontinuous.
Therefore, by (3.44) we have
d(h,F(t,x))≤lim infn→∞d(hn,F(t,x))=0,
which implies h∈F(t,x).
This proves that condition (H$F$1) (ii) holds.
So, we can use Theorem 3.5 and establish that given any ξ∈H, the solution set S(ξ) is nonempty, weakly compact in Wp(0,b) and compact in C(T,H).
The next result extends an earlier result of Cellina and Ornelas [9] for differential inclusions in ℝN with A≡0.
Proposition 3.8If hypotheses (H$A$2),(H$F$2) hold, then there exists a continuous map ϑ:H→C(T,H) such that
ϑ(ξ)∈S(ξ) for all ξ∈H.
Proof.
Consider the following auxiliary Cauchy problem:
-x′(t)∈A(t,x(t)) for almost all t∈T=[0,b],x(0)=ξ.
This problem has a unique solution x0(ξ)∈Wp(0,b) (see Proposition 2.3 and use the monotonicity of A(t,⋅) and Proposition 2.2 to check the uniqueness of this solution).
If ξ1,ξ2∈H, then
-x0′(ξ1)=h1* and -x0′(ξ2)=h2* with hk*∈a(x0(ξk)) for k=1,2.
So, using Proposition 2.2, we have
12|x0(ξ1)(t)-x0(ξ2)(t)|2+∫0t〈h1*(s)-h2*(s),x0(ξ1)(s)-x0(ξ2)(s)〉𝑑s=12|ξ1-ξ2|2 for all t∈T,
and hence (recall that A(t,⋅) is monotone)
(3.45)∥x0(ξ1)-x0(ξ2)∥C(T,H)≤|ξ1-ξ2|.
We consider the multifunction Γ0:H→Pwkc(L1(T,H)) defined by
Γ0(ξ)=SF(⋅,x0(ξ)(⋅))1 for all ξ∈H.
We have
h(Γ0(ξ),Γ0(ξ2))=sup[|σ(g,Γ0(ξ1))-σ(g,Γ0(ξ2))|:g∈L∞(T,H)=L1(T,H)*,∥g∥L∞(T,H)≤1]
≤∫0bsup|v|≤1[|σ(v,F(t,x0(ξ1)(t)))-σ(v,F(t,x0(ξ2)(z)))|]dt (see [34, Theorem 6.4.16])
=∫0bh(F(t,x0(ξ1)(t)),F(t,x0(ξ2)(t)))𝑑t
≤∫0bk(t)|x0(ξ1)(t)-x0(ξ2)(t)|𝑑t
≤∫0bk(t)|ξ1-ξ2|𝑑t (see (3.45))
=∥k∥1|ξ1-ξ2|.
Therefore, ξ↦Γ0(ξ) is h-Lipschitz.
Also, Γ0(⋅) has decomposable values.
So, we can apply the selection theorem of Bressan and Colombo [5] (see also [22, Theorem 8.7, p. 245]) and find a continuous map γ0:H→L1(T,H) such that γ0(ξ)∈Γ0(ξ) for all ξ∈H. Evidently, γ0(ξ)∈L2(T,H) for all ξ∈H.
We consider the following auxiliary Cauchy problem:
-x′(t)∈A(t,x(t))+γ0(ξ)(t) for almost all t∈T,x(0)=ξ.
This problem has a unique solution x1(ξ)∈Wp(0,b).
By induction we will produce two sequences
{xn(ξ)}n≥1⊆Wp(0,b) and {γn(ξ)}n≥1⊆L2(T,H),
which satisfy the following:
xn(ξ)∈Wp(0,b) is the unique solution of the Cauchy problem
(3.46)-x′(t)∈A(t,x(t))+γn-1(ξ)(t) for almost all t∈T,x(0)=ξ.
ξ↦γn(ξ) is continuous from H into C(T,H).
γn(ξ)(t)∈F(t,xn(ξ)(t)) for almost all t∈T and all ξ∈H.
|γn(ξ)(t)-γn-1(ξ)(t)|≤k(t)βn(ξ)(t) for almost all t∈T and all ξ∈H, where
βn(ξ)(t)=2∫0tη(ξ)(s)(τ(t)-τ(s))n-1(n-1)!𝑑s+2b(∑k=1nϵ2k+1)τ(t)n-1(n-1)!
with ϵ>0, η(ξ)(t)=a2(t)+c2|x0(ξ)(t)| and τ(t)=∫0tk(s)𝑑s.
Note that the maps ξ↦η(ξ) and ξ↦βn(ξ) are continuous from H into L1(T).
So, suppose we have produced {xk(ξ)}k=1n and {γk(ξ)}k=1n (induction hypothesis). Let xn+1(ξ)∈Wp(0,b) be the unique solution of the Cauchy problem
(3.47)-x′(t)∈A(t,x(t))+γn(ξ)(t) for almost all t∈T,x(0)=ξ.
By (3.46) and (3.47) we have
(3.48)-xn′(ξ)=hn*+γn-1(ξ) and -xn+1′(ξ)=hn+1*+γn(ξ) in Lp′(T,X*)
with
(3.49)hn*∈a(xn(ξ)),hn+1*∈a(xn+1(ξ)).
Using (3.48) we can write
〈xn+1′(ξ)(t)-xn′(ξ)(t),xn+1(ξ)(t)-xn(ξ)(t)〉
=〈hn*(ξ)(t)-hn+1*(ξ)(t),xn+1(ξ)(t)-xn(ξ)(t)〉+(γn-1(ξ)(t)-γn(ξ)(t),xn+1(ξ)(t)-xn(ξ)(t))
≤(γn-1(ξ)(t)-γn(ξ)(t),xn+1(ξ)(t)-xn(ξ)(t)) for almost all t∈T
(see hypothesis (H$A$2) (ii) and (3.49)). Therefore, from Proposition 2.2,
12ddt|xn+1(ξ)(t)-xn(ξ)(t)|2≤|γn-1(ξ)(t)-γn(ξ)(t)||xn+1(ξ)(t)-xn(ξ)(t)| for almost all t∈T,
and thus
(3.50)12|xn+1(ξ)(t)-xn(ξ)(t)|2≤∫0t|γn-1(ξ)(s)-γn(ξ)(s)||xn+1(ξ)(s)-xn(ξ)(s)|𝑑s for all t∈T.
By (3.50) and [6, Lemma A.5], we infer that
|xn+1(ξ)(t)-xn(ξ)(t)|≤∫0t|γn-1(ξ)(s)-γn(ξ)(s)|ds
≤∫0tk(s)βn(ξ)(s)𝑑s (by the induction hypothesis, see (d))
=2∫0tk(s)∫0sη(ξ)(r)(τ(s)-τ(r))n-1(n-1)!𝑑r𝑑s+2b(∑k=0nϵ2k+1)∫0tk(s)τ(s)n-1(n-1)!𝑑s
=2∫0tη(ξ)(s)∫stk(r)(τ(r)-τ(s))n-1(n-1)!𝑑r𝑑s+2b(∑k=0nϵ2k+1)∫0tddsτ(s)nn!𝑑s
=2∫0tη(ξ)(s)∫stddr(τ(r)-τ(s))nn!𝑑r𝑑s+2b(∑k=0nϵ2k+1)τ(t)nn!
=2∫0tη(ξ)(s)(τ(t)-τ(s))nn!𝑑s+2b(∑k=0nϵ2k+1)τ(t)nn!
(3.51)<βn+1(ξ)(t) for almost all t∈T
(see (d)). Using the induction hypothesis (see (c)) and hypothesis (H$F$2) (ii), we have
d(γn(ξ)(t),F(t,xn+1(ξ)(t)))≤h(F(t,xn(ξ)(t)),F(t,xn+1(ξ)(t)))
≤k(t)|xn(ξ)(t)-xn+1(ξ)(t)|
(3.52)<k(t)βn+1(ξ)(t) for almost all t∈T
(see (3.51)).
Consider the multifunction Γn+1:H→2L1(T,H) defined by
Γn+1(ξ)={f∈SF(⋅,xn+1(ξ)(⋅))1:|γn(ξ)(t)-f(t)|<k(t)βn+1(ξ)(t) for almost all t∈T}.
By (3.52) and [22, Lemma 8.3, p 239], we have that
ξ↦Γn+1(ξ) has nonempty decomposable values and it is LSC.
Thus,
ξ↦Γn+1(ξ)¯ is LSC with decomposable values.
We can apply the selection theorem of Bressan and Colombo [5, Theorem 3] to find a continuous map γn+1:H→L1(T,H) such that γn+1(ξ)∈Γn+1(ξ)¯ for all ξ∈H.
This completes the induction process and we have produced two sequences {xn(ξ)}n≥1, {γn(ξ)}n≥1 which satisfy properties (a)–(d) stated earlier.
From (3.51) we have
∫0b|γn(ξ)(t)-γn-1(ξ)(t)|dt<∫0bβn+1(ξ)(t)dt
<∫0bη(ξ)(t)(τ(b)-τ(t))nn!𝑑t+2bϵτ(b)nn!
(3.53)≤τ(b)nn![∥η(ξ)∥1+2bϵ].
Recall that ξ↦η(ξ) is continuous from H into L1(H) and maps bounded sets to bounded sets. So, from (3.53) it follows that
{γn(ξ)}n≥1⊆L1(T) is Cauchy, uniformly on bounded sets of H.
Moreover, from (3.51) and (3.53), we have
∥xn+1(ξ)-xn(ξ)∥C(T,H)≤∥γn(ξ)-γn-1(ξ)∥L1(T,H)≤τ(b)nn![∥η(ξ)∥1+2bϵ].
Thus,
{xn(ξ)}n≥1⊆C(T,H) is Cauchy, uniformly on bounded sets.
Therefore, we have
(3.54)xn(ξ)→x(ξ) in C(T,H) and γn(ξ)→γ(ξ) in L1(T,H).
Evidently, ξ↦x(ξ) is continuous from H into C(T,H), while because of hypothesis (H$F$2) (iii), we have γn(ξ)→γ(ξ) in L2(T,H). Let x^(ξ)∈Wp(0,b) be the unique solution of
-y′(t)∈A(t,y(t))+γ(ξ)(t) for almost all t∈T,y(0)=0.
As before, exploiting the monotonicity of A(t,⋅) (see hypothesis (H$A$2) (ii)), we have
12|xn+1(ξ)(t)-x^(ξ)(t)|2≤∫0t|γn(ξ)(s)-γ(ξ)(s)||xn+1(ξ)(s)-x^(ξ)(s)|𝑑s for all t∈T,
which implies (see [6, Lemma A.5])
∥xn+1(ξ)-x^(ξ)∥C(T,H)≤∥γn(ξ)-γ(ξ)∥L1(T,H).
Therefore, from (3.54),
x(ξ)=x^(ξ).
So, x(ξ)∈S(ξ) and the map ϑ:H→C(T,H) defined by ϑ(ξ)=x(ξ) is a continuous selection of the solution multifunction ξ↦S(ξ).
∎
An easy but useful consequence of Proposition 3.8 and its proof is a parametric version of the Filippov–Gronwall inequality (see [1, Theorem 1, pp. 120–121] and [19]) for differential inclusions.
So, we consider the following parametric version of problem (3.1):
-x′(t)∈A(t,x(t))+F(t,x(t),λ) for almost all t∈T,x(0)=ξ(λ).
The parameter space D is a complete metric space.
The hypotheses on the parametric vector field F(t,x,λ) and the initial condition ξ(λ) are as follows.
F:T×H×D→Pfc(H) is a multifunction such that the following hold:
t↦F(t,x,λ) is graph measurable for all (x,λ)∈H×D.
For almost all t∈T, all x,y∈H and all λ∈D, we have
h(F(t,x,λ),F(t,y,λ))≤k(t)|x-y|
with k∈L1(T)+.
For almost all t∈T, all x∈H, all λ∈D and all h∈F(t,x), we have
|h|≤a3(t)+c3|x|
with a3∈L2(T)+, c3>0, and if p=2, then β2c3<c2 (see (2.1)).
For almost all t∈T and all x∈H, the multifunction λ↦F(t,x,λ) is LSC.
The mapping λ↦ξ(λ) is continuous from D into H.
Assume that λ↦(u(λ),h(λ)) is a continuous map from D into C(T,H)×L2(T,H).
We can find a continuous map p:D→L2(T) such that
d(h(λ)(t),F(t,u(λ)(t),λ))≤p(λ)(t) for almost all t∈T,
see hypothesis (H$F$2)’ (iii).
In what follows, by e(h,λ)∈Wp(0,b) we denote the unique solution of the Cauchy problem
-u′(t)∈A(t,u(t))+h(t) for almost all t∈T,u(0)=ξ(λ)
with h∈L2(T,H).
We have the following approximation result.
Proposition 3.9Assume that hypotheses (H$A$2), (H$F$2)’, (H0) hold, λ↦(u(λ),h(λ)) is a continuous map from D into C(T,H)×L2(T,H) with u(λ)=e(h(λ),λ), ϵ>0 and p:D→L2(T)+ is a continuous map such that
d(h(λ)(t),F(t,u(λ)(t),λ))≤p(λ)(t) for almost all t∈T.
Then there exists a continuous map λ↦(x(λ),f(λ)) from D into C(T,H)×L2(T,H) such that
x(λ)=e(f(λ),λ) with f(λ)∈SF(⋅,x(λ)(⋅),λ)2
and
|x(λ)(t)-u(λ)(t)|≤bϵeτ(t)+∫0tp(λ)(s)eτ(t)-τ(s)𝑑s for all t∈T
with τ(t)=∫0tk(s)𝑑s.
Proof.
Consider the multifunction Rϵ:D→2L1(T,H) defined by
Rϵ(λ)={v∈SF(⋅,u(λ)(⋅),λ)1:|v(t)-h(λ)(t)|<p(λ)(t)+ϵ for almost all t∈T}.
This multifunction has nonempty, decomposable values and it is LSC (see [22, Lemma 8.3, p 239]).
Hence, λ↦Rϵ(λ)¯ has the same properties.
So, we can find a continuous map γ0:D→L1(T,H) such that
γ0(λ)∈Rϵ(λ)¯ for all λ∈D.
Let x1(λ)∈Wp(0,b) be the unique solution of the following Cauchy problem:
-x′(t)∈A(t,x(t))+γ0(λ)(t) for almost all t∈T,x(0)=ξ(λ).
Then as in the proof of Proposition 3.8, we can generate by induction two sequences
{xn(λ)}n≥1⊆Wp(0,b) and {γn(λ)}n≥1⊆L2(T,H),
which satisfy properties (a)–(d) listed in the proof of Proposition 3.8.
As before (see the proof of Proposition 3.8), we have
|xn+1(λ)(t)-xn(λ)(t)|≤∥γn(λ)-γn-1(λ)∥L1(T,H) for all (λ,t)∈D×T.
From this inequality and property (d) of the sequences (see the proof of Proposition 3.8), we infer that
{xn(λ)}n≥1⊆C(T,H) and {γn(λ)}n≥1⊆L1(T,H)
are both Cauchy uniformly in λ∈K⊆D compact (recall that λ↦p(λ) is continuous, hence locally bounded). So, we have
xn(λ)→x^(λ) in C(T,H) and γn(λ)→γ^(λ) in L1(T,H),
and both maps D∋λ↦x^(λ)∈C(T,H) and D∋λ↦γ^(λ)∈L1(T,H) are continuous.
Moreover, we have γ^(λ)∈SF(⋅,x^(λ)(⋅),λ)2 (see the proof of Theorem 3.5) and that λ↦γ^(λ) is continuous from D into L2(T,H).
If x(λ)=e(γ^(λ),λ), then
|xn(λ)(t)-x(λ)(t)|≤∫0b|γn-1(λ)(s)-γ^(λ)(s)|ds→0 for all t∈T,
which implies
x^(λ)=x(λ)=e(γ^(λ),λ) for all λ∈D.
From the triangle inequality, we have
|u(λ)(t)-xn(λ)(t)|≤|u(λ)(t)-x1(λ)(t)|+∑k=1n-1|xk(λ)(t)-xk+1(λ)(t)| for all t∈T.
Using property (d) (see the proof of Proposition 3.8), we have
|xk(λ)(t)-xk+1(λ)(t)|≤1k!∫0tp(λ)(s)(τ(t)-τ(s))k𝑑s+bϵk!τ(t)k for all t∈T.
So, finally we can write that
|u(λ)(t)-x(λ)(t)|≤bϵeτ(b)+∫0tp(λ)(s)eτ(t)-τ(s)𝑑s for all t∈T and all λ∈D.∎
We want to strengthen Proposition 3.8, and require that the selection ϑ(⋅) passes through a preassigned solution. We mention that an analogous result for differential inclusions in ℝN with A≡0, was proved by Cellina and Staicu [10].
We start with a simple technical lemma.
Lemma 3.10If {uk}k=0N⊆L1(T,H) and {Tk(ξ)}k=0N is a partition of T=[0,b] with endpoints which depend continuously on ξ∈H, then there exists d^∈L1(T)+ for which the following holds:
“Given ϵ>0, we can find δ>0 such that for |ξ-ξ′|≤δ,
|∑k=0NχTk(ξ)(t)uk(t)-∑k=0NχTk(ξ′)(t)uk(t)|≤d^(t)χC(t)
with C⊆T measurable and |C|1≤ϵ”.
Proof.
We have
|∑k=0NχTk(ξ)(t)uk(t)-∑k=0NχTk(ξ′)(t)(t)uk(t)|≤∑k=0N|χTk(ξ)(t)-χTk(ξ′)||uk(t)|
(3.55)=∑kχTk(ξ)ΔTk(ξ′)(t)|uk(t)|.
We set d^(t)=∑k=0N|uk(t)|∈L1(T)+.
From the hypothesis concerning the partition {Tk(ξ)}k=0N of T, we see that given ϵ>0, we can find δ>0 such that for |ξ-ξ′|≤δ,
(3.56)χTk(ξ)ΔTk(ξ′)(t)≤χC(t) for almost all t∈T and all k∈{0,…,N}
with C⊆T measurable, |C|1≤ϵ. Then, from (3.55) and (3.56),
|∑k=0NχTk(ξ)(t)uk(t)-∑k=0NχTk(ξ′)(t)uk(t)|≤χC(t)∑k=0N|uk(t)|=d^(t)χC(t) for almost all t∈T.
The proof is now complete.
∎
With this lemma, we can produce a continuous selection of the solution multifunction ξ↦S(ξ), which passes through a preassigned point.
Proposition 3.11If hypotheses (H$A$2), (H$F$2) hold, K⊆H is compact, ξ0∈K and v∈S(ξ0), then there exists a continuous map ψ:K→C(T,H) such that
ψ(ξ)∈S(ξ) for all ξ∈K 𝑎𝑛𝑑 ψ(ξ0)=v.
Proof.
Since v∈S(ξ0), we have
(3.57)-v′(t)∈A(t,v(t))+f(t) for almost all t∈T,v(0)=ξ0
with f∈SF(⋅,v(⋅))2. Given g∈L2(T,H), we consider the unique solution of the Cauchy problem
(3.58)-y′(t)∈A(t,y(t))+g(t) for almost all t∈T,y(0)=ξ∈H.
In what follows, by e(g,ξ)∈Wp(0,b) we denote the unique solution of problem (3.58) and we set μ0(ξ)=e(f,ξ). An easy application of the Yankov–von Neumann–Aumann selection theorem (see [22, Theorem 2.14, p. 158]) gives γ0(ξ)∈L2(T,H) such that
γ0(ξ)(t)∈F(t,μ0(ξ)(t)) for almost all t∈T
and
|f(t)-γ0(ξ)(t)|=d(f(t),F(t,μ0(ξ)(t)))
≤k(t)|v(t)-μ0(ξ)(t)| (see hypothesis (HF2) (ii))
=k(t)|e(f,ξ0)(t)-e(f,ξ)(t)|
≤k(t)|ξ0-ξ| for almost all t∈T,
see (3.45).
Let ϑ>0. We define
δ(ξ)={min{2-3ϑ,|ξ-ξ0|2}if ξ≠ξ0,2-3ϑif ξ=ξ0.
The family {Bδ(ξ)(ξ)}ξ∈K is an open cover of the compact set K.
So, we can find {ξk}k=0N⊆K such that {Bδ(ξk)(ξk)}k=0N is a finite subcover of K.
Let {ηk}k=0N be a locally Lipschitz partition of unity subordinated to the finite subcover.
We define
T0(ξ)=[0,η0(ξ)b] and Tk(ξ)=[(∑i=0k-1ηi(ξ))b,(∑i=0kηi(ξ))(b)] for all k∈{1,…,N}.
The endpoints in these intervals are continuous functions of ξ. We consider the following Cauchy problem:
(3.59)-y′(t)∈A(t,y(t))+∑k=0NχTk(ξ)(t)γ0(ξk)(t) for almost all t∈T,y(0)=ξ∈K.
Problem (3.59) has a unique solution μ1(ξ)∈Wp(0,b). Let
λ0(ξ)(⋅)=∑k=0NχTk(ξ)(⋅)γ0(ξk)(⋅)∈L2(T,H).
Using Lemma 3.10, we can find d^∈L1(T)+ such that, for any given ϵ>0, we can find δ>0 for which we have
(3.60)ξ,ξ′∈K,|ξ-ξ′|1≤δ⇒|λ0(ξ)(t)-λ0(ξ′)(t)|≤d^(t)χC(t) for almost all t∈T
with C⊆T measurable, |C|1≤ϵ.
We have μ1(ξ′)=e(λ0(ξ′),ξ′). As before, exploiting the monotonicity of A(t,⋅) (see hypothesis (H$A$2) (ii)) and using [6, Lemma A.5], we have
(3.61)|μ1(ξ)(t)-μ1(ξ′)(t)|≤|ξ-ξ′|+∫0t|λ0(ξ)(s)-λ0(ξ′)(s)|𝑑s for all t∈T.
Let ϵ>0 be given. By the absolute continuity of the Lebesgue integral, we can find δ1>0 such that
(3.62)∫Cd^(s)𝑑s≤ϵ2 for all C⊆T measurable with |C|1≤δ1.
Also, using (3.60), we can find δ∈(0,ϵ/2) such that
(3.63)ξ,ξ′∈K,|ξ-ξ′|1≤δ⇒|λ0(ξ)(t)-λ0(ξ′)(t)|≤d^(t)χC1(t) for almost all t∈T
with C1⊆T measurable, |C1|1≤δ1. So, returning to (3.61) and using (3.62) and (3.63), we see that
ξ,ξ′∈K,|ξ-ξ′|≤δ⇒|μ1(ξ)(t)-μ1(ξ′)(t)|≤ϵ2+∫0td^(s)χC1(s)𝑑s≤ϵ2+ϵ2=ϵ for all t∈T.
Therefore, ξ↦μ1(ξ) is continuous from H into C(T,H). Again, with an application of the Yankov–von Neumann–Aumann selection theorem, we obtain γ1(ξ)∈L2(T,H) such that
γ1(ξ)(t)∈F(t,μ1(ξ)(t)) for almost all t∈T
and
|γ0(ξ)(t)-γ1(ξ)(t)|=d(γ0(ξ)(t),F(t,μ1(ξ)(t))) for almost all t∈T and all ξ∈K.
As in the proof of Proposition 3.8, we produce inductively two sequences
{μn(ξ)}n≥0⊆Wp(0,b) and {γn(ξ)}n≥0⊆L2(T,H),ξ∈K,
which satisfy the following properties:
μn(ξ)=e(λn-1(ξ),ξ) with λn-1(ξ)=∑k=0NχTk(ξ)γn-1(ξk)(t),γ-1(ξ)=f,
ξ↦μn(ξ) is continuous from K into C(T,H),
|μn(ξ)(t)-μn-1(ξ)(t)|≤ϑ2n+2n!(∫0tk(s)𝑑s)n for all ξ∈K,
γn(ξ)(t)∈F(t,μn(ξ)(t)) for almost all t∈T and
|γn-1(ξ)(t)-γn(ξ)(t)|=d(γn-1(ξ)(t),F(t,μn(ξ)(t))) for almost all t∈T.
So, by the induction hypothesis, suppose that we have produced
{μk(ξ)}k=0n⊆Wp(0,b) and {γk(ξ)}k=0n⊆L2(T,H),
which satisfy properties (a)–(d) stated above. We set
μn+1(ξ)=e(λn(ξ),ξ) with λn(ξ)(t)=∑k=0nχTk(ξ)(t)γn(ξk)(t).
As above (see in the first part of the proof the argument concerning the map ξ↦μ1(ξ)), we can show that ξ↦μn+1(ξ) is continuous from K into C(T,H).
Also, by the monotonicity of A(t,⋅) (see hypothesis (H$A$2) (ii) and [6, Lemma A.5]), we have (see hypothesis (H$F$2) (ii) and property (d) of the induction hypothesis)
|μn+1(ξ)(t)-μn(ξ)(t)|≤∑k=0n∫0tχTk(ξ)(s)k(s)|μn(ξ)(s)-μn-1(ξ)(s)|𝑑s
≤∑k=0n∫0tχTk(ξ)(s)k(s)ϑ2n+2n!(∫0sk(s)𝑑r)n𝑑s
=∫0tϑ2n+2(n+2)!dds(∫0sk(r)𝑑r)n+1𝑑s
=ϑ2n+2(n+1)!(∫0tk(s)𝑑s)n+1.
Note that for the second inequality we used property (c) of the induction hypothesis.
Moreover, a standard measurable selection argument produces a measurable map γn+1(ξ):T→H, ξ∈K, such that
γn+1(ξ)(t)∈F(t,μn+1(ξ)(t)) and |γn(ξ)(t)-γn+1(ξ)(t)|=d(γn(ξ)(t),F(t,μn+1(ξ)(t)))
for almost all t∈T.
This completes the induction process.
Note that from property (c),
∥γn+1(ξ)-μn(ξ)∥C(T,H)≤ϑ2n+3e∥k∥1.
Therefore, we can say that
(3.64)μn(ξ)→ψ(ξ) in C(T,H) as n→∞, uniformly in ξ∈K.
It follows that ξ↦ψ(ξ) is continuous from K into C(T,H).
Note that T0(ξ0)=T=[0,b] and so μ0(ξ0)=e(f,ξ0)=v (see (3.57)). Hence, ψ(ξ0)=v. It remains to show that ψ is a selection of the solution multifunction ξ↦S(ξ).
By property (d) and hypothesis (H$F$2) (ii), we have
|γn(ξ)(t)-γn+1(ξ)(t)|≤k(t)|μn(ξ)(t)-μn+1(ξ)(t)| for almost all t∈T,
and thus
(3.65)γn+1(ξ)→γ^(ξ) in L2(T,H).
Let
μ^(ξ)=e(∑k=0N∫0tχTk(ξ)(s)γ^(ξk)(s)𝑑s,ξ).
Since, from (3.65),
∑k=0NχTk(ξ)γn(ξk)→∑k=0NχTk(ξ)γ^(ξk) in L2(T,H),
we have μn(ξ)→μ^(ξ) in C(T,H). Therefore, from (3.64),
ψ(ξ)=μ^(ξ)∈S(ξ) for all ξ∈K.∎
4 Optimal control problems
In this section we deal with the sensitivity analysis of the optimal control problem (1.1).
Let Q(ξ,λ)⊆Wp(0,b)×L2(T,Y) be the admissible “state-control” pairs. First we investigate the dependence of this set on the initial condition ξ∈H and the parameter λ∈E. Recall that the control space Y is a separable reflexive Banach space and the parameter space E is a compact metric space. To have a useful result on the dependence of Q(ξ,λ) on (ξ,λ)∈H×E, we introduce the following conditions on the data of the evolution inclusion in problem (1.1) (the dynamical constraint of the problem).
A:T×X×E→2X*∖{∅} is a multifunction such that
t↦Aλ(t,x) is graph measurable for every (x,λ)∈X×E.
x↦Aλ(t,x) is maximal monotone for almost all t∈T, all λ∈E.
For almost all t∈T, all x∈X, all λ∈E and all h*∈Aλ(t,x), we have
∥h*∥*≤aλ(t)+cλ∥x∥p-1
with {aλ}λ∈E⊆Lp′(T) bounded, {cλ}λ∈E⊆(0,+∞) bounded and 2≤p<∞.
For almost all t∈T, all x∈X, all λ∈E and all h*∈Aλ(t,x), we have
〈h*,x〉≥c^∥x∥p-a^(t)
with c^>0,a^∈L1(T)+.
If λn→λ in E, then ddt+aλn→PGddt+aλ as n→∞.
Hypotheses (H$A$3) (i)–(iv) are the same as hypotheses (H$A$2) (i)–(iv) for every map Aλ, λ∈E. The new condition is hypothesis (H$A$3) (v), which requires elaboration. In the examples that follow, we present characteristic situations where this hypothesis is satisfied.
Example 4.1(a) First, we present a situation which will be used in Section 5.
So, let Ω⊆ℝN be a bounded domain with Lipschitz boundary ∂Ω. Let X=W01,p(Ω) (2≤p<∞), H=L2(Ω) and X*=W-1,p′(Ω).
Evidently, (X,H,X*) is an evolution triple (see Definition 2.1) with compact embeddings.
We consider a map a(t,z,ξ) satisfying the following conditions:
a:T×Ω×ℝN→ℝN is a map such that the following hold:
|a(t,z,0)|≤c0 for almost all (t,z)∈T×Ω.
(t,z)↦a(t,z,ξ) is measurable for every ξ∈ℝN.
For almost all (t,z)∈T×Ω and all ξ1,ξ2∈ℝN, we have
|a(t,z,ξ1)-a(t,z,ξ2)|≤c^(1+|ξ1|+|ξ2|)p-1-α|ξ1-ξ2|α
with c^1>0, α∈(0,1].
For almost all (t,z)∈T×Ω and all ξ1,ξ2∈ℝN, ξ1≠ξ2, we have
(a(t,z,ξ1)-a(t,z,ξ2),ξ1-ξ2)ℝN≥c^2|ξ1-ξ2|p
with c^2>0.
We consider the operator A:T×X→X* defined by
〈A(t,x),h〉=∫Ω(a(t,z,Dx),Dh)ℝN𝑑z for all (t,x,h)∈T×X×X.
Using the nonlinear Green’s identity (see [20, p. 210]), we have
A(t,x)=-div(B(t,x)),
with B(t,x)(⋅)=a(t,⋅,Dx(⋅))∈Lp′(Ω,ℝN) for all (t,x)∈T×X.
Now consider a sequence {an(t,z,ξ)}n≥1 of such maps satisfying
|an(t,z,ξ)-an(s,z,ξ)|≤ϑ(t-s)(1+|ξ|p-1)
for almost all z∈Ω, all t,s∈T, all ξ∈ℝN and all n∈ℕ,
with ϑ:ℝ+→ℝ+ being an increasing function which is continuous at r=0 and ϑ(0)=0.
We assume that for almost all t∈T, an(t,⋅,⋅)→Ga(t,⋅,⋅) in the sense of Defranceschi [12]. By [40] we have
ddt+an→PGddt+a.
(b) We can allow multivalued maps, provided that we drop the t-dependence.
So, we consider multivalued maps a(z,ξ) which satisfy the following conditions:
a:Ω×ℝN→2ℝN∖{∅} is a measurable map such that the following hold:
ξ↦a(z,ξ) is maximal monotone for almost all z∈Ω.
For almost all z∈Ω, all ξ∈ℝN and all y∈a(z,ξ), we have
|y|p′≤m1(z)+c~1(y,ξ)with m1∈L1(Ω),c~1(y,ξ)>0,
|ξ|p≤m2(z)+c~2(y,ξ)with m2∈L1(Ω),c~2(y,ξ)>0(2≤p<∞).
We again consider the evolution triple
X=W01,p(Ω),H=L2(Ω),X*=W-1,p′(Ω)(2≤p<∞),
and consider the multivalued map A:X→2X*∖{∅} defined by
A(x)={-divg:g∈Sa(⋅,Dx(⋅))p′}.
We consider a sequence {an(z,ξ)}n≥1 of such maps and assume that an→Ga in the sense of Defranceschi [12]. Then by [16] we have
ddt+an→PGddt+a.
(c) A third situation leading to hypothesis (H$A$3) (v) is the following one. We consider maps Aλ(t,x) satisfying the following conditions:
A:T×X×E→X* is a map such that the following hold:
For all t,t+τ∈T, all x∈X and all λ∈E, we have
∥Aλ(t+τ,x)-Aλ(t,x)∥≤O(τ)(1+∥x∥p-1).
x↦Aλ(t,x) is semicontinous for all (t,λ)∈T×E.
For all t∈T, all x,u∈X and all λ∈E, we have
〈Aλ(t,x)-Aλ(t,u),x-u〉≥c~∥x-u∥p
with c~>0.
If λn→λ in E, then Aλn(t,⋅)→GAλ(t,⋅)
for all t∈T. (This means that Aλn-1(t,x*)→𝑤Aλ-1(t,x) for all x*∈X*, see [14, Definition 3.8.20].)
Under these conditions, by [26], we have
ddt+aλn→PGddt+aλ.
Next, we introduce the conditions on the multifunctions F and G involved in the dynamics of (1.1).
F:T×H×E→Pfc(H) is a multifunction such that the following hold:
t↦F(t,x,λ) is graph measurable for all (x,λ)∈H×E.
for almost all t∈T, all x,y∈H and all λ∈E, we have
h(F(t,x,λ),F(t,y,λ))≤k(t)|x-y|
with k∈L1(T)+.
For almost all t∈T, all x∈H and all λ∈E, we have
|F(t,x,λ)|≤aλ(t)+cλ|x|
with {aλ}λ∈E⊆L2(T) and {cλ}λ∈E⊆(0,+∞) bounded.
For almost all t∈T, all x∈H and all λ,λ′∈E, we have
h(F(t,x,λ),F(t,x,λ′))≤β(d(λ,λ′))w(t,|x|)
with β(r)→0+ as r→0+ and w(t,⋅) bounded on bounded sets.
G:T×Y×E→Pfc(H) is a multifunction such that the following hold:
t↦G(t,u,λ) is graph measurable for all (u,λ)∈Y×E.
For almost all t∈T, all λ∈E, u↦G(t,u,λ) is concave (that is, GrG(t,⋅,λ)⊆Y×H is concave, see [22, Definition 1.1 and Remark 1.2, p. 585]) and (u,λ)↦G(t,u,λ) is h-continuous.
For almost all t∈T, all u∈U(t,λ) and all λ∈E, we have
|G(t,u,λ)|≤a^λ(t)
with {a^λ}λ∈E⊆L2(T) bounded.
Remark 4.2A typical situation resulting to a concave multifunction u↦G(t,u,λ) is when
G(t,u,λ)=Bλ(t)u+C(t,λ) for all (t,u,λ)∈T×Y×E
with Bλ(t)∈ℒ(Y,H) and C(t,λ)∈Pfc(H) for all (t,λ)∈T×E.
Another situation, leading to the concavity of G(t,⋅,λ), is when H is an ordered Hilbert space and gλ,g~λ:T×Y→H are two Carathéodory maps such that for almost all t∈T
gλ(t,⋅) is order convex and g~λ(t,⋅) is order concave.
We set G(t,u,λ)={h∈H:gλ(t,u)≤h≤g~λ(t,u)}. Then G(t,⋅,λ) is concave.
Finally, we impose conditions on the control constraint U(t,λ).
U:T×E→Pfc(Y) is a multifunction such that the following hold:
t↦U(t,λ) is graph measurable for all λ∈E.
λ↦U(t,λ) is h-continuous for almost all t∈T.
|U(t,λ)|≤a~λ(t) for almost all t∈T and all λ∈E with {a~λ}λ∈E⊆L2(T) bounded.
Proposition 4.3If hypotheses (H$A$3), (H$F$3), (H$G$), (H$U$) hold and (ξn,λn)→(ξ,λ) in H×E, then
Kseq(s×w)-lim supn→∞Q(ξn,λn)⊆Q(ξ,λ)in Lp(T,H)×L2(T,Y),
K(s×s)-lim infn→∞Q(ξn,λn)⊇Q(ξ,λ)in C(T,H)×L2(T,Y).
Proof.
Let (x,u)∈Kseq(s×w)-lim supn→∞Q(ξn,λn). By definition (see Section 2), we can find a subsequence {m} of {n} and (xm,um)∈Q(ξm,λm), m∈ℕ such that
(4.1)xm→x in Lp(T,H) and um→𝑤u in L2(T,Y) as m→∞.
For every m∈ℕ, we have
(4.2)-xm′(t)∈Aλm(t,xm(t))+fm(t)+gm(t) for almost all t∈T,xm(0)=ξm,
with fm,gm∈L2(T,H) such that
(4.3)fm(t)∈F(t,xm(t),λm) and gm(t)∈G(t,um(t),λm) for almost all t∈T.
We deduce by hypotheses (H$F$3) (iii), (H$G$) (iii) and Theorem 3.5 and its proof that
{xm}m∈ℕ⊆Wp(0,b) is bounded and {xm}m∈ℕ⊆C(T,H) is relatively compact.
So, from (4.1) we obtain
(4.4)xm→𝑤x in Wp(0,b) and xm→x in C(T,H) as m→∞.
By (4.3) and hypotheses (H$F$3) (iii), (H$G$) (iii), it is clear that
{fm}m∈ℕ,{gm}m∈ℕ⊆L2(T,H) are bounded.
Hence, we may assume (at least for a subsequence), that
(4.5)fm→𝑤f and gm→𝑤g in L2(T,H) as m→∞.
Proposition 6.6.33 of [34], implies that
(4.6)f(t)∈conv¯w-lim supm→∞F(t,xm(t),λm) for all t∈T∖N,|N|1=0.
Fix t∈T∖N and let y∈w-lim supm→∞F(t,xm(t),λm).
By definition, we know that there exists a subsequence {k} of {m}, and yk∈F(t,xk(t),λk) for all k∈ℕ such that yk→𝑤y in H as k→∞.
The function v↦d(v,F(t,x(t),λ)) is continuous and convex, hence weakly lower semicontinuous. Therefore,
(4.7)d(y,F(t,x(t),λ))≤lim infk→∞d(yk,F(t,x(t),λ)).
On the other hand, we have
(4.8)d(yk,F(t,x(t),λ))≤h(F(t,xk(t),λk),F(t,x(t),λ)).
Using hypotheses (H$F$3) (ii) and (iv), we have
h(F(t,xk(t),λk),F(t,x(t),λ))≤h(F(t,xk(t),λk),F(t,x(t),λk))+h(F(t,x(t),λk),F(t,x(t),λ))
≤k(t)|xk(t)-x(t)|+β(d(λk,λ))w(t,|x(t)|),
and so, from (4.4),
h(F(t,xk(t),λk),F(t,x(t),λ))→0 as k→∞.
Then, from (4.7) and (4.8), we obtain d(y,F(t,x(t),λ))=0, hence
y∈F(t,x(t),λ).
Therefore,
w-lim supm→∞F(t,xm(t),λm)⊆F(t,x(t),λ) for all t∈T∖N,|N|1=0,
which implies (see (4.6) and recall that F is convex-valued)
f(t)∈F(t,x(t),λ) for all t∈T∖N,|N|1=0.
Next, for each m∈ℕ, we have
gm∈SG(⋅,um(⋅),λm)2.
Let h∈L2(T,H) and let (⋅,⋅)L2(T,H) denote the inner product of L2(T,H) (recall that L2(T,H)*=L2(T,H)).
Then (see [34, Theorem 6.4.16])
(4.9)(h,gm)L2(T,H)≤σ(h,SG(⋅,um(⋅),λm)2)=∫0bσ(h(t),G(t,um(t),λm))𝑑t.
The concavity of G(t,⋅,λ) (see hypothesis (H$G$) (ii)), implies that the function u↦σ(h(t),G(t,u,λ)) is concave. Since E is a complete metric space, it can be isometrically embedded, by the Arens–Eells theorem (see [21, Theorem 4.143]), as a closed subset of a separable Banach space (recall that E is compact). So, by [3], we have (see hypothesis (H$G$) (ii))
lim supm→∞∫0bσ(h(t),F(t,um(t),λm))𝑑t≤∫0bσ(h(t),F(t,u(t),λ))𝑑t,
and thus
lim supm→∞σ(h,SG(⋅,um(⋅),λm)2)≤σ(h,SG(⋅,u(⋅),λ)2).
Therefore, from (4.5) and (4.9),
(h,g)L2(T,H)≤σ(h,SG(⋅,u(⋅),λ)2).
Since h∈L2(T,H) is arbitrary, it follows that
g∈SG(⋅,u(⋅),λ)2, hence
g(t)∈G(t,u(t),λ) for almost all t∈T.
Let ym∈Wp(0,b) be the unique solution of the Cauchy problem
(4.10)-ym′(t)∈Aλm(t,ym(t))+f(t)+g(t) for almost all t∈T,ym(0)=ξ.
Hypothesis (H$A$3) (v) implies that
(4.11)ym→𝑤y in Wp(0,b)
with y∈Wp(0,b) being the unique solution of the Cauchy problem
(4.12)-y′(t)∈Aλ(t,y(t))+f(t)+g(t) for almost all t∈T,y(0)=ξ,
see Section 2.
From (4.2), (4.10) and the monotonicity of Aλm(t,⋅) (see hypothesis (H$A$3) (ii)), we have
〈xm′(t)-ym′(t),xm(t)-ym(t)〉≤(f(t)+g(t)-fm(t)-gm(t),xm(t)-ym(t)) for almost all t∈T.
Therefore, by Proposition 2.2,
12|xm(t)-ym(t)|2≤12|ξm-ξ|2+∫0t(f(s)+g(s)-fm(s)-gm(s),xm(s)-ym(s))𝑑s for all t∈T,
which yields
∥xm-ym∥C(T,H)→0 as m→∞,
and hence, by (4.4) and (4.11), x=y.
Recalling that
f(t)∈F(t,x(t),λ) and g(t)∈G(t,u(t),λ) for almost all t∈T,
it follows from (4.12) that
(x,u)∈Q(ξ,λ),
which implies
Kseq(s×w)-lim supn→∞Q(ξn,λn)⊆Q(ξ,λ) in Lp(T,H)×L2(T,Y).
Next, we will prove the second convergence of the proposition.
So, let (x,u)∈Q(ξ,λ). By definition we have
-x′(t)∈Aλ(t,x(t))+F(t,x(t),λ)+g(t) for almost all t∈T,x(0)=ξ
with g∈L2(T,H) satisfying
g(t)∈G(t,u(t),λ) for almost all t∈T.
For every v∈L2(T,Y), we have (see [34, Theorem 6.4.16])
d(v,SU(⋅,λn)2)=∫0bd(v(t),U(t,λn))𝑑t
Hypothesis (H$U$) (ii) and the dominated convergence theorem imply that
∫0bd(v(t),U(t,λn))𝑑t→∫0bd(v(t),U(t,λ))𝑑t,
and so
d(v,SU(⋅,λn)2)→d(v,SU(⋅,λ)2).
Hence, [34, Proposition 6.6.22] implies that we can find un∈SU(⋅,λn)2 (n∈ℕ) such that
un→u in L2(T,Y) as n→∞.
Then hypothesis (H$G$) (ii) guarantees that we can find
gn∈L2(T,H),gn(t)∈G(t,un(t),λn) for almost all t∈T and all n∈ℕ
such that
gn→g in L2(T,H) as n→∞.
Given ξ′∈H, let S(ξ′)⊆Wp(0,b) be the set of solutions of the Cauchy problem
-y′(t)∈Aλn(t,y(t))+F(t,y(t),λ)+g(t) for almost all t∈T,y(0)=ξ′.
Let K={ξn,ξ}n≥1⊆H.
This is a compact set in H. Invoking Proposition 3.11 (with ξ0=ξ), we produce a continuous map ψ:K→C(T,H) such that
(4.13)y^=ψ(ξ^)∈S(ξ^) for all ξ^∈H,ψ(ξ)=x.
Let yn=ψ(ξn) (n∈ℕ) and use Proposition 3.9 to find xn∈Wp(0,b) solution of the Cauchy problem
-xn′(t)∈Aλn(t,xn(t))+F(t,xn(t),λn)+gn(t) for almost all t∈T,xn(0)=ϵn,
for which, we have
(4.14)|xn(t)-yn(t)|≤bϵeτ(t)+∫0tηn(s)eτ(t)-τ(s)𝑑s for all t∈T
with ϵ>0, τ(t)=∫0tk(s)𝑑s, ηn∈L1(T), ηn→0 in L1(T).
So, we obtain (see (4.14))
lim supn→∞∥xn-yn∥C(T,H)≤bϵeτ(b).
Since ϵ>0 is arbitrary, it follows that
(4.15)∥xn-yn∥C(T,H)→0 as n→∞.
Finally, from (4.13) we have
∥xn-x∥C(T,H)≤∥xn-yn∥C(T,H)+∥yn-x∥C(T,H)=∥xn-yn∥C(T,H)+∥ψ(ξn)-ψ(ξ)∥C(T,H).
Therefore, from (4.15) and the fact that ψ(⋅) is continuous,
∥xn-x∥C(T,H)→0.
Since (xn,un)∈Q(ξn,λn) (n∈ℕ) and un→u in L2(T,Y), we conclude that
Q(ξ,λ)⊆K(s×s)-lim infn→∞Q(ξn,λn) in C(T,H)×L2(T,Y).∎
An immediate consequence of the above proposition is the following corollary concerning the multifunction (ξ,λ)↦Q(ξ,λ) of admissible state-control pairs.
Corollary 4.4If hypotheses (H$A$3), (H$F$3), (H$G$), (H$U$) hold, then the multifunction
Q:H×E→2C(T,H)×L2(T,Y)∖{∅}
is LSC and sequentially closed in C(T,H)×L2(T,Y)w (that is, GrQ⊆H×E×C(T,H)×L2(T,Y)w is sequentially closed).
Now we bring the cost functional into the picture. The hypotheses on the integrands L(t,x,λ) and H(t,u,λ) are as follows.
L:T×H×E→ℝ is an integrand such that the following hold:
t↦L(t,x,λ) is measurable for every (x,λ)∈H×E.
If λn→λ in E, then for all x∈H, we have L(⋅,x,λn)→𝑤L(⋅,x,λ) in L1(T).
For almost all t∈T, all x,y∈H and all λ∈E, we have
|L(t,x,λ)-L(t,y,λ)|=(1+|x|∨|y|)ρ(t,|x-y|),
where |x|∨|y|=max{|x|,|y|} and ρ(t,r) is a Carathéodory function on T×ℝ+ with values in (0,+∞) such that
ρ(t,0)=0for almost all t∈T
and sup0≤r≤ϑ[ρ(t,r)]≤βϑ(t)for almost all t∈T
with βϑ∈L1(T)+,ϑ>0.
H:T×Y×E→ℝ is an integrand such that the following hold:
t↦H(t,u,λ) is measurable for all (u,λ)∈Y×E.
u↦H(t,u,λ) is convex for almost all t∈T and all v∈E, and λ↦H(t,u,λ) is continuous for almost all t∈T and all u∈Y.
For almost all t∈T and all (u,λ)∈Y×E, we have
H(t,u,λ)≤a(t)(1+∥u∥Y2) with a∈L∞(T).
ψ^:H×E→ℝ is a continuous function.
Using the direct method of the calculus of variations, we can produce optimal admissible state-control pairs for problem (1.1).
Proposition 4.5If hypotheses (H$A$3), (H$F$3), (H$G$), (H$U$), (H$L$), (H$H$) and (H${\hat{\psi}}$) hold, then for every (ξ,λ)∈H×E we can find (x*,u*)∈Q(ξ,λ) such that
J(x*,u*,ξ,λ)=m(ξ,λ).
Proof.
Let {(xn,un)}n≥1⊆Q(ξ,λ) be a minimizing sequence for problem (1.1). So, we have
J(xn,un,ξ,λ)↓m(ξ,λ) as n→∞.
Theorem 3.5 and hypothesis (H$U$) imply that
{(xn,un)}n≥1⊆Wp(0,b)×L2(T,Y) (respectively, ⊆C(T,H)×L2(T,Y))
is relatively w×w-compact (respectively, s×w-compact).
So, by the Eberlein–Smulian theorem and by passing to a suitable subsequence if necessary, we can say that
(4.16)xn→𝑤x* in Wp(0,b),xn→x* in C(T,H),un→𝑤u* in L2(T,Y).
Then (4.16) and Proposition 4.3 imply that
(4.17)(x*,u*)∈Q(ξ,λ).
Also, (4.16), hypothesis (H$L$) (iii) and the dominated convergence theorem, imply that
(4.18)∫0bL(t,xn(t),λ)𝑑t→∫0bL(t,x*(t),λ)𝑑t.
In addition, as before (see the proof of Proposition 4.3), using [3, Theorem 2.1], we obtain
(4.19)∫0bH(t,u*(t),λ)𝑑t≤lim infn→∞∫0bH(t,un(t),λ)𝑑t.
Finally, (4.16) and hypothesis H(ψ^) imply that
(4.20)ψ^(ξ,xn(b),λ)→ψ^(ξ,x*(b),λ).
We deduce from (4.17), (4.18), (4.19) and (4.20) that
J(x*,u*,ξ,λ)=m(ξ,λ) with (x*,u*)∈Q(ξ,λ).
This concludes the proof.
∎
We are now ready for the main sensitivity results concerning problem (1.1). The first one establishes the Hadamard well-posedness of the problem.
Proof.
Let (ξn,λn)→(ξ,λ) in H×E. Let (x,u)∈Q(ξ,λ) such that (see Proposition 4.5)
J(x,u,ξ,λ)=m(ξ,λ).
Invoking Proposition 4.3, we can find (xn,un)∈Q(ξn,λn) for all n∈ℕ such that
(4.21)xn→x in C(T,H) and un→u in L2(T,Y).
We claim that
(4.22)|∫0bL(t,xn(t),λn)dt-∫0bL(t,x(t),λ)dt|→0 as n→∞.
To this end, note that
|∫0bL(t,xn(t),λn)dt-∫0bL(t,x(t),λ)dt|
(4.23)≤|∫0bL(t,xn(t),λn)dt-∫0bL(t,x(t),λn)dt|+|∫0bL(t,x(t),λn)dt-∫0bL(t,x(t),λ)dt|
for all n∈ℕ.
First, we estimate the first summand in the right-hand side of (4.23). Using hypothesis (H$L$) (iii), we have
|∫0bL(t,xn(t),λn)-L(t,x(t),λn)dt|≤∫b0(1+|xn(t)|∨|x(t)|)ρ(t,|xn(t)-x(t)|)dt.
Let M=supn≥1∥xn∥C(T,H)<+∞ (see (4.21)). Then, from (4.21) and hypothesis (H$L$) (iii), we have
(4.24)|∫0bL(t,xn(t),λn)dt-∫0bL(t,x(t),λn)dt|≤(1+M)∫b0ρ(t,|xn(t)-x(t)|)dt→0 as n→∞.
Next, we estimate the second term on the right-hand side of (4.23).
Let ϑ>2∥x∥C(T,H) and let βϑ∈L1(T)+ as postulated by hypothesis (H$L$) (iii). Given ϵ>0, we can find δ>0 such that
(4.25)“if C⊆T is measurable with |C|1≤δ, then ∫Cβϑ(t)𝑑t≤ϵ2(1+ϑ).”
Here, we use the absolute continuity of the Lebesgue integral. Invoking the Scorza–Dragoni theorem (see [34, Theorem 6.2.9]), we can find T1⊆T closed with |T∖T1|≤δ2 and such that ρ|T1×ℝ+ is continuous.
Since ρ(t,0)=0, we can find δ1>0 such that
(4.26)“if r∈[0,δ1], then |ρ(t,r)|≤ϵ2b(1+ϑ) for all t∈T1.”
Recall that simple functions are dense in Lp(T,H).
Using this fact, the property that Lp(T,H)-convergence implies pointwise convergence for almost all t∈T for at least a subsequence, and invoking Egorov’s theorem, we can find T2⊆T closed and a simple function s:T→H such that
(4.27)∥s∥∞≤∥x∥C(T,H),|T∖T2|1≤δ2 and |x(t)-s(t)|≤δ1 for all t∈T2.
We set T3=T1∩T2. This is a closed subset of T with |T∖T3|1≤δ. We have (see hypothesis (H$L$) (iii) and (4.27))
|∫0bL(t,x(t),λn)dt-∫0bL(t,s(t),λn)dt|≤(1+∥x∥C(T,H))∫b0ρ(t,|x(t)-s(t)|)dt
≤(1+ϑ)[∫T3ρ(t,|x(t)-s(t)|)𝑑t+∫T∖T3ρ(t,|x(t)-s(t)|)𝑑t]
(4.28)≤ϵ2+ϵ2=ϵ,
see (4.25), (4.26) and (4.27).
Similarly, we show that
(4.29)|∫0bL(t,s(t),λ)dt-∫0bL(t,x(t),λ)dt|≤ϵ.
Let s(t)=∑k=1NvkχCk(t) with vk∈H,Ck⊆T measurable.
Using hypothesis (H$L$) (ii), we can find n0∈ℕ such that
(4.30)|∫0b(L(t,s(t),λn)-L(t,s(t),λ))dt|≤∑k=1N|∫Ck(L(t,vk,λn)-L(t,vk,λ))dt|≤ϵ for all n≥n0.
From (4.28), (4.29) and (4.30) it follows that
∫0bL(t,x(t),λn)𝑑t→∫0bL(t,x(t),λ)𝑑t as n→∞.
This convergence and (4.24) imply that (4.22) (our claim) is true.
Next, we consider the integral functional
Φ(u,λ)=∫0bH(t,u(t),λ)𝑑t for all (u,λ)∈L2(T,Y)×E.
For every λ∈E,u↦Φ(u,λ) is convex (see hypothesis (H$H$) (ii)).
Also, hypothesis (H$H$) (iii) implies that in a neighborhood of every u∈L2(T,Y),
{Φ(⋅,λ)}λ∈E
is equibounded above, hence
{Φ(⋅,λ)}λ∈E is equi-locally Lipschitz
(see [34, Theorem 1.2.3]). Therefore, it follows that (see (4.21))
(4.31)Φ(un,λn)→Φ(u,λ) as n→∞.
Finally, (4.21) and hypothesis (H${\hat{\psi}}$) imply that
(4.32)ψ^(ξn,xn(b),λn)→ψ^(ξ,x(b),λ).
By (4.22), (4.31), (4.32), we have
∫0bL(t,xn(t),λn)𝑑t+∫0bH(t,un(t),λn)𝑑t+ψ^(ξn,xn(b),λn)→J(x,u,ξ,λ)=m(ξ,λ),
which implies
(4.33)lim supn→∞m(ξn,λn)≤m(ξ,λ).
From Proposition 4.5 we know that for every n∈ℕ, we can find (xn,un)∈Q(ξn,λn) such that
(4.34)J(xn,un,ξn,λn)=m(ξn,λn).
As in the proof of Theorem 3.5, we can show that {xn}n≥1⊆Wp(0,b) is bounded.
In addition, hypothesis (H$U$) implies that {un}n≥1⊆L2(T,Y) is bounded. So, by passing to a suitable subsequence if necessary, we may assume that
(4.35)xn→𝑤x in Wp(0,b) and un→𝑤u in L2(T,Y) as n→∞.
By (4.35) and (2.3), we also have
(4.36)xn→x in Lp(T,H) as n→∞.
Then (4.35), (4.36) and Proposition 4.3 imply that
(x,u)∈Q(ξ,λ).
Moreover, reasoning as in the proof of Theorem 3.5, we can show that
{xn}n≥1⊆C(T,H) is relatively compact, hence (see (4.36))
(4.37)xn→x in C(T,H).
By (4.37) and the first part of the proof, we have
∫0bL(t,xn(t),λn)𝑑t→∫0bL(t,x(t),λ)𝑑t.
In addition, (4.35) and hypotheses (H$H$) (ii), (H${\hat{\psi}}$) imply (see[3])
∫0bH(t,u(t),λ)𝑑t≤lim infn→∞∫0bH(t,un(t),λn)𝑑t,
and thus
ψ^(ξn,xn(b),λn)→ψ^(ξ,x(b),λ).
Therefore, from (4.34) we see that
∫0bL(t,x(t),λ)𝑑t+∫0bH(t,u(t),λ)𝑑t+ψ^(ξ,x(b),λ)≤lim infn→∞m(ξn,λn),
and thus
(4.38)m(ξ,λ)≤lim infn→∞m(ξn,λn).
We infer from (4.33) and (4.38) that
m(ξn,λn)→m(ξ,λ),
hence m:H×E→ℝ is continuous.
∎
For every (ξ,λ)∈H×E, we introduce the set Σ(ξ,λ) of optimal state-control pairs, that is,
Σ(ξ,λ)={(x,u)∈Q(ξ,λ):J(x,u,ξ,λ)=m(ξ,λ)}.
By Proposition 4.5, we know that Σ(ξ,λ)≠∅ for every (ξ,λ)∈H×E. For this multifunction we can prove the following useful continuity property.
Proof.
Let C⊆C(T,H)×L2(T,Y)w be sequentially closed. We need to show that
Σ-(C)={(ξ,λ)∈H×E:V(ξ,λ)∩C≠∅}
is closed in H×E (see Section 2).
To this end, let {(ξn,λn)}n≥1⊆Σ-(C), and assume that
(ξn,λn)→(ξ,λ) in H×E.
Let (xn,un)∈Σ(ξn,λn)∩C, n∈ℕ. We know from the proof of Theorem 4.6 that at least for a subsequence, we have
(4.39)xn→𝑤x in Wp(0,b),xn→x in C(T,H),un→𝑤u in L2(T,Y) as n→∞.
By (4.39) and Proposition 4.3, we have
(4.40)(x,u)∈Q(ξ,λ).
Also, we know from the proof of Theorem 4.6 that
J(x,u,ξ,λ)≤lim infn→∞J(xn,un,ξn,λn)=lim infn→∞m(ξn,λ)=m(ξ,λ).
Therefore, from (4.40), J(x,u,ξ,λ)=m(ξ,λ),
and thus
(x,u)∈Σ(ξ,λ).
Moreover, from (4.39) and since C⊆C(T,H)×L2(T,Y)w is sequentially closed, we deduce that (x,u)∈Σ(ξ,λ)∩C. Therefore, Σ-(C)⊆H×E is closed and this proves the desired sequential upper semicontinuity of the multifunction (ξ,λ)↦Σ(ξ,λ).
∎
5 Application to distributed parameter systems
In this section we present an application to a class of multivalued parabolic optimal control problems.
So, let T=[0,b] and let Ω⊆ℝN be a bounded domain with a Lipschitz boundary ∂Ω.
We examine the following nonlinear, multivalued parabolic optimal control problem:
(5.1){J(x,u,ξ,λ)=∫0b∫ΩL1(t,z,x(t,z))𝑑z𝑑t+∫0b∫ΩH1(t,z,u(t,z))𝑑z𝑑t→inf=m(ξ,λ),-∂x∂t∈-divaλ(z,Du)+F1(t,z,x(t,z),λ)+g(t,z,λ)u(t,z) on T×Ω,x|T×∂Ω=0,x(0,z)=ξ(z) for almost all z∈Ω,∥u(t,⋅)∥L2(Ω)≤r(t,λ) for almost all t∈T.
Here, aλ:Ω×ℝN→2ℝN (λ∈E) is a family of multifunctions as in Example 4.1 (b).
For the other data of problem (5.1), we introduce the following conditions:
F1:T×Ω×ℝ×E→Pfc(ℝ) is a multifunction such that the following hold:
(t,z)↦F1(t,z,x,λ) is measurable for all (x,λ)∈ℝ×E.
For almost all (t,z)∈T×Ω, all x,y∈ℝ and all λ∈E, we have
h(F1(t,z,x,λ),F1(t,z,y,λ))≤k1(t,z)|x-y|
with k1∈L1(T,L∞(Ω)).
For almost all (t,z)∈T×Ω, all x∈ℝ and all λ∈E, we have
|F1(t,z,x,λ)|≤a^1(t,z)+c^|x|
with a^1∈L2(T×Ω),c^1>0.
For almost all (t,z)∈T×Ω, all x∈ℝ and all λ,λ′∈E, we have
h(F1(t,z,x,λ),F1(t,z,x,λ′))≤β(d(λ,λ′))w(z,|x|)
with β(r)→0 as r→0+ and w∈Lloc∞(Ω×ℝ+).
Remark 5.1Consider the multifunction F(t,z,x,λ) defined by
F(t,z,x,λ)=[f(t,z,x,λ),f^(t,z,x,λ)],
where f,f^:T×Ω×ℝ×E→ℝ are two functions with the following properties:
(t,z)↦f(t,z,x,λ),f^(t,z,x,λ) are both measurable for all (x,λ)∈ℝ×E.
For almost all (t,z)∈T×Ω, all x,x′∈ℝ and all λ,λ′∈E, we have
|f(t,z,x,λ)-f(t,z,x′,λ′)|≤k(t,z)[|x-x′|+d(λ,λ′)]
|f^(t,z,x,λ)-f^(t,z,x′,λ′)|≤k^(t,z)[|x-x′|+d(λ,λ′)],
with k,k^∈L1(T,L∞(Ω)).
Then this multifunction satisfies hypotheses (H${F_{1}}$).
g:T×Ω×E→ℝ is a Carathéodory function (that is, (t,z)→g(t,z,λ) is measurable for all λ∈E and λ→g(t,z,λ) is continuous for almost all (t,z)∈T×Ω) and for almost all (t,z)∈T×Ω and all λ∈E, we have |g(t,z,λ)|≤M with M>0.
r:T×E→ℝ+ is a Carathéodory function (that is, t↦r(t,λ) is measurable for all λ∈E and λ→r(t,λ) is continuous for almost all t∈T) and for almost all t∈T and all λ∈E, we have
0≤r(t,λ)≤a(t) with a∈L2(T).
Now, we introduce the conditions on the two integrands involved in the cost functional problem (5.1).
L:T×Ω×ℝ×E→ℝ is an integrand such that the following hold:
(t,z)↦L1(t,z,x,λ) is measurable for all (x,λ)∈ℝ×E.
If λn→λ in E, then for all x∈L2(Ω) we have L1(⋅,⋅,x(⋅),λn)→𝑤L1(⋅,⋅,x(⋅),λ) in L1(T×Ω).
For almost all (t,z)∈T×Ω, all x,y∈ℝ and all λ∈E,
|L1(t,z,x,λ)-L1(t,z,y,λ)|≤c(1+|x|∨|y|)ρ(t,z,|x-y|),
with ρ(t,z,r) Carathéodory, ρ(t,z,0)=0 for almost all (t,z)∈T×Ω and for almost all (t,z), all r∈[0,ϑ] we have
0≤ρ(t,z,r)≤βϑ(t,z)
with βϑ∈L1(T×Ω).
H1:T×Ω×ℝ×E→ℝ is an integrand such that the following hold:
(t,z)↦H1(t,z,x,λ) is measurable for all (x,λ)∈ℝ×E.
For almost all (t,z)∈T×Ω, u↦H1(t,z,u,λ) is convex for all λ∈E, while λ↦H1(t,z,u,λ) is continuous for all u∈ℝ.
For almost all (t,z)∈T×Ω, all |u|≤rλ(t,z) and all λ∈E, we have
|H1(t,z,u,λ)|≤a^λ(t,z)
with {a^λ}λ∈E⊆L2(T×Ω) bounded.
We consider the following evolution triple:
X=W01,p(Ω),H=L2(Ω),X*=W-1,p′(Ω).
Since 2≤p<∞, the Sobolev embedding theorem implies that in this triple the embeddings are compact.
For every λ∈E, let Aλ:X→2X*∖{∅} be the multivalued map defined by
Aλ(x)={-divg:g∈Lp′(Ω,ℝN),g(z)∈aλ(z,Dx(z)) for almost all z∈Ω}.
This map is maximal monotone and if λn→λ in E, then (see Example 4.1 (b))
ddt+aλn→PGddt+aλ.
So, hypotheses (H$A$3) hold. In fact, we can have t-dependence at the expense of assuming that aλ is single-valued. So, we assume that aλ(t,z,ξ) satisfies the conditions of Example 4.1 (a). Then the map Aλ:T×X→X* is defined by
Aλ(t,x)(⋅)=-divaλ(t,⋅,Dx(⋅)).
In fact, by the nonlinear Green’s identity (see [20, p. 210]), we have
〈Aλ(t,x),h〉=∫Ω(aλ(t,z,Dx),Dh)ℝN𝑑z for all x,h∈W01,p(Ω).
As we have already mentioned in Example 4.1 (a), we know from [40] that if λn→λ in E, then
ddt+aλn→PGddt+aλ,
and so hypotheses (H$A$3) hold.
As a special case of interest, we consider the situation where the elliptic differential operator is a weighted p-Laplacian, that is,
div(aλ(t,z)|Dx|p-2Dx) for all x∈W01,p(Ω).
Here, for every λ∈E,aλ:T×Ω→ℝ is a measurable function with the following properties:
0<c^1≤aλ(t,z)≤c^2 for almost all (t,z)∈T×Ω and all λ∈E.
If λn→λ in E, then for almost all t∈T,
1aλn(t,⋅)p′-1→𝑤1aλ(t,⋅)p′-1 inL1(Ω).
For this case we consider the following parametric (with parameter λ∈E) family of convex (in ξ∈ℝN) integrands:
φλ(t,z,ξ)=aλ(t,z)p|ξ|p.
Then the convex conjugate of φλ(t,z,⋅) is given by
φλ*(t,z,ξ*)=1p′aλ(t,z)p′-1|ξ*|p′.
By hypothesis we have that λn→λ in E, hence
(5.2)φλn*(t,⋅,ξ*)→φλ*(t,⋅,ξ*) in L1(Ω) for almost all t∈T and all ξ*∈ℝN.
We introduce the integral functional Φλ defined by
Φλ(t,x)=∫Ωφλ(t,z,Dx)𝑑z for all (t,x)∈T×W01,p(Ω).
By [29], we know that (5.2) implies
Φλ(t,x)=Γseq(w)-Φλn(t,x)
with Γseq(w) denoting the sequential Γ-convergence of Φλn(t,⋅) on W01,p(Ω)w (see [7]). Then, from [12, Theorem 3.3], it follows that
aλn(t,⋅,⋅)→Gaλ(t,⋅,⋅) for almost all t∈T,
and so from [40], we conclude that
ddt+aλn→PGddt+aλ.
Also, let Y=H=L2(Ω) and
F(t,x,λ)=SF1(t,⋅,x(⋅),λ)2,G(t,u,λ)={g(t,⋅,λ)u(⋅):∥u∥L2(Ω)≤r(t,λ)}
U(t,λ)={u∈L2(Ω):∥u∥L2(Ω)≤r(t,λ)}.
Then hypotheses (H$F$1), (H${g}$), (H${r}$) imply that conditions (H$F$3), (H$G$), (H$U$) hold. So, the dynamics of (5.1) are described by an evolution inclusion similar to the one in problem (1.1).
Finally, let
L(t,x,λ)=∫ΩL1(t,z,x(z),λ)𝑑z for all x∈L2(Ω),
H(t,u,λ)=∫ΩH1(t,z,u(z),λ)𝑑z for all u∈L2(Ω).
Hypotheses (H${L_{1}}$), (H${H_{1}}$) imply that conditions (H$L$),(H$H$), respectively, hold.
So, we can apply Theorems 4.6 and 4.7 and obtain the following result concerning the variational stability of problem (5.1).
Proposition 5.2If the maps aλ are as above and hypotheses (H${F_{1}}$), (H${g}$), (H${r}$), (H${L_{1}}$), (H${H_{1}}$) hold, then for every (ξ,λ)∈L2(Ω)×E, problem (5.1) admits optimal pairs (that is, Σ(ξ,λ)≠∅) and
(ξ,λ)↦m(ξ,λ) is continuous on L2(Ω)×E,
(ξ,λ)↦Σ(ξ,λ) is sequentially USC from L2(Ω)×E into C(T,L2(Ω))×L2(T×Ω)w.
and
Dušan D. Repovš
