Weak Pullback Attractors of Non-Autonomous Difference Inclusions

Weak pullback attractors of nonautonomous difference inclusions∗ P.E. Kloeden FB Mathematik, Johann Wolfgang Goethe Universit¨at, D-60054 Frankfurt am Main, Germany email: klo[email protected] and P. Mar´ın-Rubio Dpto. de Ecuaciones Diferenciales y An´alisis Num´erico, Universidad de Sevilla, Apdo. de Correos 1160, 41080 Sevilla, Spain email: [email protected] Abstract Weak pullback attractors are defined for nonautonomous difference inclusions and their existence and upper semi continuous convergence under perturbation is established. Unlike strong pullback attractors, invariance and pullback attraction here are required only for (at least) a single trajectory rather than all trajectories at each starting point. The concept is thus useful, in particular, for discrete time control systems. AMS Subject Classification: 37B25, 37B55, 58C06 Key words: difference inclusion, pullback attraction, weak invariance, weak attractor. Dedicated to George Sell on the occasion of his 65th birthday ∗Partially supported by the DAAD (Germany) and the Ministerio de Ciencia y Tecnolog´ıa (Spain) under the bi-lateral programme of Acciones Integradas. 1 1 Introduction A nonautonomous difference inclusion xt+1 ∈Ft(xt) (1) arises naturally in a variety of ways. An important source of applications is based on single valued control systems of the form xt+1 =ft(xt, ut) (2) with controls uttaking values in a nonempty compact set Ut, thus (2) generates a nonautonomous difference inclusion of the form (1) with Ft(x) := ft(x, Ut). Other sources of examples are the discretization or time–1 mappings of differential control systems or differential equations without uniqueness [3, 7, 8, 9]. The mappings Ft, which are usually assumed to be compact valued and upper semi continuous, may vary in some regular or completely arbitrarily fashion. The discrete–time system generated by (1) is thus nonautonomous and no longer enjoys a setvalued semigroup property, so many of the concepts of autonomous systems are either too restrictive or inappropriate for an investigation of their asymptotic behaviour. The concept of a nonautonomous pullback attractor, which consists of a family of nonempty compact subsets rather than a single subset and “pullback” attracts from asymptotically earlier starting times, was introduced in [7] for nonautonomous difference inclusion. In the autonomous case this pullback attractor reduces to as single set which is attracting in the usual forward sense. Szeg¨o and Treccani [10] call it a strong attractor for their continuous time setvalued semigroups. They also distinguish another type of attractor for autonomous setvalued systems, which they call a weak attractor. The difference is that only one or more trajectories for each starting point must be attracted to or remain in the weak attractor rather than all trajectories in the case of the strong attractors. This situation is of particular interest in control systems. Our aim in this paper is to introduce and investigate a pullback version of a weak attractor for setvalued difference processes generated by nonautonomous difference inclusions. We define these setvalued difference processes in Section 2 and recall the definitions and basic results of strong autonomous and nonautonomous attractors in Section 3. Then in Section 4 we introduce the concept of a weak pullback attractor in terms of weakly invariant and weakly pullback attracting families of nonempty compact subsets of the state space. We state our first main result in Section 5, that 2 the existence of a weak pullback attractor follows from that of a weakly positively invariant weakly pullback absorbing family of nonempty compact subsets, and indicate why other seemingly more natural constructions of the components subsets are not appropriate. Our second main result on the upper semi continuous convergence of weak pullback attractors is stated in Section 6. We then give five examples in Section 7 to illustrate some of the features and peculiarities of weak pullback attractors. The proofs of our main results and some supporting lemmata are given at the end of the paper in Section 8. We require the following definitions and terminology [1]. The distance of a point x∈Rdfrom a nonempty compact set Ais defined by dist(x, A) = min a∈Akx−ak. The Hausdorff separation H∗(A, B) of nonempty compact subsets A,Bof Rdis defined by H∗(A, B) := max a∈Adist(a, B) = max a∈Amin b∈Bka−bk and H(A, B) = max {H∗(A, B), H∗(B, A)}denotes the Hausdorff metric on the space H(Rd) of nonempty compact subsets of Rd. An open –neighbourhood of A∈ H(Rd) is defined by N(A) = {x∈Rd: dist(x, A)< }and a closed – neighbourhood of Aby N[A] = {x∈Rd: dist(x, A)≤}. A mapping F:Rd7→ H(Rd) is upper semi continuous at x0if for all ε > 0 there exists a δ=δ(ε, x0)>0 such that F(x)⊂Nε(F(x0)) for all x∈Nδ({x0}) or alternatively if lim xn→x0 H∗(F(xn), F(x0)) = 0 for all sequences xn→x0. Denote the space of all upper semi continuous mappings F:Rd7→ H(Rd) by USC(Rd,H(Rd)) and define Z2 +={(i, j)∈Z2|i≥j}. For any A∈ H(Rd) define F(A) := ∪a∈AF(a) and define the set composition of two mappings F,G:Rd7→ H(Rd) as F◦G(x) := F(G(x)) for all x∈Rd. Note that F◦G∈USC(Rd,H(Rd)) if F,G∈USC(Rd,H(Rd)). For simplicity we shall present our results for a Euclidean state space Rd, though they are in fact valid for more general metric or Banach state spaces since we do not use the local compactness property of Rdin our proofs. 2 Setvalued difference processes As in the singlevalued case, a natural nonautonomous generalization of an autonomous system defined in terms of a semigroup of mappings is a two–parameter 3 semigroup or process. Definition 2.1 A mapping Φ : Z2 +×Rd7→ H (Rd)is called a setvalued difference process on Rdif Φ(t, t0,·)∈USC(Rd,H(Rd)) for all (t, t0)∈Z2 +and Φ(t0, t0, x) = {x},(3) Φ(t2, t0, x) = Φ (t2, t1,Φ(t1, t0, x)) ,(4) for all t0≤t1≤t2in Zand all x∈Rd. Thus nonautonomous difference inclusion (1) with mappings Ft∈USC(Rd,H(Rd)) for t∈Zgenerates a setvalued difference process with the mappings Φ(t, t0,·)∈ USC(Rd,H(Rd)) defined by Φ(t0, t0, x) := {x}and Φ(t, t0, x) := Ft−1◦ · · · ◦ Ft0(x) for all x∈Rdand t0< t in Z. Conversely, a setvalued difference process Φ generates a nonautonomous difference inclusion (1) with mappings Fndefined by Ft(x) := Φ(t+ 1, t, x) for all x∈Rdand t∈Z. A trajectory of a setvalued difference process Φ is a single valued mapping φ: [T0, T1]∩Z7→ Rd, for some T0< T1in Z, which satisfies φ(t)∈Φ(t, t0, φ(t0)) for all T0≤t0≤t≤T1. Note that, due to (4) , the concatenation of trajectories on adjacent time sets [T0, T1]∩Zand [T1, T2]∩Zforms a trajectory on the union [T0, T2]∩Zof these times sets. A trajectory defined on all of Zis called an entire trajectory. 3 Attractors of setvalued difference processes An attractor for an autonomous difference inclusion, i.e., (1) with Ft≡F, is a nonempty compact subset Aof Rdwhich is invariant, i.e., satisfies F(A) = A, and is attracting in the sense that lim n→∞ H∗(Fn(D), A) = 0 for every nonempty bounded set Dof Rd; here Fndenotes the nfold composition of Fwith itself. As for singlevalued dynamical systems, the existence of an attractor is implied by that of a more easily determinable absorbing set, i.e., a nonempty 4 compact subset Bof Rdsuch that for every nonempty bounded set Dof Rdthere exists a nonnegative integer NDsuch that Fn(D)⊆Bfor all n≥ND. The following theorem is a setvalued generalization of a well known result for singlevalued semigroups. Theorem 3.1 Let F∈USC(Rd,H(Rd)) and suppose that the autonomous difference inclusion with mapping Fhas an absorbing set B∈ H(Rd). Then it has a unique attractor Adefined by A=\ m≥0[ n≥m Fn(B). The concepts of an absorbing set and attractor are somewhat more complicated in the nonautonomous difference case, with the obvious generalisations being too restrictive for most situations. As in the case of singlevalued difference equations [5], families of sets rather than individual sets should be considered. Definition 3.2 A family B={Bt, t ∈Z}of nonempty compact subsets of Rdis called a pullback absorbing family for a setvalued difference process Φon Rdif for every t0∈Zand every nonempty bounded subset Dof Rdthere exists an Nt0,D ∈ Z+such that Φ (t0, t0−n, D)⊆Bt0 for all n≥Nt0,D. Definition 3.3 A family A={At, t ∈Z}of nonempty compact subsets of Rdis called a pullback attractor of a setvalued difference process Φon Rdif it is strictly invariant, i.e., Φ (t, t0, At0) = Atfor any t≥t0,(5) and pullback attracts bounded sets, i.e., lim n→∞ H∗(Φ(t0, t0−n, D), At0) = 0 (6) for every t0∈Zand every bounded subset Dof Rd. Property (5) is a generalisation of the positive invariance property of a semigroup. Note that the pullback convergence property (6) does not describe the convergence of Φ(t, t0, D) as t→ ∞. See [4, 5, 6] for a discussion on these properties in the context of singlevalued processes. The following theorem from [7, 9] is a generalisation of Theorem 3.1 to the nonautonomous case. 5 Theorem 3.4 Let Φbe a setvalued difference process with a positive invariant and uniformly bounded pullback absorbing family B={Bt, t ∈Z}. Then there exists the minimal negatively invariant (strictly if Φis lower semicontinuous) pullback attractor A={At, t ∈Z}which is determined by At0=\ m≥0[ n≥m Φ (t0, t0−n, Bt0−n) (7) for each t0∈Z. If the pullback absorbing family Bis not assumed or known to be positively invariant then one can also obtain the minimal negatively invariant attractor for each t0∈Z by [ bounded D⊂Rd\ m≥0[ n≥m Φ (t0, t0−n, D) (8) It was shown in [4] for singlevalued difference equations that a pullback attractor always has a forward invariant pullback absorbing family. The proof can be adapted to the difference inclusion case under consideration here. 4 Weak attractors of difference inclusion processes The preceding concepts of invariance and attraction for setvalued difference processes are nonautonomous counterparts of what Szeg¨o and Treccani [10] called strong invariance and attractors of continuous time autonomous setvalued systems or semigroups. Essentially, the invariance and attraction holds with respect to all possible trajectories emanating from each starting point. For setvalued systems arising from control systems, one is often interested in situations where just one, or a few, rather than all trajectories emanating from each starting point satisfy a given property. This is also of interest for systems generated by differential equations without uniqueness such as x0=x1/3, for which the set {0} is only “weakly” positively invariant due to the nonuniqueness of solutions with the initial value x(0) = 0. Szeg¨o and Treccani also introduced corresponding concepts of weak invariance and weak attraction for such continuous time autonomous setvalued systems. In the discrete time case under consideration here these read as follows: A nonempty compact subset Ais weakly positively invariant if for each x0∈Athere exists a trajectory φwith φ(0) = x0such that φ(t)∈Afor all t≥0. A nonempty compact subset Ais weakly attracting if for each x0∈Rd, there exists a trajectory φwith φ(0) 6 =x0such that dist(φ(t), A)→0 as t→ ∞. Finally, a nonempty compact subset Ais called a weak attractor if it is weakly positively invariant and weakly attracting. Our aim in this paper is to introduce and investigate pullback versions of these weak concepts for setvalued difference processes. As with the strong concepts of invariance and attraction above, it is also less restrictive here to consider families of sets rather than individual sets. Definition 4.1 A family A={At, t ∈Z}of nonempty compact subsets of Rdis said to be weakly positively invariant for a setvalued difference process Φon Rdif for every t0∈Zand every x0∈At0there exists a trajectory φ: [t0,∞)∩Z→Rd of Φwith φ(t0) = x0such that φ(t)∈Atfor all t≥t0. The family A={At, t ∈Z} is said to be weakly invariant if for every t0∈Zand every x0∈At0there exists an entire trajectory φ:Z→Rdof Φwith φ(t0) = x0such that φ(t)∈Atfor all t∈Z. Definition 4.2 A weakly invariant family A={At, t ∈Z}of nonempty compact subsets of Rdis called a weak pullback attractor of a setvalued difference process Φ on Rdif it is weakly pullback attracting, i.e., for any t0∈Z, any nonempty bounded subset Dof Rdand any sequence dn∈Dthere exist sequences of integers tn→ ∞ as n→ ∞ and trajectories φn: [t0−tn, t0]∩Z→Rdof Φwith φn(t0−tn) = dn such that lim n→∞ dist (φn(t0), At0)=0.(9) Note that a strong pullback attractor, if it exists, is also a weak pullback attractor. 5 Existence of weak pullback attractors Our first main result is to show that the existence of a weak pullback attractor follows from that of a more easily determined weak pullback absorbing family. Definition 5.1 A weakly positively invariant family B={Bt, t ∈Z}of nonempty compact subsets of Rdis called a weak pullback absorbing family of a setvalued difference process Φon Rdif for t0∈Zand any bounded subset Dof Rdthere exists an integer Nt0,D such that for each n≥Nt0,D and dn∈Dthere exists a trajectory φn: [t0−n, t0]∩Z→Rdof Φwith φn(t0−n) = dnand φn(t0)∈Bt0. Note, by the weak positive invariance of Bthe trajectories φncan be extended to remain in Bfor t≥t0, i.e., with φn(t)∈Btfor each t≥t0. 7 Theorem 5.2 Let Φbe a setvalued difference process with a weak pullback absorbing family B={Bt, t ∈Z}. Then Φhas a maximal weak pullback attractor A= {At, t ∈Z}relative to B, which is uniquely determined by At0=na0∈Rd;∃tn→ ∞ as n→ ∞ and trajectories φn: [t0−tn, t0]∩Z→Rdsuch that φn(t)∈Bt(10) for t∈[t0−tn, t0]∩Zand lim n→∞ φn(t0) = a0o for each t0∈Z. The maximal weak pullback attractor A={At, t ∈Z}here apparently consists of the entire trajectories that “move” or remain in Bfor the entire time set Z, i.e., satisfy φ(t)∈Btfor each t∈Z; we will prove this in Lemma 8.1. It is thus the maximal weakly invariant family which is contained in Band is unique in this sense. However, as our examples in Section 7 will show, a setvalued inclusion process may have several different pullback absorbing families either with overlapping or with disjoint component sets. Each of these absorbing families contains a maximal weak pullback attractor relative to itself, the component sets of which may overlap or be disjoint. Uniqueness of weak pullback attractors is thus not a universal property as in the case of a strong pullback attractor, the uniqueness of the weak pullback attractor being only relative to its given absorbing family. In particular, this means that steady state or periodic trajectories need not be contained in a given weak pullback attractor. Some of our examples are in fact an autonomous system, so this is a peculiarity of weak attractors in general rather than a characteristic of nonautonomity. Our assumption that a weakly pullback absorbing family is weakly positively invariant is an interesting distinction between strong and weak pullback absorbing families. As we saw above, the strong positive invariance of a strong pullback absorbing is not essential to ensure the existence of a strong pullback attractor. In the weak case we need it to ensure the existence of a weak pullback attractor which is weakly invariant. A closely related issue is our construction (10) of the component sets of the weak pullback attractor, with other seemingly natural definitions failing to be weakly invariant with or without the assumed weak positively of the weakly pullback absorbing family. For instance, suppose that the weak pullback absorbing family B={Bt, t ∈Z} is weakly positive invariant. Then the family A∗={A∗ t, t ∈Z}defined by A∗ t0:= \ m≥0 [ n≥m Φ(t0, t0−n, Bt0−n)\Bt0!(11) 8 is weakly pullback attracting with At⊂A∗ t⊂Btfor all t∈Z, where Atis defined by (10). The positive weak invariance of A∗follows from this inclusion, but the inclusion A∗ t⊂Φ(t, t0, A∗ t0)TBtis generally false, so negative weak invariance usually does not hold, cf. Lemma 8.1. On the other hand, if we do not assume that B={Bt, t ∈Z}is positively weakly invariant, then we might define the family A∗={A∗ t, t ∈Z}by A∗ t0=[ bounded D⊂Rd\ m≥0 [ n≥m Φ(t0, t0−n, D)\Bt0!. This family A∗weakly pullback attracts all bounded subsets of Rdand satisfies A∗ t ⊂Btfor all t∈Z, but need not be either positively or negatively weakly invariant. (A sufficient condition for the positive weak invariance is that Φ(t, t0,·) be lower semi continuous for any pair (t, t0)∈Z2 +, see e.g. [2]). 6 Upper semi continuity of weak pullback attractors Our second main result is to establish the upper semi continuous dependence of weak pullback attractors under perturbation. For this we consider a perturbed nonautonomous difference inclusion xt+1 ∈F t(xt) (12) with the F t∈USC(Rd,H(Rd)) such that H∗(F t(x), Ft(x)) ≤(13) for all x∈Rdand n∈Z. Let Φbe the setvalued difference process generated by the perturbed nonautonomous difference inclusion (12). Theorem 6.1 Suppose that the setvalued difference process Φgenerated by the unperturbed nonautonomous difference inclusion (1) has a weakly positive invariant weakly pullback absorbing family B={Bt, t ∈Z}and suppose that the perturbed setvalued difference process Φgenerated by the perturbed nonautonomous difference inclusion (12) satisfying (13) has a weakly positive invariant weakly pullback absorbing family B={B t, t ∈Z}such that lim →0H∗B t0, Bt0= 0 (14) 9 8.2 Proof of Theorem 6.1 Let A={At, t ∈Z}be the maximal weak pullback attractor in B={Bt, t ∈Z} of the unperturbed setvalued difference process Φ and let A={A t, t ∈Z}be the maximal weak pullback attractor in B={B t, t ∈Z}of the perturbed setvalued difference process Φ. Suppose for some t0∈Zthat lim →0H∗A t0, At06= 0. Then there exists an η0>0 and a subsequence j→0 as j→ ∞ such that H∗Aj t0, At0≥η0(16) for all j∈Z+. We will show that this leads to a contradiction. Let aj∈A t0be such that dist (aj, At0) = H∗Aj t0, At0, so dist (aj, At0)≥η0 for j∈Z+, which is possible since A t0is compact. By Lemma 8.1 there is an entire trajectory φjof the perturbed setvalued difference process Φjsuch that φj(t)∈ Aj t⊂Bj tfor each t∈Zwith φj(t0) = aj. Since for each t, the Bj tand Btare compact with H∗Bj t, Bt→0 as j→0 , by Lemma 8.2 there exists a convergent (diagonal) subsequence φ0 j(t)→¯ φ(t)∈Btas 0 j→0 for each t∈Z. Obviously aj =φ0 j(t0)→¯ φ(t0), so from (16) we have dist ¯ φ(t0), At0≥η0/2.(17) We will show that ¯ φis a trajectory of the unperturbed setvalued difference process Φ. We have dist ¯ φ(t+ 1), Ft(¯ φ(t))≤   ¯ φ(t+ 1) −φ0 j(t+ 1)  + dist φ0 j(t+ 1), F0 j t(φ0 j(t)) +H∗F0 j t(φ0 j(t)), Ft(¯ φ(t)) =   ¯ φ(t+ 1) −φ0 j(t+ 1)  +H∗F0 j t(φ0 j(t)), Ft(¯ φ(t)) for each t≥t0, since φ0 j(t+ 1) ∈F0 j t(φj(t)) for the trajectories φ0 jof F0 j. ¿From above φ0 j(t+ 1) →¯ φ(t+ 1), φ0 j(t)→¯ φ(t) as 0 j→0. Since the setvalued mappings F0 j tand Ftare upper semi continuous and the F0 j t converge upper semi continuously to Ft, it follows by Lemma 8.3 that H∗F0 j t(φ0 j(t)), Ft(¯ φ(t))−→ 0 as 0 j→0. 16 Thus dist ¯ φ(t+ 1), Ft(¯ φ(t))= 0 for all t∈Z, i.e., ¯ φ(t+ 1) ∈Ft(¯ φ(t)) for all t∈Z, which means that ¯ φis an entire trajectory of the unperturbed setvalued difference process Φ with ¯ φ(t)∈Btfor each t∈Z. By Lemma 8.1 it follows that ¯ φ(t)∈At for each t∈Z. However, this contradicts (17) and hence (16). This contradiction means that the A tconverge upper semi continuously to Atfor each t∈Z. References [1] J.-P. 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