Weak pullback attractors of setvalued processes

Weak pullback attractors of setvalued processes T. Caraballo a, P.E. Kloeden b,∗, P. Mar´ın–Rubio a aDpto. de Ecuaciones Diferenciales y An´alisis Num´erico, Universidad de Sevilla, Apdo. de Correos 1160, 41080 Sevilla, Spain bFB Mathematik, Johann Wolfgang Goethe Universit¨at, D-60054 Frankfurt am Main, Germany Abstract Weak pullback attractors are defined for nonautonomous setvalued processes 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 one trajectory rather than all trajectories at each starting point. The concept is useful in, for example, continuous time control systems and is related to that of viability. Key words: setvalued processes, pullback attraction, weak invariance 1991 MSC: 34D45, 37B25, 37B75, 58C06 1 Introduction Typical and important examples of setvalued processes are dynamical systems without uniqueness generated by ordinary differential equations without uniqueness and ordinary differential control systems (i.e., ˙x=f(t, x, u) where u∈U) or, more generally, inclusion equations (i.e., ˙x∈F(x)). Obviously, control systems have more significant applications and thus provide a powerful motivation for studying dynamical systems without uniqueness, although historically the original motivation came from ordinary differential equations without uniqueness. Many interesting systems are in fact nonautonomous, ∗Corresponding author Email addresses: [email protected] (T. Caraballo), [email protected] (P.E. Kloeden), [email protected] (P. Mar´ın–Rubio). Preprint submitted to Journal of Mathematical Analysis and Applications although most concepts have been developed only in the more convenient setting of autonomous systems. It is of practical importance as well as intellectual interest to see how such concepts generalize to nonautonomous systems. In many applications, physical, economical and industrial, such problems are typically stated on finite time intervals, e.g. in Viability Theory (see Aubin [3,4] and Aubin and Frankowska [7] and the references there), capture basins represent the points from which at least one trajectory reaches the target in finite time, thus in a weak sense. This arises in minimal time control problems (cf. Saint-Pierre [19]), or in tracking control, cf. Chen et al. [10], where the target is the graph of a single or multi-valued map (obtained by condition (2.8), see their proof of Theorem 2.2); and in controllability theory (e.g. Johnson and Nerurkar [11]), in which the dramatical influence of parameters on the controllability of the system is particularly worth noting. At the same time, the asymptotic behavior of such weakly invariant systems has also been intensively investigated with many meaningful interpretations in biology such as persistence and extinction, and applications in population genetics (cf. Vuillermot [22]), in minimization problems (cf. Attouch and Cominetti [1]) and stabilization in Mechanics (cf. Attouch and Czarnecki [2]). Attractors provide an important means of characterizing the long time behaviour of dynamical systems. They have been extensively investigated, in particular, global attractors in the autonomous case and its pullback version for general nonautonomous situations [9], which are known as strong attractors for setvalued systems. In the autonomous context, Szeg¨o and Treccani [21] introduced the concept of a weak attractor for the continuous time setvalued semigroup generated by differential inclusions. The key difference here is that only at least one trajectory for each starting point must be attracted to or remain in the weak attractor rather than all trajectories as in the case of the usual (strong) attractor. The concept of a weak attractor has been found to be very useful for autonomous control systems as well as for some optimization systems, so the corresponding concept should thus also be of practical usefulness in the nonautonomous case. In this paper we will introduce the concept of weak pullback attractor for the setvalued processes generated by different types of nonautonomous dynamical systems such as ordinary differential equations without uniqueness, nonautonomous contingent or inclusion equations, nonautonomous ordinary differential control systems, etc. Here the attractor consists of a family of sets invariant, i.e. carried into each other under the dynamics. Thus forward convergence is to a moving target, whereas pullback convergence is to a fixed target, a particular member of the family. Although similar concepts were introduced recently in [15] for nonautonomous difference inclusions, the techniques needed here to prove the existence of the weak pullback attractor in the continuous time case are somewhat different and more complicated, in particular, requiring Barbashin’s results on the compactness of set of trajectories and its generalization for a single setvalued process to a setvalued convergent 2 sequence of processes. Moreover, we establish our results for a more general Banach state space, thus removing a long standing restriction to locally compact state spaces in earlier publications. In sections 2 and 3 we introduce the usual notation for the setvalued framework, and the analogous tool of semiflows and semigroups through what are known as the setvalued process or general(ized) dynamical systems. In section 4 we establish our main results and highlight some of their features with several examples in Section 5. Proofs are given at the end of the paper. 2 Terminology Let be given a general Banach space (X, k·k). Recall that dist(x, A) = min a∈Akx−ak is the distance of a point x∈Xfrom a nonempty compact set Aand that the Hausdorff separation H∗(A, B) of nonempty compact subsets A,Bof X is defined as H∗(A, B) := max a∈Adist(a, B) = max a∈Amin b∈Bka−bk, while H(A, B) = max {H∗(A, B), H∗(B, A)}is the Hausdorff metric on the space (X) of nonempty compact subsets of X. Define an open –neighbourhood of A∈(X) by N(A) = {x∈X: dist(x, A)< }and closed –neighbourhood of Aby N[A] = {x∈X: dist(x, A)≤}. A mapping F:X7→ (X) 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→x0H∗(F(xn), F(x0)) = 0 for all sequences xn→x0. For any A∈(X) define F(A) := ∪a∈AF(a) and define the set composition of two mappings F,G:X7→ (X) as F◦G(x) := F(G(x)) for all x∈X. Note that F◦Gis upper semi continuous and compact valued if Fand Gare (see [6]). 3 3 Setvalued processes Barbashin [8] investigated setvalued generalized or general dynamical systems generated by ordinary differential equations without uniqueness. Roxin [18] showed that nonautonomous contingent or inclusion equations generated nonautonomous general dynamical systems, as did nonautonomous ordinary differential control systems in which case he called the generated system a general control system [17]. See also [12,13]. We will use the name setvalued process for all such nonautonomous setvalued systems without assumed backwards extendability in time. Definition 1 A setvalued process on a state space Xis defined in terms of an attainability set mapping (t, t0, x0)7→ Φ(t, t0, x0)for all t≥t0in Rand x0 ∈Xwhich satisfies the following properties: 1. Compactness Φ(t, t0, x0)is a nonempty compact subset of Xfor all t≥t0 in Rand all x0∈X; 2. Initial condition Φ(t0, t0, x0) = {x0} for all t0∈Rand x0∈X; 3. Time evolution Φ(t2, t0, x0) = Φ (t2, t1,Φ(t1, t0, x0)) for all t0≤t1≤t2in Rand all x0∈X; 4. Continuity in time lim s→tH(Φ(s, t0, x0),Φ(t, t0, x0)) = 0 for all s,t≥t0and all t0∈Rand x0∈X; 5. Upper semi continuity in initial conditions lim t(n) 0→t0,x(n) 0→x0 H∗Φ(t, t(n) 0, x(n) 0),Φ(t, t0, x0)= 0 uniformly in t∈[T0, T1]for any T0< T1<∞with T0≥t(n) 0,t0and for all t0∈Rand x0∈X. Simple examples of differential equations without uniqueness (e.g., Example 1 on page 122 of [13]) show that Condition 5 cannot in general be strengthened 4 to continuity in the initial variables, i.e., with the Hausdorff metric Hinstead of the semi–metric H∗. Definition 2 A trajectory of a setvalued process Φis a single valued mapping φ: [T0, T1]→Xwhich satisfies φ(t)∈Φ(t, s, φ(s)) for all T0≤s≤t≤T1 for some T0< T1in R. A trajectory φis called an entire trajectory if it is defined on all of R. Trajectories are in fact continuous functions. See Lemma 6.1 in [17], or Theorem 4.2 in [12] for the systems without assumed backwards extendability under consideration here. Barbashin [8] proved 1existence of at least one trajectory φ: [t0, t1]→Xwith φ(t0) = x0and φ(t1) = x1for any x0and x1with x1∈Φ(t1, t0, x0). Barbashin [8] also proved a result on the compactness of trajectories of a setvalued process Φ. The following generalization is due to Roxin [17]; see also [12]). Theorem 3 (Barbashin) Let Bbe a nonempty compact subset of Xand let φn: [t0, t1]→Xbe a sequence of trajectories of a setvalued process Φwith φn(t0)∈Bfor given t0< t1∈R. Then there exists a subsequence φnjand a trajectory ¯ φ: [t0, t1]→Xof Φwith ¯ φ(t0)∈Bsuch that φnj(t)→¯ φ(t)as nj → ∞ uniformly in t∈[t0, t1]. We will also state, prove and use a further generalization of this theorem for sequences of trajectories belonging to a sequence of upper semi continuously convergent setvalued processes; see Theorem 16 in Section 6. 4 Weak attractors of setvalued processes 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. Szeg¨o and Treccani [21] introduced concepts of weak invariance and weak attractors for such situations in the autonomous case. Our aim in this paper is to introduce and investigate pullback versions of these weak concepts for setvalued processes. As with the strong concepts of 1Although he worked in finite dimensional spaces, the extension to a general Banach space Xis straightforward due to the fact that the constructed objects are contained in the compact integral funnel. 5 invariance and attraction, it is also less restrictive here to consider families of sets rather than individual sets. Definition 4 A family A={At, t ∈R}of nonempty compact subsets of X is said to be weakly positively invariant for a setvalued process Φon Xif for every t0∈Rand every x0∈At0there exists a trajectory φ: [t0,∞)→Xof Φwith φ(t0) = x0such that φ(t)∈Atfor all t≥t0. It is called weakly invariant if, for every t0∈Rand every x0∈At0, there is an entire trajectory φwith φ(t0) = x0and φ(t)∈Atfor all t∈R. Definition 5 A weakly invariant family A={At, t ∈R}of nonempty compact subsets of Xis called a weak pullback attractor of a setvalued process Φ on Xif it is weakly pullback attracting, i.e., for any t0∈R, any nonempty bounded subset Dof Xand any sequence dn∈Dthere exist sequences of positive numbers τn→ ∞ as n→ ∞ and trajectories φn: [t0−τn, t0]→Xof Φ with φn(t0−τn) = dnsuch that lim n→∞ dist (φn(t0), At0)=0.(1) Note that a strong pullback attractor, when it exists, is also a weak pullback attractor. Now, one of our main results will be to show that the existence of a weak pullback attractor follows from that of a more easily determined weak pullback absorbing family of sets. Definition 6 A weakly positively invariant family B={Bt, t ∈R}of nonempty compact subsets of Xis called a weak pullback absorbing family of a setvalued process Φon Xif for t0∈Rand any bounded subset Dof Xthere exists a Tt0,D ∈R+such that for each τn≥Tt0,D and dn∈Dthere exists a trajectory φn: [t0−τn, t0]→Xof Φwith φn(t0−τn) = dnand φn(t0)∈Bt0. Note that by the weak positive invariance of Bthe trajectories φncan be extended, using the concatenation property given by the time evolution property 3, to remain in Bfor t≥t0, i.e. φn(t)∈Btfor each t≥t0. Theorem 7 Let Φbe a setvalued process with a weak pullback absorbing family B. Then Φhas a maximal weak pullback attractor A={At, t ∈R}relative to B, which is uniquely determined by At0={a0∈X;there exist τn→ ∞ as n→ ∞, and trajectories φn: [t0−τn, t0]→X(2) with φn(t)∈Btfor t∈[t0−τn, t0] 6 and lim n→∞ φn(t0) = a0 for each t0∈R. Remark 8 A weak pullback attractor consists of trajectories that exist and remain in Bfor the entire time set R, but it does not necessarily contain all such trajectories. See Lemma 13 in the Section 6. Remark 9 As well as being weakly invariant, a weak pullback attractor is also negatively strongly invariant, i.e., satisfies At⊂Φ (t, t0, At0)for all t≥ t0and t0∈R. Remark 10 The uniqueness and maximality of a weak attractor cannot be understood in the usual sense, but rather with respect to an absorbing family Bof sets under discussion. This is an intrinsic property of weak pullback attractors and is not contradicted by the existence of other weak pullback attractors, with or without intersecting component set, with respect to different families B. This is transparent in the examples of weak pullback attractors for nonautonomous difference equations in [15]. A similar example for setvalued processes will be given in Section 5. The proof of the following basic continuity property of a weak pullback attractor is not as immediate a consequence of definitions as in the strong case. It is given in Section 8. Proposition 11 Let A={At, t ∈R}be a weak pullback attractor. Then the setvalued mapping t7→ Atis continuous. Our second objective is to prove some results on the structure of weak pullback attractors for setvalued processes. In fact, we are interested in some kind of upper semi continuous behaviour produced by some perturbations appeared in the model. Theorem 12 Suppose that the setvalued process Φhas a weak pullback absorbing family B={Bt, t ∈R}and suppose that each perturbed setvalued process Φhas a weak pullback absorbing family B={B t, t ∈R}for  > 0 such that max 0≤δ≤1H∗(Φ(t+δ, t, x),Φ(t+δ, t, x)) ≤for all t∈R, x ∈X(3) and H∗B t0, Bt0≤for all t0∈R.(4) Then the maximal weak pullback attractors A={A t, t ∈R}w.r.t. Bof the 7 perturbed processes Φconverge upper semi continuously to the maximal weak pullback attractor A={At, t ∈R}w.r.t. Bof Φin the sense that lim →0H∗A t0, At0= 0.(5) for each t0∈R. The following structural condition on the unperturbed setvalued process Φ provides a simple (if rather strong) condition for X=Rdensuring the existence of a nearby uniform weak pullback absorbing family: assume that B= {Bt, t ∈R}is a family of nonempty compact sets of Rdand that there exists aγ:R+→[0,1] such that min y∈Φ(t,t0,x)dist(y, Bt)≤γ(t−t0) dist(x, Bt0) for all x∈Rdand t≥t0in Rand that for all bounded Dand all fixed time t: lim t0→−∞ γ(t−t0) sup x∈D dist(x, Bt0) = 0. We can take N[Bt] := {x∈Rd: dist(x, Bt)≤}for  > 0 small enough. Then the family N[Bt] is weakly positively invariant and weakly pullback absorbing. 5 Examples Our first example involves nonautonomous setvalued process generated by the nonautonomous differential inclusion x0∈F(t, x) :=          {−x}if t < 0 {−x, 0}if t≥0 , x ∈R. 8 The setvalued process here is given by Φ(t, t0, x0) :=                              nx0e−(t−t0)oif t0≤t≤0, x0∈R hx0e−(t−t0), x0iif 0 ≤t0≤t, x0≥0 hx0, x0e−(t−t0)iif 0 ≤t0≤t, x0≤0 and the composition of these cases. The family A={At, t ∈R}with At≡ {0}for all t∈Ris strongly invariant and hence weakly invariant. It is a weak pullback (and forward) attractor with respect to any absorbing family set B ={Bt, t ∈R}with component sets Bt≡[−R, R] for all t∈Rand any R≥ 0. As a second example we consider the nonautonomous setvalued process generated by the nonautonomous differential inclusion x0∈F(t, x) :=          {−x}if t < 0, {−x, 1}if t≥0. The setvalued process here is given by Φ(t, t0, x0) =                                nx0e−(t−t0)oif t0≤t≤0, {x0}if 0 ≤t0=t, hx0e−(t−t0), x0+t−t0iif 0 ≤t0< t, x0≥t−t0 et0−t−1, hx0+t−t0, x0e−(t−t0)iif 0 ≤t0< t, x0≤t−t0 et0−t−1. and the composition of these cases. The family A={At, t ∈R}with At≡ {0}for all t∈Ris weakly invariant, but not strongly invariant. It is a weak pullback (and forward) attractor with respect to any absorbing family set B ={Bt, t ∈R}with component sets Bt≡[−R, R] for all t∈Rand any 0 ≤ R≤1. Our third example illustrates the ambiguity concerning the existence and uniqueness of weak pullback attractors alluded to in Remark 10. It is based 9 0≤H∗At, Asnj ≤dist φnj(t), φnj(snj) ≤dist φnj(t),¯ φ(t)+ dist ¯ φ(t),¯ φ(snj)+ dist ¯ φ(snj), φnj(snj) ≤dist φnj(t),¯ φ(t)+ dist ¯ φ(t),¯ φ(snj)+ sup t−1≤s≤t+1 dist ¯ φ(s), φnj(s) →0 as j→ ∞ by the uniform convergence of the subsequence in the first and third terms and the continuity of the trajectory ¯ φin the second term. But this is a contradiction, so we must have H∗(At, As)→0 as s→t. Secondly, consider the limit H∗(As, At)→0 as s→t. If this does not hold there would exist an 0>0 and a sequence sn→tsuch that 0≤H∗(Asn, At), n ∈N. We will show that this leads to a contradiction. Since Asnis compact, there exists an an∈Asnsuch that H∗(Asn, At) = dist (an, At)≤dist (an, a) for all a∈At. By the weak invariance of Athere exists a trajectory φnwith φn(sn) = anand φn(s)∈Asfor all s∈Rand n∈N. Thus H∗(Asn, At)≤dist (φn(sn), φn(t)) , n ∈N. Now an∈Asn⊂Φ ([t−1, t + 1], t −1, At−1), which is compact. Here we have used the negative strong invariance of the weak pullback attractor, see Remark 9. Thus we can apply Barbashin’s Theorem to obtain the existence of a subsequence of trajectories φnjwhich converges to a trajectory ¯ φuniformly on the interval [t−1, t + 1]. Thus 0≤H∗Asnj, At ≤dist φnj(snj), φnj(t) ≤dist φnj(snj),¯ φ(snj)+ dist ¯ φ(snj),¯ φ(t) ≤sup t−1≤s≤t+1 dist φnj(s),¯ φ(s)+ dist ¯ φ(snj),¯ φ(t) →0 as j→ ∞ by the uniform convergence of the subsequence in the first term 16 and the continuity of the trajectory ¯ φin the second term. But this is a contradiction, so we must have H∗(As, At)→0 as s→t. Combining the two cases gives the desired result, i.e., H(As, At)→0 as s→ t. 9 Proof of Theorem 12 Let A={At, t ∈R}be the weak pullback attractor in Bgiven by (2) for the unperturbed setvalued process Φ and let A={A t, t ∈R}be the weak pullback attractor in Bfor the perturbed setvalued process Φ. Suppose for some t0∈Rthat 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(8) for all j∈Z+. We will show that this leads to a contradiction. Let aj∈Aj t0be such that dist (aj, At0) = H∗Aj t0, At0, so dist (aj, At0)≥ η0for j∈Z+. This is possible since Aj t0is compact. By Lemma 13 there is an entire trajectory φjof the perturbed setvalued process Φjsuch that φj(t)∈ Aj t⊂Bj tfor each t∈Rwith φj(t0) = aj. Since the Bj t0and Bt0are compact with H∗Bj t0, Bt0→0 as j→0, by Lemma 14 there exists a convergent subsequence a0 j=φ0 j(t0)→¯a0∈Bt0as 0 j→0. ¿From (8) we have dist (¯a0, At0)≥η0/2.(9) By Theorem 16 (Generalized Barbashin Theorem) applied to the interval [t0, t0+ 1], there exists a trajectory ¯ φof Φ on [t0, t0+ 1] with ¯ φ(t0) = ¯a0 and a subsubsequence φ00 jwith φ00 j(t)→¯ φ(t) as 00 j→0 uniformly in t∈ [t0, t0+ 1]. Moreover, by Lemma 14 we have ¯ φ(t)∈Btfor each t∈[t0, t0+ 1]. We repeat this construction on successive subintervals [t0+n, t0+n+ 1] for n = 1, 2, . . . to obtain a trajectory ¯ φof Φ on [t0,∞) and a diagonal subsequence 17 (denoted the same as before) φ00 jwith φ00 j(t)→¯ φ(t)∈Btas 00 j→0 for all t∈[t0,∞). We can also work backwards in time on successive subintervals [t0−n−1, t0−n] for n= 1, 2, . . . to obtain a trajectory ¯ φof Φ on (−∞, t0] with ¯ φ(t0) = a0and a further diagonal subsequence (denoted the same as before) φ00 jwith φ00 j(t)→¯ φ(t)∈Btas 00 j→0 for all t∈(−∞, t0]. Thus ¯ φis an entire trajectory of the unperturbed setvalued process Φ with ¯ φ(t)∈Btfor each t∈R. By Lemma 13 it follows that ¯ φ(t)∈Atfor each t ∈R. In particular, ¯ φ(t0)∈At0. However, this contradicts (9) and hence (8). This contradiction means that the A tconverge upper semi continuously to At for each t∈R. 10 Proof of Theorem 16 For convenience, we consider without loss of generality the interval [0,1] instead of [t0, t1]. By assumption, there is a sequence of trajectories φjof Φj on [t0, t1] with φj(0) = x0,j →x0as j→0. Write φ(0) = x0. By the upper semi continuous convergence (3) and the upper semi continuity of Φ(t, 0,·) uniformly in t∈[0,1], we have H∗(Φj(t, 0, x0,j),Φ(t, 0, x0)) ≤H∗(Φj(t, 0, x0,j),Φ(t, 0, x0,j)) +H∗(Φ(t, 0, x0,j),Φ(t, 0, x0)) ≤j+H∗(Φ(t, 0, x0,j),Φ(t, 0, x0)) −→ 0 as j→0 uniformly in t∈[0,1]. Hence for every  > 0 and taking jsufficiently small, Φj(t, 0, x0,j)⊂N[Φ([0,1],0, x0)] , for all t∈[0,1]. The set Φ([0,1],0, x0) is compact by the continuity of Φ(·,0, x) because of the Properties 1 and 4 of a setvalued process, so from φj(1) ∈B∗, there exists a convergent subsequence φ0 j(1) = x1,j →x1∈Φ([0,1],0, x0) as 0 j→0. Write φ(1) = x1. Moreover, φ(1) ∈Φ(1,0, x0). This follows from the fact that dist (φ(1),Φ(1,0, φ(0))) ≤  φ(1) −φ0 j(1)  + dist φ0 j(1),Φ0 j(1,0, φ0 j(0)) +H∗Φ0 j(1,0, φ0 j(0)),Φ(1,0, φ(0)) =  φ(1) −φ0 j(1)  +H∗Φ0 j(1,0, φ0 j(0)),Φ(1,0, φ(0)), 18 since φ0 j(1) ∈Φ0 j(1,0, φ0 j(0)) for the trajectories φ0 jof Φ0 j. Thus φ0 j(1) →φ(1), φ0 j(0) →φ(0) as 0 j→0. Since the setvalued mappings Φ0 j(1,0,·) and Φ(1,0,·) are upper semi continuous and the Φ0 j(1,0,·) converge upper semi continuously to Φ(1,0,·) due to (3), it follows by Lemma 15 that H∗Φ0 j(1,0, φ0 j(0)),Φ(1,0, φ(0))−→ 0 as 0 j→0. Thus dist (φ(1),Φ(1,0, φ(0))) = 0, i.e., φ(1) ∈Φ (1,0, φ(0)). Consider the time instant t=1 2. We repeat the above argument on the interval [0,1 2], to construct φ1 2∈Φ1 2,0, φ(0)using a subsequence of the above one that converges at t=1 2as well as at t= 0 and 1. Using this same sequence on the interval [1 2,1] we also obtain φ(1) ∈Φ1,1 2, φ 1 2. The construction for φ(t) for dyadic t∈Sq=0,1,2,... np 2q:p= 0,1,2, . . . , qo follows recursively, taking subsequences of the previous ones that also converge at the new points under consideration. Suppose that for a given qwe have constructed all of the φp 2qsuch that φp+ 1 2q∈Φp+ 1 2q,p 2q, φ p 2q, p = 0,1,2, . . . , q −1.(10) Consider the time instant 2p+1 2q+1 , which is the midpoint of the interval [ p 2q,p+1 2q]. The construction of φ2p+1 2q+1 with φ2p+ 1 2q+1 ∈Φ2p+ 1 2q+1 ,p 2q, φ p 2q and φp+ 1 2q∈Φp+ 1 2q,2p+ 1 2q+1 , φ 2p+ 1 2q+1  follows exactly the same as in the case of p= 0 and q= 1, i.e., for φ1 2from φ(0) and φ(1). It follows from the 2–parameter semigroup property of Φ, i.e., the time evolution property 3, and the inclusions (10), we have φ(t)∈Φ (t, s, φ(s)) for all 19 dyadic s,t∈[0,1] with s≤t. 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