Extremal Selections of Multifunctions
이러한 다항 함수 집합 F(t,x)에서 extremal 점들을 찾는 문제를 다루고 있다. extremal 점은 다항 함수 집합의 극단점으로, 이 극단점들은 다른 모든 점보다 더 가까운 두 점 사이에 위치한다.
F(t,x)는 compact set이므로, extremal 점을 찾기 위해서는 다항 함수 집합 F(t,x)의 compact subset S에서 extremal 점들을 찾으면 된다.
Lemma 3은 이 문제를 푸는데 도움을 줄 수 있는 lemma로, S가 compact subset 인 경우, F의 selection인 φ가 주어졌을 때, 근사 extremal selection g와 compact subset S의 covering {Γi}이 존재한다고 가정한다.
g는 piecewise Lipschitz selection이고, 이 선택함수 g는 다음과 같은 성질을 만족한다.
(i) g는 piecewise Lipschitz selection이다.
(ii) g는 다음 조건을 만족한다.
a) g는 compact subset S의 covering {Γi}에 대한 pairwise disjoint sets ∆i 에서 piecewise linear로 정의된다.
b) g는 다음 조건을 만족한다.
∀(t,x) ∈ ∆i, ϕij(yj(t,x)) ≤ ε,
∀z ∈ F(t,x), ϕij(z) ≥ h(z,F(t,x)).
(iii) g는 다음 조건을 만족한다.
∀u ∈ Y, ∀τ ≤ s < τ',
Z τ' τ φ(s,u(s)) - g(s,u(s)) ds ≤ ε.
이러한 성질은 다항 함수 집합 F(t,x)의 extremal 점을 근사하는 선택함수 g를 찾는 데 도움을 줄 수 있다.
다만, 이 선택함수의 extremal 점에 대한 보장은 존재하지 않으며, 이러한 문제점을 해결하기 위해 추가적인 작업이 필요하다.
한글 요약 끝
Extremal Selections of Multifunctions
arXiv:funct-an/9209001v1 9 Sep 1992Extremal Selections of MultifunctionsGenerating a Continuous FlowAlberto Bressan and Graziano CrastaS.I.S.S.A. – Via Beirut 4, Trieste 34014, ITALY1 - IntroductionLet F : [0, T] × IRn 7→2IRn be a continuous multifunction with compact, not neces-sarily convex values.
If F is Lipschitz continuous, it was shown in [4] that there exists ameasurable selection f of F such that, for every x0, the Cauchy problem˙x(t) = f(t, x(t)),x(0) = x0has a unique Caratheodory solution, depending continuously on x0.In this paper, we prove that the above selection f can be chosen so that f(t, x) ∈extF(t, x) for all t, x. More generally, the result remains valid if F satisfies the followingLipschitz Selection Property:(LSP) For every t, x, every y ∈coF(t, x) and ε > 0, there exists a Lipschitz selection φ ofcoF, defined on a neighborhood of (t, x), with |φ(t, x) −y| < ε.We remark that, by [7,9], every Lipschitz multifunction with compact values satisfies(LSP).
Another interesting class, for which (LSP) holds, consists of those continuous mul-tifunctions F whose values are compact and have convex closure with nonempty interior.1
Indeed, for any given t, x, y, ε, choosing y′ ∈int coF(t, x) with |y′ −y| < ε, the constantfunction φ ≡y′ is a local selection from coF satisfying the requirements.In the following, Ω⊆IRn is an open set, B(0, M) is the closed ball centered at theorigin with radius M, B(D; MT) is the closed neighborhood of radius MT around theset D, while AC the Sobolev space of all absolutely continuous functions u : [0, T] 7→IRn,with norm ∥u∥AC =R T0|u(t)| + | ˙u(t)|dt.Theorem 1. Let F : [0, T]×Ω7→2IRn be a bounded continuous multifunction with compactvalues, satisfying (LSP).
Assume that F(t, x) ⊆B(0, M) for all t, x and let D be a compactset such that B(D; MT) ⊂Ω. Then there exists a measurable function f, withf(t, x) ∈extF(t, x)∀t, x,(1.1)such that, for every (t0, x0) ∈[0, T] × D, the Cauchy problem˙x(t) = f(t, x(t)),x(t0) = x0(1.2)has a unique Caratheodory solution x(·) = x(·, t0, x0) on [0, T], depending continuously ont0, x0 in the norm of AC.Moreover, if ε0 > 0 and a Lipschitz continuous selection f0 of coF are given, then onecan construct f with the following additional property.
Denoting by y(·, t0, x0) the uniquesolution of˙y(t) = f0(t, y(t)),y(t0) = x0,(1.3)for every (t0, x0) ∈[0, T] × D one hasy(t, t0, x0) −x(t, t0, x0) ≤ε0∀t ∈[0, T]. (1.4)The proof of the above theorem, given in section 3, starts with the construction of asequence fn of selections from coF, which are piecewise Lipschitz continuous in the (t, x)-space.
For every u : [0, T] 7→IRn in a class of Lipschitz continuous functions, we thenshow that the composed maps t 7→fn(t, u(t)) form a Cauchy sequence in L1[0, T]; IRn,converging pointwise almost everywhere to a map of the form f(·, u(·)), taking valueswithin the extreme points of F. This convergence is obtained through an argument whichis considerably different from previous works.Indeed, it relies on a careful use of the2
likelihood functional introduced in [3], interpreted here as a measure of “oscillatory non-convergence” of a set of derivatives.Among various corollaries, Theorem 1 yields an extension, valid for the wider class ofmultifunctions with the property (LSP), of the following results, proved in [5], [4] and [6],respectively. (i) Existence of selections from the solution set of a differential inclusion, dependingcontinuously on the initial data.
(ii) Existence of selections from a multifunction, which generate a continuous flow. (iii) Contractibility of the solution sets of ˙x ∈F(t, x) and ˙x ∈extF(t, x).These consequences, together with an application to bang-bang feedback controls, aredescribed in section 4.2 - PreliminariesAs customary, ¯A and co A denote here the closure and the closed convex hull of Arespectively, while A\B indicates a set–theoretic difference.
The Lebesgue measure of aset J ⊂IR is m(J). The characteristic function of a set A is written as χA.In the following, Kn denotes the family of all nonempty compact convex subsets ofIRn, endowed with Hausdorffmetric.
A key technical tool used in our proofs will be thefunction h : IRn × Kn 7→IR ∪{−∞}, defined byh(y, K) .= supZ 10|w(ξ) −y|2 dξ12 ;w : [0, 1] 7→K,Z 10w(ξ) dξ = y(2.1)with the understanding that h(y, K) = −∞if y ̸∈K.Observe that h2(y, K) can beinterpreted as the maximum variance among all random variables supported inside K,whose mean value is y. The following results were proved in [3]:Lemma 1.
The map (y, K) 7→h(y, K) is upper semicontinuous in both variables; for eachfixed K ∈Kn the function y 7→h(y, K) is strictly concave down on K. Moreover, one hash(y, K) = 0if and only ify ∈extK,(2.2)3
h2(y, K) ≤r2(K) −y −c(K)2,(2.3)where c(K) and r(K) denote the Chebyschev center and the Chebyschev radius of K,respectively.For the basic theory of multifunctions and differential inclusions we refer to [1]. As in[2], given a map g : [0, T] × Ω7→IRn, we say that g is directionally continuous along thedirections of the cone ΓN =(s, y); |y| ≤Nsifg(t, x) = limk→∞g(tk, xk)for every (t, x) and every sequence (tk, xk) in the domain of g such that tk →t and|xk −x| ≤N(tk −t) for every k. Equivalently, g is ΓN-continuous iffit is continuous w.r.t.the topology generated by the family of all conical neighborhoodsΓN(ˆt,ˆx,ε).=(s, y) ;ˆt ≤s ≤ˆt + ε, |y −ˆx| ≤N(s −t).
(2.4)A set of the form (2.4) will be called an N–cone.Under the assumptions on Ω, D made in Theorem 1, consider the set of LipschitzeanfunctionsY .=u : [0, T] 7→B(D, MT);|u(t) −u(s)| ≤M|t −s|∀t, s.The Picard operator of a map g : [0, T] × Ω7→IRn is defined asPg(u)(t) .=Z t0g(s, u(s)) dsu ∈Y.The distance between two Picard operators will be measured byPf −Pg = supZ t0[f(s, u(s)) −g(s, u(s))] ds ;t ∈[0, T],u ∈Y. (2.5)The next Lemma will be useful in order to prove the uniqueness of solutions of the Cauchyproblems (1.2).Lemma 2.
Let f be a measurable map from [0, T] × Ωinto B(0, M), with Pf continuouson Y . Let D be compact, with B(D, MT) ⊂Ω, and assume that the Cauchy problem˙x(t) = f(t, x(t)),x(t0) = x0,t ∈[0, T](2.6)4
has a unique solution, for each (t0, x0) ∈[0, T] × D.Then, for every ǫ > 0, there exists δ > 0 with the following property. If g : [0, T]×Ω→B(0, M) satisfiesPg −Pf ≤δ, then for every (t0, x0) ∈[0, T] × D, any solution of theCauchy problem˙y(t) = g(t, y(t))y(t0) = x0t ∈[0, T](2.7)has distance < ε from the corresponding solution of (2.6).
In particular, the solution setof (2.7) has diameter ≤2ε in C0[0, T]; IRn.Proof. If the conclusion fails, then there exist sequences of times tν, t′ν, maps gνwithPgν −Pf →0, and couples of solutions xν, yν : [0, T] 7→B(D; MT) of˙xν(t) = f(t, xν(t)),˙yν(t) = gν(t, yν(t))t ∈[0, T],(2.8)withxν(tν) = yν(tν) ∈D,xν(t′ν) −yν(t′ν) ≥ε∀ν.
(2.9)By taking subsequences, we can assume that tν →t0, t′ν →τ, xν(t0) →x0, while xν →xand yν →y uniformly on [0, T]. From (2.8) it followsy(t) −x0 −Z tt0f(s, y(s)) ds ≤y(t) −yν(t) +x0 −yν(t0)+Z tt0f(s, y(s)) −f(s, yν(s))ds +Z tt0f(s, yν(s)) −gν(s, yν(s))ds .
(2.10)As ν →∞, the right hand side of (2.10) tends to zero, showing that y(·) is a solution of(2.6). By the continuity of Pf, x(·) is also a solution of (2.6), distinct from y(·) because|x(τ) −y(τ)| = limν→∞|xν(τ) −yν(τ)| = limν→∞xν(t′ν) −yν(t′ν) ≥ε.This contradicts the uniqueness assumption, proving the lemma.3 - Proof of the main theoremObserving that extF(t, x) = extcoF(t, x) for every compact set F(t, x), it is clearlynot restrictive to prove Theorem 1 under the additional assumption that all values of F areconvex.
Moreover, the bounds on F and D imply that no solution of the Cauchy problem˙x(t) ∈F(t, x(t)),x(t0) = x0,t ∈[0, T],5
with x0 ∈D, can escape from the set B(D, MT). Therefore, it suffices to construct theselection f on the compact set Ω† .= [0, T]×B(D, MT).
Finally, since every convex valuedmultifunction satisfying (LSP) admits a globally defined Lipschitz selection, it suffices toprove the second part of the theorem, with f0 and ε0 > 0 assigned.We shall define a sequence of directionally continuous selections of F, converging a.e.to a selection from extF. The basic step of our constructive procedure will be provided bythe next lemma.Lemma 3.
Fix any ε > 0. Let S be a compact subset of [0, T] × Ωand let φ : S →IRn bea continuous selection of F such thathφ(t, x), F(t, x)< η∀(t, x) ∈S,(3.1)with h as in (2.1).
Then there exists a piecewise Lipschitz selection g : S →IRn of F withthe following properties:(i) There exists a finite covering {Γi}i=1...,ν, consisting of ΓM+1–cones, such that, if wedefine the pairwise disjoint sets ∆i .= Γi\Sℓ
., n, such thatg∆i =nXj=0ψij χAij. (3.2)where each Aij is a finite union of strips of the form[t′, t′′) × IRn∩∆i.
(b) For every j = 0, . .
., n there exists an affine map ϕij(·) = ⟨aij, ·⟩+ bij such thatϕijψij(t, x)≤ε,ϕij(z) ≥h(z, F(t, x)),∀(t, x) ∈∆i,z ∈F(t, x). (3.3)(ii) For every u ∈Y and every interval [τ, τ ′] such that (s, u(s)) ∈S for τ ≤s < τ ′, thefollowing estimates hold:Z τ ′τφ(s, u(s)) −g(s, u(s))ds ≤ε,(3.4)Z τ ′τφ(s, u(s)) −g(s, u(s)) ds ≤ε + η(τ ′ −τ).
(3.5)6
Remark 1. Thinking of h(y, K) as a measure for the distance of y from the extremepoints of K, the above lemma can be interpreted as follows.
Given any selection φ of F,one can find a ΓM+1-continuous selection g whose values lie close to the extreme points ofF and whose Picard operator Pg, by (3.4), is close to Pφ. Moreover, if the values of φ arenear the extreme points of F, i.e.
if η in (3.1) is small, then g can be chosen close to φ.The estimate (3.5) will be a direct consequence of the definition (2.1) of h and of H¨older’sinequality.Remark 2.Since h is only upper semicontinuous, the two assumptions yν →y andh(yν, K) →0 do not necessarily imply h(y, K) = 0. As a consequence, the a.e.
limit ofa convergent sequence of approximately extremal selections fν of F need not take valuesinside extF.To overcome this difficulty, the estimates in (3.3) provide upper boundsfor h in terms of the affine maps ϕij.Since each ϕij is continuous, limits of the formϕij(yν) →ϕij(y) will be straightforward.Proof of Lemma 3.For every (t, x) ∈S there exist values yj(t, x) ∈F(t, x) andcoefficients θj(t, x) ≥0, withφ(t, x) =nXj=0θj(t, x)yj(t, x),nXj=0θj(t, x) = 1,hyj(t, x), F(t, x)< ε/2.By the concavity and the upper semicontinuity of h, for every j = 0, . .
., n there exists anaffine function ϕ(t,x)j(·) = ⟨a(t,x)j, ·⟩+ b(t,x)jsuch thatϕ(t,x)j(yj(t, x)) < hyj(t, x), F(t, x)+ ε2 < ε,ϕ(t,x)j(z) > hz, F(t, x)∀z ∈F(t, x).By (LSP) and the continuity of each ϕ(t,x)j, there exists a neighborhood U of (t, x) togetherwith Lipschitzean selections ψ(t,x)j: U 7→IRn, such that, for every j and every (s, y) ∈U,ψ(t,x)j(s, y) −yj(t, x) < ε4T ,(3.6)ϕ(t,x)jψ(t,x)j(s, y)< ε. (3.7)7
Using again the upper semicontinuity of h, we can find a neighborhood U′ of (t, x) suchthatϕ(t,x)j(z) ≥hz, F(s, y)∀z ∈F(s, y),(s, y) ∈U′,j = 0, . .
., n.(3.8)Choose a neighborhood Γt,x of (t, x), contained in U ∩U′, such that, for every point (s, y)in the closure Γt,x, one has|φ(s, y) −φ(t, x)| < ε4T ,(3.9)It is not restrictive to assume that Γt,x is a (M + 1)-cone, i.e. it has the form (2.4) withN = M +1.
By the compactness of S we can extract a finite subcoveringΓi; 1 ≤i ≤ν,with Γi.= Γti,xi.Define ∆i.= Γi \ Sj 8Mν2Tε(3.10)and divide [0, T] into N equal subintervals J1, .
. ., JN, withJk =tk−1, tk,tk = kTN .
(3.11)For each i, k such thatJk × IRn∩∆i ̸= ∅, we then split Jk into n + 1 subintervalsJik,0, . .
., Jik,n with lengths proportional to θi0, . .
., θin, by settingJik,j =tk,j−1, tk,j,tk,j = TN ·k +jXℓ=0θiℓ,tk,−1 = TkN .For any point (t, x) ∈∆i we now set( gi(t, x) .= ψij(t, x)¯gi(t, x) = yijif t ∈N[k=1Jik,j. (3.12)The piecewise Lipschitz selection g and a piecewise constant approximation ¯g of g can nowbe defined asg =νXi=1giχ∆i,¯g =νXi=1¯giχ∆i.
(3.13)By construction, recalling (3.7) and (3.8), the conditions (a), (b) in (i) clearly hold.8
It remains to show that the estimates in (ii) hold as well. Let τ, τ ′ ∈[0, T] and u ∈Ybe such thatt, u(t)∈S for every t ∈[τ, τ ′], and defineEi =t ∈I ;(t, u(t)) ∈∆i,i = 1, .
. ., ν.From our previous definition ∆i .= Γi \Sj
We can thus writeEi =[Jk⊂EiJk∪ˆEi,with Jk given by (3.11) andm( ˆEi) ≤2iTN≤2νTN . (3.14)Sinceφ(ti, xi) =nXj=0θijyij,(3.15)the definition of ¯g at (3.12), (3.13) impliesZJkφ(ti, xi) −¯g(s, u(s))ds = m(Jk) ·φ(ti, xi) −nXj=0θijyij= 0.Therefore, from (3.9) and (3.6) it followsZJkφ(s, u(s)) −g(s, u(s))ds ≤ZJkφ(s, u(s)) −φ(ti, xi)ds+ZJkφ(ti, xi) −¯g(s, u(s))ds +ZJk¯g(s, u(s)) −g(s, u(s))ds≤m(Jk) ·h ε4T + 0 + ε4Ti= m(Jk) · ε2T .The choice of N at (3.10) and the bound (3.14) thus implyZ τ ′τφ(s, u(s)) −g(s, u(s))ds ≤2M · mν[i=1ˆEi+ (τ ′ −τ) ε2T ≤2Mν · 2νTN+ ε2 ≤ε,proving (3.4).We next consider (3.5).
For a fixed i ∈{1, . .
., ν}, let Ei be as before and defineξ−1 = 0,ξj =jXℓ=0θiℓ,wi(ξ) =nXj=0yijχ[ξj−1, ξj].9
Recalling (3.15), the definition of h at (2.1) and H¨older’s inequality together implyhφ(ti, xi), F(ti, xi)≥Z 10φ(ti, xi) −wi(ξ)2 dξ12≥Z 10φ(ti, xi) −wi(ξ) dξ =nXj=0θijφ(ti, xi) −yij.Using this inequality we obtainZJkφ(ti, xi)−¯g(s, u(s)) ds = m(Jk) ·nXj=0θijφ(ti, xi) −yij≤m(Jk) · hφ(ti, xi), F(ti, xi)≤η · m(Jk),and therefore, by (3.9) and (3.6),ZJkφ(s, u(s)) −g(s, u(s)) ds≤ZJkφ(s, u(s)) −φ(ti, xi) ds +ZJk¯g(s, u(s)) −g(s, u(s)) ds+ZJkφ(ti, xi) −¯g(s, u(s))≤m(Jk) ·h ε4T + ε4T + ηi= m(Jk) · ε2T + η.Using again (3.14) and (3.10), we concludeZ τ ′τφ(s, u(s)) −g(s, u(s)) ds ≤(τ ′ −τ) ε2T + η+ 2Mν · 2νTN≤ε + (τ ′ −τ)η.Q.E.D.Using Lemma 3, given any continuous selection ˜f of F on Ω†, and any sequence(εk)k≥1 of strictly positive numbers, we can generate a sequence (fk)k≥1 of selections fromF as follows.To construct f1, we apply the lemma with S = Ω†, φ = f0, ε = ε1. This yields apartitionAi1; i = 1, .
. ., ν1of Ω† and a piecewise Lipschitz selection f1 of F of the formf1 =ν1Xi=1f i1χAi1.10
In general, at the beginning of the k-th step we are given a partition of Ω†, sayAik;i = 1, . .
., νk, and a selectionfk =νkXi=1f ikχAik,where each f ik is Lipschitz continuous and satisfieshfk(t, x), F(t, x)≤εk∀(t, x) ∈Aik.We then apply Lemma 3 separately to each Aik, choosing S = Aik, ε = εk, φ = f ik. Thisyields a partitionAik+1;i = 1, .
. ., νk+1of Ω† and functions of the formfk+1 =νk+1Xi=1f ik+1χAik+1,ϕik+1(·) = ⟨aik+1, ·⟩+ bik+1,where each f ik+1 : Aik+1 7→IRn is a Lipschitz continuous selection from F, satisfying thefollowing estimates:ϕik+1(z) > hz, F(t, x)∀(t, x) ∈Aik+1,(3.16)ϕik+1f ik+1(t, x)≤εk+1∀(t, x) ∈Aik+1,(3.17)Z τ ′τfk+1(s, u(s)) −fk(s, u(s))ds ≤εk+1,(3.18)Z τ ′τfk+1(s, u(s)) −fk(s, u(s)) ds ≤εk+1 + εk(τ ′ −τ),(3.19)for every u ∈Y and every τ, τ ′, as long as the values (s, u(s)) remain inside a single setAik, for s ∈[τ, τ ′).Observe that, according to Lemma 3, each Aik is closed-open in the finer topologygenerated by all (M +1)–cones.
Therefore, each fk is ΓM+1–continuous. By Theorem 2 in[2], the substitution operator Sfk : u(·) 7→fk(·, u(·)) is continuous from the set Y definedat (2.5) into L1[0, T]; IRn.
The Picard map Pfk is thus continuous as well.Furthermore, there exists an integer Nk with the following property. Given any u ∈Y ,there exists a finite partition of [0, T] with nodes 0 = τ0 < τ1 < · · · < τn(u) = T, withn(u) ≤Nk, such that, as t ranges in any [τℓ−1, τℓ), the point (t, u(t)) remains inside onesingle set Aik.
Otherwise stated, the number of times in which the curve t 7→(t, u(t))11
crosses a boundary between two distinct sets Aik, Ajk is smaller that Nk, for every u ∈Y .The construction of the Aik in terms of (M + 1)–cones implies that all these crossings aretransversal. Since the restriction of fk to each Aik is Lipschitz continuous, it is clear thatevery Cauchy problem˙x(t) = fk(t, x(t)),x(t0) = x0has a unique solution, depending continuously on the initial data (t0, x0) ∈[0, T] × D.From (3.18), (3.19) and the property of Nk it followsZ t0fk+1(s, u(s)) −fk(s, u(s))ds ≤ˆℓXℓ=1Z τℓτℓ−1fk+1(s, u(s)) −fk(s, u(s))ds≤Nkεk+1,(3.20)where 0 = τ0 < τ1 < · · · < τˆℓ= t are the times at which the map s →(s, u(s)) crossesa boundary between two distinct sets Aik, Ajk.
Since (3.20) holds for every t ∈[0, T], weconcludePfk+1 −Pfk ≤Nkεk+1. (3.21)Similarly, for every u ∈Y one hasfk+1(·, u(·)) −fk(·, u(·))L1([0,T ]; IRn) ≤n(u)Xℓ=1Z τℓτℓ−1fk+1(s, u(s)) −fk(s, u(s)) ds≤n(u)Xℓ=1εk+1 + εk(τℓ−τℓ−1)≤Nkεk+1 + εkT.
(3.22)Now consider the functions ϕk : IRn × Ω† →IR, withϕk(y, t, x) .= ⟨aik, y⟩+ bikif(t, x) ∈Aik. (3.23)From (3.16), (3.17) it followsϕk(y, t, x) ≥h(y, F(t, x))∀(t, x) ∈Ω†,y ∈F(t, x),(3.24)ϕkfk(t, x), t, x≤εk∀(t, x) ∈Ω†.
(3.25)12
For every u ∈Y , (3.18) and the linearity of ϕk w.r.t. y implyZ T0ϕkfk+1(s, u(s)), s, u(s)−ϕkfk(s, u(s)), s, u(s)ds≤n(u)Xℓ=1max|a1k|, .
. ., |aνkk |·Z τℓτℓ−1fk+1(s, u(s)) −fk(s, u(s))ds≤Nk · max|a1k|, .
. ., |aνkk |· εk+1.
(3.26)Moreover, for every ℓ≥k, from (3.19) it followsZ T0ϕkfℓ+1(s, u(s)), s, u(s)−ϕkfℓ(s, u(s)), s, u(s) ds≤max|a1k|, . .
., |aνkk |·Z T0fℓ+1(s, u(s)) −fℓ(s, u(s)) ds≤max|a1k|, . .
., |aνkk |·Nℓεℓ+1 + εℓT. (3.27)Observe that all of the above estimates hold regardless of the choice of the εk.
We nowintroduce an inductive procedure for choosing the constants εk, which will yield the con-vergence of the sequence fk to a function f with the desired properties.Given f0 and ε0, by Lemma 2 there exists δ0 > 0 such that, if g : Ω† 7→B(0, M)andPg −Pf0 ≤δ0, then, for each (t0, x0) ∈[0, T] × D, every solution of (2.7) remainsε0-close to the unique solution of (1.3). We then choose ε1 = δ0/2.By induction on k, assume that the functions f1, .
. ., fk have been constructed, to-gether with the linear functions ϕiℓ(·) = ⟨aiℓ, ·⟩+ biℓand the integers Nℓ, ℓ= 1, .
. ., k. Letthe values δ0, δ1, .
. .
, δk > 0 be inductively chosen, satisfyingδℓ≤δℓ−12ℓ= 1, . .
., k,(3.28)and such thatPg −Pfℓ ≤δℓimplies that for every (t0, x0) ∈[0, T] × D the solution setof (2.7) has diameter ≤2−ℓ, for ℓ= 1, . .
., k. This is possible again because of Lemma 2.For k ≥1 we then chooseεk+1 .= min(δk2Nk, 2−kNk,2−kNk · max|aiℓ|;1 ≤ℓ≤k, 1 ≤i ≤νℓ). (3.29)Using (3.28), (3.29) in (3.21), with N0 .= 1, we now obtain∞Xk=pPfk+1 −Pfk ≤∞Xk=pNk · δk2Nk≤∞Xk=p2p−kδp2≤δp(3.30)13
for every p ≥0. From (3.22) and (3.29) we further obtain∞Xk=1fk+1(·, u(·))−fk(·, u(·))L1 ≤∞Xk=1Nk · 2−kNk+ 21−kTNk≤∞Xk=12−k+21−kT≤1+2T.
(3.31)Definef(t, x) .= limk→∞fk(t, x)(3.32)for all (t, x) ∈Ω† at which the sequence fk converges. By (3.31), for every u ∈Y thesequence fk(·, u(·)) converges in L1[0, T]; IRnand a.e.
on [0, T]. In particular, consider-ing the constant functions u ≡x ∈B(D, MT), by Fubini’s theorem we conclude that f isdefined a.e.
on Ω†. Moreover, the substitution operators Sfk : u(·) 7→fk(·, u(·)) convergeto the operator Sf : u(·) 7→f(·, u(·)) uniformly on Y .
Since each Sfk is continuous, Sf isalso continuous. Clearly, the Picard map Pf is continuous as well.
By (3.30) we havePf −Pfk ≤∞Xk=pPfk+1 −Pfk ≤δp∀p ≥1.Recalling the property of δp, this implies that, for every p, the solution set of (2.7) hasdiameter ≤2−p. Since p is arbitrary, for every (t0, x0) ∈[0, T] × D the Cauchy problemcan have at most one solution.
On the other hand, the existence of such a solution isguaranteed by Schauder’s theorem. The continuous dependence of this solution on theinitial data t0, x0, in the norm of AC, is now an immediate consequence of uniquenessand of the continuity of the operators Sf, Pf.Furthermore, for p = 0, (3.30) yieldsPf −Pf0 ≤δ0.
The choice of δ0 thus implies (1.4).It now remains to prove (1.1). Since every set F(t, x) is closed, it is clear that f(t, x) ∈F(t, x).
For every u ∈Y and k ≥1, by (3.24)–(3.27) the choices of εk at (3.29) yieldZ T0hf(s, u(s)), F(s, u(s))ds ≤Z T0ϕkf(s, u(s)), s, u(s)ds≤Z T0ϕkfk(s, u(s)), s, u(s)ds+Z T0ϕkfk+1(s, u(s)), s, u(s)−ϕkfk(s, u(s)), s, u(s)ds+∞Xℓ=k+1Z T0ϕkfℓ+1((s, u(s)), s, u(s)−ϕkfℓ(s, u(s)), s, u(s) ds≤21−kT + 2−k +∞Xℓ=k+12−ℓ+ 21−ℓT. (3.33)14
Observing that the right hand side of (3.33) approaches zero as k →∞, we conclude thatZ T0hf(t, u(t)), F(t, u(t))dt = 0.By (2.2), given any u ∈Y , this implies f(t, u(t)) ∈extF(t, u(t)) for almost every t ∈[0, T].By possibly redefining f on a set of measure zero, this yields (1.1).4 - ApplicationsThroughout this section we make the following assumptions. (H) F : [0, T] × Ω7→B(0, M) is a bounded continuous multifunction with compact valuessatisfying (LSP), while D is a compact set such that B(D, MT) ⊂Ω.An immediate consequence of Theorem 1 isCorollary 1.
Let the hypotheses (H) hold. Then there exists a continuous map (t0, x0) 7→x(·, t0, x0) from [0, T] × D into AC, such that( ˙x(t, t0, x0) ∈extFt, x(t, t0, x0)∀t ∈[0, T],x(t0, t0, x0) = x0∀t0, x0.Another consequence of Theorem 1 is the contractibility of the sets of solutions ofcertain differential inclusions.
We recall here that a metric space X is contractible if thereexist a point ˜u ∈X and a continuous mapping Φ : X × [0, 1] →X such that:Φ(v, 0) = ˜u,Φ(v, 1) = v,∀v ∈X.The map Φ is then called a null homotopy of X.Corollary 2. Let the assumptions (H) hold.
Then, for any ¯x ∈D, the sets M, Mext ofsolutions ofx(0) = ¯x,˙x(t) ∈F(t, x(t))t ∈[0, T],x(0) = ¯x,˙x ∈extF(t, x(t))t ∈[0, T],are both contractible in AC.15
Proof.Let f be a selection from extFwith the properties stated in Theorem 1. Asusual, we denote by x(·, t0, x0) the unique solution of the Cauchy problem (1.2).
Definethe null homotopy Φ : M × [0, 1] →M by settingΦ(v, λ)(t) .=v(t)ift ∈[0, λT],x(t, λT, v(λT))ift ∈[λT, T].By Theorem 1, Φ is continuous. Moreover, setting ˜u(·) .= u(·, 0, ¯x), we obtainΦ(v, 0) = ˜u,Φ(v, 1) = v,Φ(v, λ) ∈M∀v ∈M,proving that M is contractible.
We now observe that, if v ∈Mext, then Φ(v, λ) ∈Mextfor every λ. Therefore, Mext is contractible as well.Our last application is concerned with feedback controls.Let Ω⊆IRn be open,U ⊂IRm compact, and let g : [0, T] × Ω× U →IRn be a continuous function.
By a wellknown theorem of Filippov [8], the solutions of the control system˙x = g(t, x, u),u ∈U,(4.1)correspond to the trajectories of the differential inclusion˙x ∈F(t, x) .=g(t, x, ω)ω ∈U. (4.2)In connection with (4.1), one can consider the “relaxed” system˙x = g#(t, x, u#),u# ∈U #,(4.3)whose trajectories are precisely those of the differential inclusion˙x ∈F #(t, x) .= coF(t, x).The control system (4.3) is obtained defining the compact setU # .= U × · · · × U × ∆n = U n+1 × ∆n,where∆n .=(θ = (θ0, .
. ., θn) ;nXi=0θi = 1,θi ≥0∀i)16
is the standard simplex in IRn+1, and settingg#(t, x, u#) = g#t, x, (u0, . .
., un, (θ0, . .
., θn)) .=nXi=0θif(t, x, ui).Generalized controls of the form u# = (u0, . .
., un, θ) taking values in the set U n+1 × ∆nare called chattering controls.Corollary 3.Consider the control system (4.1), with g : [0, T] × Ω× U 7→B(0, M)Lipschitz continuous. Let D be a compact set with B(D; MT) ⊂Ω.
Let u#(t, x) ∈U # bea chattering feedback control such that the mapping(t, x) 7→g#(t, x, u#(t, x)) .= f0(t, x)is Lipschitz continuous.Then, for every ε0 > 0 there exists a measurable feedback control ¯u = ¯u(t, x) with thefollowing properties:(a) For every (t, x), one has g(t, x, ¯u(t, x)) ∈extF(t, x), with F as in (4.2). (b) for every (t0, x0) ∈[0, T] × D, the Cauchy problem˙x(t) = gt, x(t), ¯u(t, x(t)),x(t0) = x0has a unique solution x(·, t0, x0),(c) if y(·, t0, x0) denotes the (unique) solution of the Cauchy problem˙y = f0(t, y(t)),y(t0) = x0,then for every (t0, x0) one hasx(t, t0, x0) −y(t, t0, x0) < ε0,∀t ∈[0, T].Proof.
The Lipschitz continuity of g implies that the multifunction F in (4.2) is Lipschitzcontinuous in the Hausdorffmetric, hence it satisfies (LSP). We can thus apply Theorem17
1, and obtain a suitable selection f of extF, in connection with f0, ε0. For every (t, x),the setW(t, x) .=ω ∈U ;g(t, x, ω) = f(t, x)⊂IRmis a compact nonempty subset of U.
Let ¯u(t, x) ∈W(t, x) be the lexicographic selection.Then the feedback control ¯u is measurable, and it is trivial to check that ¯u satisfies allrequired properties.References[1] J. P. Aubin and A. Cellina, “Differential Inclusions”, Springer-Verlag, Berlin, 1984. [2] A. Bressan, Directionally continuous selections and differential inclusions, Funkc.
Ek-vac. 31 (1988), 459-470.
[3] A. Bressan, The most likely path of a differential inclusion, J. Differential Equations88 (1990), 155-174. [4] A. Bressan, Selections of Lipschitz multifunctions generating a continuous flow, Diff.& Integ.
Equat. 4 (1991), 483-490.
[5] A. Cellina, On the set of solutions to Lipschitzean differential inclusions, Diff.&Integ. Equat.
1 (1988), 495-500. [6] F. S. De Blasi and G. Pianigiani, On the solution set of nonconvex differential inclu-sions, preprint.
[7] A. LeDonne and M. V. Marchi, Representation of Lipschitz compact convex valuedmappings, Rend. Accad.
Naz. Lincei 68 (1980), 278-280.
[8] A. F. Filippov, On certain questions in the theory of optimal control, SIAM J. Control1 (1962), 76-84. [9] A. Ornelas, Parametrization of Caratheodory multifunctions, Rend.
Sem. Mat.
Univ.Padova 83 (1990), 33-44.18
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