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Alberto Bressan and Benedetto Piccoli

arXiv:funct-an/9205001v1 6 May 1992A Baire Category Approachto the Bang-Bang PropertyAlberto Bressan and Benedetto PiccoliS.I.S.S.A., Via Beirut 4,34014 Trieste, ItalyAbstract - Aim of this paper is to develop a new technique, based on the Baire categorytheorem, in order to establish the closure of reachable sets and the existence of optimaltrajectories for control systems, without the usual convexity assumptions. The bang-bang property is proved for a new class of “concave” multifunctions, characterized bythe existence of suitable linear selections.

The proofs rely on Lyapunov’s theorem inconnection with a Baire category argument.Ref. S.I.S.S.A.70/92/M1

1 - Introduction.Aim of this paper is to develop a new technique, based on the Baire category theorem,in order to establish the closure of reachable sets and the existence of optimal trajectoriesfor control systems, without the usual convexity assumptions.Most of our results will be formulated within the framework of differential inclusions.Let F : IR × IRn 7→2IRn be a continuous multifunction with compact convex values anddenote by extF(t, x) the set of extreme points of F(t, x). We say that F has the bang-bangproperty if, for every interval [a, b] and every Caratheodory solution x(·) of˙x(t) ∈F(t, x(t))t ∈[a, b],(1.1)there exists also a solution of˙y(t) ∈extF(t, y(t))t ∈[a, b](1.2)such thaty(a) = x(a),y(b) = x(b).

(1.3)If A, B are respectively n × n and n × m matrices, and U ⊂IRm is compact convex, thewell known Bang-Bang Theorem [8, 10, 15] implies that the above property holds for the“linear” multifunctionF(t, x) =A(t)x + B(t)u;u ∈U⊂IRn. (1.4)In the present paper, the bang-bang property is proved for a new class of “concave” mul-tifunctions, characterized by the existence of suitable linear selections.

The proofs relyon Lyapunov’s theorem in connection with a Baire category argument. As applications,we obtain some closure theorems for the reachable set of a differential inclusion with non-convex right hand side, and new existence results for optimal control problems in Mayeras well as in Bolza form.Roughly speaking, the Baire category method consists in showing that the set SextFof solutions of (1.2) is the intersection of countably many relatively open and dense subsetsof the family SF of all solutions of (1.1).

Since SF is closed, Baire’s theorem thus impliesSextF ̸= ∅. The effectiveness of such an argument, in connection with the Cauchy problemfor a differential inclusion, was suggested by Cellina [5] and demonstrated in [4, 9, 19] andin other papers.

Here, this basic technique will be combined with Lyapunov’s theorem andapplied to the two-point boundary value problem (1.2), (1.3).2

The use of a Lyapunov-type theorem, in order to prove existence of optimal solutionsfor non-convex control problems, was introduced by Neustadt [12] and later applied in [1,13, 16] to a variety of optimization problems, always in connection with evolution equationsand cost functionals which are linear w.r.t. the state variable.

In [6], Cellina and Colomboshowed that the linear cost functional can be replaced by one which is concave w.r.t. thestate variable.

Extensions and applications to partial differential equations have recentlyappeared in [7, 14]. We remark that, if a variational problem of the type considered in [6] isreformulated as a Mayer problem of optimal control, then the corresponding multifunctionsatisfies our concavity assumptions.

The present results can thus be regarded as a naturalextension of the theorem in [6], for optimization problems which are “fully concave”: intheir dynamics as well as in the cost functional.2 - Preliminaries.In this paper, | · | is the euclidean norm in IRn, B(x, r) denotes the open ball centeredat x with radius r, while B(A, ε) denotes the open ε-neighborhood around the set A. Wewrite A and coA respectively for the closure and the closed convex hull of A, while A \ Bindicates a set-theoretic difference. The Lebesgue measure of a set J ⊂IR is meas(J).

Werecall that a subset A ⊆S is a Gδ if A is the intersection of countably many relativelyopen subsets of S.In the following, Kn denotes the family of all nonempty compact convex subsets ofIRn, endowed with the Hausdorffmetric. A key technical tool used in our proofs will bethe function h : IRn × Kn 7→IR ∪{−∞}, defined byh(y, K) = sup(Z 10|f(x) −y|2 dx 12;f : [0, 1] →K,Z 10f(x) dx = 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.

From the above definition, it is clear thath(ξ + y, ξ + K) = h(y, K),h(λy, λK) = λh(y, K),∀ξ ∈IRn, λ > 0. (2.2)For the basic theory of multifunctions and differential inclusions we refer to [1].

Given3

a solution x(·) of (1.1), following [3] we define its likelihood asL(x) = Z bah2( ˙x(t), F(t, x(t))dt! 12= ∥h( ˙x, F(·, x)∥L2.

(2.3)The following results were proved in [3]:Lemma 1. For every y, K, one has h(y, K) ≤r(K), where r(K) is the radius of thesmallest ball containing K (i.e., the Cebyshev radius).

Moreover, h(y, K) = 0 iffy ∈extK.Therefore, a solution x(·) of (1.1) satisfies also (1.2) iffL(x) = 0.Lemma 2. The map (y, K) 7→h(y, K) is upper semicontinuous in both variables andconcave w.r.t.

y. The map x(·) 7→L(x) is upper semicontinuous on the set of solutions of(1.1), endowed with the C0 norm.3 - The main results.In the following, we denote by Sb,qa,p the set of all Caratheodory solutions of the two-point boundary value problem˙x(t) ∈F(t, x(t)),x(a) = p,x(b) = q.

(3.1)Theorem 1. Let F : IR × IRn 7→2IRn be a continuous multifunction with compact, convexvalues.

The following conditions are equivalent:(1) For every interval [a, b] and every p, q ∈IRn, if Sb,qa,p ̸= ∅, then the set of solutions of˙y(t) ∈extF(t, y(t)),x(a) = p,x(b) = q(3.2)is a dense Gδ in Sb,qa,p. (2) F has the bang-bang property(3) For every interval [a, b] and every p, q ∈IRn, if Sb,qa,p ̸= ∅, then for every ε > 0 thereexists a solution x(·) of (3.1) such thatL2(x) .=Z bah2˙x(t), F(t, x(t))dt < ε.(3.3)Proof.

(1) ⇒(2)If Sb,qa,p ̸= ∅, then by (1) the set of solutions of (3.2), being dense, isnonempty. Hence (2) holds.4

(2) ⇒(3)If Sb,qa,p ̸= ∅then by (2) there exists a solution y(·) of (3.2). This implies (3),because by Lemma 1Z bah2( ˙y(t), F(t, y(t))) dt = 0 < ε.

(3) ⇒(1) Consider the sets Am .=x ∈Sb,qa,p ;L(x) <1m. By Lemma 2, L is uppersemicontinuous, hence each Am is open.

Now fix any x(·) ∈Sb,qa,p, ε > 0. DefineΩ.=(t, z);t ∈[a, b],|z −x(t)| ≤εand choose a constant M so large thatF(t, x) ⊆B(0, M)∀(t, x) ∈Ω.

(3.4)Split the interval [a, b] into k equal subintervals Ji = [ti−1, ti], inserting the points ti .=a + (i/k)(b −a), choosing k so large that 2M(b −a)/k ≤ε.By the assumption (3), for each i there exists a solution yi : [ti−1, ti] 7→IRn of thetwo-point boundary value problem˙y(t) ∈F(t, y(t)),y(ti−1) = x(ti−1),y(ti) = x(ti),(3.5)withL2(yi) =Z titi−1h2˙yi(t), F(t, yi(t))dt <1m2k . (3.6)Define y(·) as the solution of (3.1) whose restriction to each Ji coincides with yi.

Givenany t ∈[a, b], if, say, t∈Ji, then (3.4), (3.5) imply|y(t) −x(t)| ≤Z tti| ˙yi(s) −˙x(s)| ds ≤2M(b −a)k≤ε.Hence ∥y −x∥C0 ≤ε. Moreover, y ∈Am becauseL2(y) =kXi=1Z titi−1h2˙yi(t), F(t, yi(t))dt 0 were arbitrary, this proves that each Am is dense in Sb,qa,p.

By Baire’stheorem, it follows that A = Tm Am is a Gδ dense subset of Sb,qa,p. If y ∈A, then L(y) = 0and hence ˙y(t) ∈extF(t, y(t)) almost everywhere.In the previous theorem, the implication (3) ⇒(1) determines the strength of thecategory method.

In order to prove that “most” solutions of (3.1) actually solve (3.2) as5

well, it suffices to show (for every a, b, p, q) the existence of some solution of (3.1) witharbitrarily small likelihood.Roughly speaking, this requires the construction of somesolution y of (3.1) whose derivative remains close to the extreme points of F(t, y) duringmost of the time.In practice, the condition (3) may often be easier to verify. We now show that this isindeed the case, if the multifunction F satisfies suitable concavity conditions.Theorem 2.Let F : IR × IRn 7→2IRn be a Hausdorffcontinuous multifunction withcompact, convex values.

Assume that:(C1) For each (t, x) and every y ∈F(t, x), there exists a linear function z 7→Az + csatisfyingy = Ax + c,Az + c ∈F(t, z)∀z ∈B(x, ρ(t, x)),(3.7)where the radius ρ = ρ(t, x) remains uniformly positive on compact sets. (C2) For each (t, x), every y ∈F(t, x) and ε > 0, there exist δ > 0 and n+1 linear functionsz 7→A′z + ci,i = 0, .

. ., n, such thaty ∈coA′x + c0 , .

. .

, A′x + cn,(3.8)h(A′x + ci, F(t, x)) ≤ε∀i,(3.9)A′z + ci ∈F(t, z)∀z ∈B(x, δ), ∀i. (3.10)Then F has the bang-bang property.We refer to (C1), (C2) as concavity conditions because they require, for each point(t, x, y) of the graph of F, the existence of suitable linear (non-homogeneous) selections.A similar property is shared by the epigraph of a concave scalar function, which admitsglobal linear selections through each of its points.Proof of Theorem 2.We will prove that F has property (3) stated in Theorem 1.

Letx∗(·) be a solution of (3.1), for some interval [a, b] and some points p, q ∈IRn. Let anyε > 0 be given, and defineη =inft∈[a,b] ρ(t, x∗(t)),V =(t, z);t ∈[a, b],|z −x∗(t)| ≤η.

(3.11)6

By assumption, η > 0. Choose M so large thatF(t, z) ⊆B(0, M)∀(t, z) ∈V.

(3.12)By Lemma 1, this impliesh(y, F(t, z)) ≤M∀y,∀(t, z) ∈V.(3.13)1. As a first step, we construct measurable, bounded functions A, c, such that˙x∗(t) = A(t)x(t) + c(t)for a.e.

t ∈[a, b],(3.14)A(t)z + c(t) ∈F(t, z)∀z ∈B(x∗(t), η). (3.15)Since ˙x∗(·) is measurable, by Lusin’s theorem there exists a sequence of disjoint compactsets (Jν)ν≥1 such thatmeas[a, b] \[ν≥1Jν= 0(3.16)and such that the restriction of ˙x∗to each Jν is continuous.

Define the multifunctionG : [a, b] 7→2IRn×n×IRn by settingG(t) .=(A, c) :˙x∗(t) = Ax + c,Az + c ∈F(t, z)∀z ∈B(x∗(t), η).Because of (C1) and of the choice of η, G(t) ̸= ∅for a.e. t. One easily checks that therestriction of G to each Jν has closed graph, because of the continuity of ˙x∗, x∗and F.Hence, G is a measurable multifunction on [a, b] with closed, nonempty values.

By [11],it admits a measurable selection t 7→A(t), c(t), which clearly satisfies (3.14), (3.15).Observe that the matrices A(t) and the vectors c(t) must be uniformly bounded, becauseof (3.15), (3.12).2. As a second step, we construct measurable functions A′, c0, .

. .cn, θ0, .

. ., θn, δ, suchthat, for almost every t ∈[a, b], the following holds:δ(t) > 0,θi(t) ∈[0, 1],nXi=0θi(t) = 1,(3.17)˙x∗(t) = A′(t)x∗(t) +nXi=0θi(t)ci(t),(3.18)7

A′(t)z + ci(t) ∈F(t, z),h2A′(t)z + ci(t), F(t, z)≤ε∀i,∀z ∈Bx∗(t), δ(t). (3.19)By Lemma 2, h is upper semicontinuous, hence there exists a nonincreasing sequence(hm)m≥1 of continuous functions such thath(y, K) = infm≥1 hm(y, K)∀(y, K) ∈IRn × Kn.

(3.20)For each m ≥1, define the multifunctionHm(t) .= A′, c0, . .

., cn, θ0, . .

., θn, δ;δ = 1m,θi ∈[0, 1],˙x∗(t) =nXi=0θi(A′x(t) + ci),nXi=0θi = 1,A′z + ci ∈F(t, z),h2mA′z + ci, F(t, z)≤ε∀z ∈Bx(t), 1m.If (Jν)ν≥1 is the same sequence of compact sets considered at (3.16), the continuity of˙x∗, x∗, hm and F implies that the restriction of Hm to each Jν has closed graph, withuniformly bounded, possibly empty values.Defining Im = {t; Hm(t) ̸= ∅}, it is clear that on each Im the multifunction Hmis measurable with closed, nonempty values. By [11], it admits a measurable selection,say t 7→Φm(t).By (C2), (3.20) and the continuity of each hm, for every ν we haveSm≥1 Im ⊇Jν.

Therefore, the selectionA′(t), c0(t), . .

., cn(t), θ0(t), . .

., θn(t), δ(t) .= Φm(t)ifft ∈Im \[ℓ

By construction, the conditions (3.17)-(3.19)hold.3. We can now complete the proof of the theorem.

Since δ(·) is measurable and positive,there exists an integer m∗such that1m∗< η,measJ′m∗≥(b −a) −ε,(3.21)whereJ′m.=nt ∈[a, b];δ(t) ≥1m,|A′(t)|, |ci(t)| ≤mo. (3.22)Split [a, b] into k equal subintervals Ij = [tj−1, tj], inserting the points tj .= a+(j/k)(b−a)and choosing k so large that:2M b −ak<1m∗.

(3.23)8

Using the selections A, c and A′, ci, θi constructed in the previous steps, define:A∗(t) =A(t)ift /∈J′m∗,A′(t)ift ∈J′m∗,f(t) =c(t)ift /∈J′m∗,Pni=0 θi(t)ci(t)ift ∈J′m∗.By (3.14), (3.18), ˙x∗(t) = A∗(t)x(t)+f(t). Calling W(·, ·) the matrix fundamental solutionof the bounded linear system ˙v = A∗(t)v, we thus have the representationx∗(t) = W(t, tj−1)x∗(tj−1) +Z ttj−1W(t, s)f(s) ds,t ∈[tj−1, tj].Applying Lyapunov’s theorem on each interval Ij, for every j we obtain a measurablepartitionIj,0, .

. ., Ij,nof Ij and an absolutely continuous function wj satisfying thetwo-point boundary value problem˙wj(t) =A∗(t)wj(t) + c(t)ift ∈Ij \ J′m∗,A∗(t)wj(t) + cℓ(t)ift ∈Ij,ℓ∩J′m∗,ℓ= 0, .

. ., n,wj(tj) = x∗(tj),wj(tj+1) = x∗(tj+1).We claim that|wj(t) −x∗(t)| ≤1m∗∀t ∈Ij.

(3.24)If not, there would exist a first time τ ∈Ij such that|wj(τ) −x∗(τ)| =1m∗. (3.25)Recalling (3.15), (3.19) and using (3.12) and (3.23), we then have:|wj(τ) −x∗(τ)| ≤Z τtj−1| ˙wj(t) −˙x∗(t)| dt ≤2M b −ak<1m∗,a contradiction with (3.25).

This proves (3.24). In particular, by (3.15), (3.19) we concludethat ˙wj(t) ∈F(t, wj(t)) for a.e.

t ∈Ij.Now consider the solution w(·) of (3.1) whose restriction to each Ij coincides with wj.Recalling (3.13), (3.19), (3.21), its likelihood is computed byL2(w) =ZJ′m∗h2( ˙w(t), F(t, w(t))) dt +Z[a,b]\J′m∗h2( ˙w(t), F(t, w(t))) dt≤ε · measJ′m∗+ M 2 · meas[a, b] \ J′m∗≤ε((b −a) + M 2).9

Since ε was arbitrary, this establishes the property (3) in Theorem 1, which is equivalentto the bang-bang property.Remark 1. The previous theorems, with the same proofs, remain valid if F is defined onsome open set Ω⊂IR × IRn.4 - Examples of concave multifunctions.Aim of this section is to exhibit some classes of multifunctions which satisfy theconcavity properties (C1), (C2) stated in Theorem 2.Proposition 1.

Let ϕ : IR × IRn 7→]0, ∞[ be a continuous function, with x 7→ϕ(t, x)convex for every t. Let U ⊂IRn be compact, convex, containing the origin. Then themultifunctionF(t, x) = ϕ(t, x)U(4.1)has the bang-bang property.Proof.

In order to apply Theorem 2, we first verify the concavity condition (C1). Fix any(t, x) and any y = ϕ(t, x)u ∈F(t, x).

Since ϕ is continuous and strictly positive and itssubdifferential ∂xϕ w.r.t. x is uniformly bounded on compact sets, we haveϕ(t, x) + ξ · (z −x) ≥0∀ξ ∈∂xϕ(t, x),∀z ∈B(x, ρ(t, x)),(4.2)for some function ρ = ρ(t, x) uniformly positive on compact sets.Choose any vectorξ ∈∂xϕ(t, x) and define the linear mapz 7→Az + c .= (ξ · z)u +ϕ(t, x)u −(ξ · x)u.If |z −x| ≤ρ(t, x), we need to show the existence of some ω ∈U such that(ξ · z)u +ϕ(t, x)u −(ξ · x)u= ϕ(t, z)ω.

(4.3)From (4.2) and the convexity of ϕ it followsω = ϕ(t, x) + ξ · (z −x)ϕ(t, z)u = αu(4.4)for some α ∈[0, 1]. The assumptions on U thus imply ω ∈U.10

We now turn to the condition (C2). Let (t, x), y = ϕ(t, x)u ∈F(t, x) and ε > 0 be given.We can assumeu =νXi=1θiui,θi ∈(0, 1],νXi=1θi = 1(4.5)for some ν ∈{1, .

. ., n + 1}, ui ∈extU.

Select ξ ∈∂xϕ(t, x) as before and defineu′i = ui + ε′(u −ui),(4.6)choosing ε′ ∈(0, 1) so small thath(u′i, U) <εϕ(t, x)∀i. (4.7)This is possible because h(ui, U) = 0 and h is upper semicontinuous.

Then defineA′z = (ξ · z)u,ci = ϕ(t, x)u′i −(ξ · x)u.Recalling (2.2)2, (4.7) yieldshA′x + ci, F(t, x)= hϕ(t, x)u′i, ϕ(t, x)U< ε.Hence, for z in a small neighborhood of x, the upper semicontinuity of h implieshA′z + ci, ϕ(t, z)U< ε∀i.Moreover, by (4.5), (4.6),y = ϕ(t, x)u =νXi=1θiϕ(t, x)u′i ∈coA′x + ci,i = 1, . .

., ν.It remains to prove that A′z + ci ∈ϕ(t, z)U for |z −x| small enough. For each fixed i,define the subspace Ei .= span{u, ui}.If Ei has dimension 2, consider the triangle ∆= co{0, u, ui} and call n1, n2, n3 theunit vectors in Ei which are outer normals to the sides ui −u, ui, u, respectively.

Observethatϕ(t, z)co{0, u, ui} =y ∈Ei;n1 · y ≤ϕ(t, z)(n1 · u),n2 · y ≤0,n3 · y ≤0⊆F(t, z). (4.8)Since u′i is a strict convex combination of u and ui, one has n2 · u′i < 0, n3 · u′i < 0.

Bycontinuity, for z sufficiently close to x we still havenj ·ϕ(t, x)u′i + ξ · (z −x)u= nj ·A′z + ci< 0,j = 2, 3. (4.9)11

Moreover, since n1 · u = n1 · u′i > 0, the convexity of ϕ impliesϕ(t, z)(n1 · u) ≥ϕ(t, x) + ξ · (z −x)(n1 · u)= n1 ·ϕ(t, x)u′i +ξ · (z −x)u= n1 · [A′z + ci]. (4.10)By (4.8), the inequalities (4.9), (4.10) together imply A′z + ci ∈F(t, z).Finally, consider the case where Ei has dimension ≤1.

Then, either u = ui, henceν = 1 and u ∈extF(t, x). In this case, the same argument as in (4.3), (4.4) can be used.Or else u′i lies in the relative interior of the segment S .= co{0, u, ui}.

In this case, the mapz 7→A′z + ci takes values inside Ei, with ϕ−1(t, x)A′x + ci∈rel intS. By continuity,ϕ−1(t, z)A′z + ci∈S for |z −x| small enough.

This proves again that A′z + ci ∈F(t, z)for every z in a neighborhood of x.An application of Theorem 2 now yields the desired conclusion.The next application is concerned with a multifunction F whose values are polytopes,with variable shape but constant number of vertices.More precisely, we assume thatF(t, x) admits the representationsF(t, x) = coy1(t, x), . .

., yN(t, x),(4.11)where y1, . .

., yN are the (distinct) vertices of F(t, x), as well asF(t, x) =ny ∈IRn ;wj(t) · y ≤ψj(t, x) =maxω∈F (t,x)wj(t) · ω,j = 1, .., k.(4.12).On the product set of indices {1, . .

., N} × {1, . .

., k}, we consider the “incidence” relationi ∼jiffwj(t) · yi(t, x) = ψj(t, x). (4.13)Proposition 2.

Let F be a multifunction admitting the representations (4.11), (4.12).Assume that(i) wj : IR 7→IRn, ψj : IR × IRn 7→IR are continuous functions, |wj(t)| ≡1, each mapx 7→ψj(t, x) is convex. (ii) The relation ∼defined at (4.13) is independent of (t, x).12

Then F has the bang-bang property.Proof. For each i ∈{1, .

. ., N}, consider the set of indicesJi =j;wj(t) · yi(t, x) = ψj(t, x).

(4.14)By (ii), this set does not depend on t, x.We begin by checking the condition (C1) in Theorem 2. First, assume y ∈extF(t, x),say y = yi(t, x).

In this case we can choose n independent vectors, say wj1(t), . .

., wjn(t),with j1, . .

., jn ∈Ji. Define the dual vectors w∗jℓ, requiring thatw∗jℓ· wjm =1ifℓ= m,0ifℓ̸= m.(4.15)By convexity, each function z 7→ψj(t, z) is differentiable almost everywhere.

Therefore,there exists a sequence of points xν →x such that the gradients ∇xψ(t, xν) exist for eachν, together with the limitslimν→∞∇xψj(t, xν) = ξj ∈∂xψj(t, x)∀j. (4.16)Now defineAiz =nXℓ=1(ξjℓ· z)w∗jℓ,ci = yi(t, x) −nXℓ=1(ξjℓ· x)w∗jℓ.

(4.17)Clearly, Aix + ci = yi. Using the representation (4.12) we now check thatAiz + ci ∈F(t, z)∀z ∈B(x, ρ(t, x))(4.18)for some ρ = ρ(t, x) uniformly positive on compact sets.If j ∈Ji, then there exist unique coefficients αℓsuch that wj(t) = P αℓwjℓ(t).

Theassumption (ii) together with (4.16) now impliesψj(t, z) = wj(t) · yi(t, z) =nXℓ=1αℓψjℓ(t, z)∀z,nXℓ=1αℓξjℓ∈∂xψj(t, x). (4.19)From (4.19) and the convexity of ψj it followswj(t) ·Aiz + ci=nXℓ=1αℓwjℓ(t) ·"yi(t, x) +nXh=1ξjh · (z −x)w∗jh#= ψj(t, x) +nXℓ=1αℓξjℓ· (z −x) ≤ψj(t, z).13

On the other hand, if j /∈Ji then wj(t) ·Aix + ci< ψj(t, x). Hence, by continuity westill havewj(t) ·Aiz + ci< ψj(t, z)∀z ∈Bx, ρ(t, x).By the assumption (ii), the continuity of the functions wj, ψj and the local boundedness ofthe subgradients ∂xψj, it follows that ρ can be taken uniformly positive on bounded sets.This proves (C1) in the case y ∈extF(t, x).When y is an arbitrary element in F(t, x), there exist extreme points yi and coefficientsθi ∈[0, 1] such thaty =NXi=1θiyi(t, x),NXi=1θi = 1.If Ai, ci are the matrices and vectors defined at (4.17), then the convex combinationsA = P θiAi, c = P θici satisfyAx + c = y,Az + c ∈F(t, z)∀z ∈B(x, ρ(t, x)).Next, consider the condition (C2).

Let (t, x), y ∈F(t, x), ε > 0 be given. Write y asa convex combination of points y1, .

. ., yν ∈extF(t, x), sayy =νXi=1θiyiθi ∈(0, 1]νXi=1θi = 1,and definey′i = yi + ε′(y −yi),choosing ε′ ∈(0, 1] so small thathy′i, F(t, x)< ε∀i.

(4.20)Consider the vector spaceE .= spanwj(t);wj(t) · y = ψj(t, x).Choose a basis {wj1, . .

., wjµ} of E and define the dual basis {w∗j1, . .

., w∗jµ} as in (4.15).Select vectors ξj ∈∂xψj(t, x) as in (4.16) and defineA′z =µXℓ=1(ξjℓ· z)w∗jℓ,ci = y′i −µXℓ=1(ξjℓ· x)w∗jℓ. (4.21)14

The above definitions implyy =νXi=1y′i ∈coA′x + ci;i = 1, . .

., µ.Moreover, by (4.20) and the upper semicontinuity of h, for |z −x| small enough we havehA′z + ci, F(t, z)< εUsing the representation (4.12), we now prove that A′z + ci ∈F(t, z). If wj(t) ∈E, thenwj(t) · y = wj(t) · y′i = ψj(t, x)∀i = 1, .

. ., ν.Moreover, there exist coefficients αℓsuch that wj(t) = P αℓwjℓ(t).

The assumption (ii)together with (4.16) now impliesψj(t, z) = wj(t) · yi(t, z) =µXℓ=1αℓψjℓ(t, z)∀z,(4.22)µXℓ=1αℓξjℓ∈∂xψj(t, x). (4.23)From (4.22), (4.23) and the convexity of ψj it followswj(t) ·A′z + ci=µXℓ=1αℓwjℓ(t) ·"y′i +µXh=1ξjh · (z −x)w∗jh#= ψj(t, x) +µXℓ=1αℓξjℓ· (z −x)≤ψj(t, z).On the other hand, if wj(t) /∈E, thenwj(t) · [A′x + ci] = wj(t) · y′i < ψj(t, x).By continuity, for |z −x| sufficiently small we still havewj(t) · [A′z + ci] < ψj(t, z).This completes the proof of condition (C2).

An application of Theorem 2 now yields thedesired result.15

Remark 2. Assume that A, b are a n×n matrix and a n-vector, depending continuously ont, and that F is a continuous, compact convex valued multifunction satisfying the concavityconditions (C1), (C2).

Then the multifunctionG(t, x) = A(t)x + b(t) + F(t, x)satisfies all assumptions in Theorem 2 as well. In particular, from Proposition 1 it followsthat the bang-bang property holds for a control system of the form˙x = A(t)x + b(t) + ϕ(t, x)u,u(t) ∈U,with U compact, convex, containing the origin and ϕ > 0 convex w.r.t.

x.5 - A nonconvex optimal control problem.This section is concerned with an application of Theorem 2 to an optimal controlproblem of Bolza. The analysis will clarify the connections between the concavity condi-tions (C1), (C2) and the assumptions made in [6, 12].

Given the linear control system onIRn˙x(t) = A(t)x + f(t, u(t))u(t) ∈U(5.1)with initial and terminal constraintsx(0) = ¯x(T, x(T)) ∈S,(5.2)consider the minimization problem:minZ T0α(t, x(t)) + β(t, u(t)) dt. (5.3)Theorem 3.

Let the functions A, f, α, β be continuous, with α concave w.r.t. x. Assumethat the control set U ⊆IRm is compact and that the terminal set S is closed and containedin [0, T0] × IRn, for some T0.

If some solution of (5.1), (5.2) exists, then the minimizationproblem (5.3) admits an optimal solution.Proof. We begin by adding an extra variable x0, writing the problem in Mayer form:min x0(T)(5.4)16

(˙x(t) = A(t)x(t) + f(t, u(t))˙x0(t) = α(t, x(t)) + β(t, u(t))u(t) ∈U,(5.5)(x, x0)(0) = (¯x, 0)(T, x(T)) ∈S. (5.6)The continuity of A, f and the compactness of U imply that all trajectories of the differ-ential inclusion˙x(t) ∈A(t)x(t) + cof(t, u);u ∈U,x(0) = ¯x,t ∈[0, T0],(5.7)are contained in some bounded open set Ω⊂IR × IRn.

Define the constantM .= 1 + supα(t, x) + β(t, u);(t, x) ∈Ω,u ∈U(5.8)and the multifunction (independent of x0)F(t, x, x0) .= co(y, y0);y = A(t)x+f(t, u),α(t, x)+β(t, u) ≤y0 ≤M for some u ∈U. (5.9)Observe that F admits the representationF(t, x) =(y, y0);y = A(t)x +n+1Xi=0θif(t, ui),y0 =hα(t, x) +n+1Xi=0θiβ(t, ui)i(1 −v) + Mv,(θ0, .

. ., θn+1) ∈∆n+1,u0, .

. ., un+1 ∈U,v ∈[0, 1],(5.10)where∆n+1 =n(θ0, .

. .

, θn+1);θi ∈[0, 1],n+1Xi=0θi = 1o.Since F is continuous with compact convex values, the optimization problem (5.4)subject to the boundary conditions (5.6) with dynamics( ˙x, ˙x0) ∈F(t, x)(5.11)admits an optimal solution. The existence of a solution to the original problem (5.1)-(5.3) will be proved by showing that the multifunction F has the bang-bang property, for(t, x, x0) ∈Ω× IR.To verify the concavity condition (C1), define the constant ρ > 0 by1ρ = sup|ξ|;ξ ∈∂xα(t, x),(t, x) ∈Ω.

(5.12)17

Given (t, x) ∈Ω, (y, y0) ∈F(t, x), assume first y0 < M −1. Choose any ξ ∈∂xα(t, x) andconsider the linear mapΦ(z) .=y + A(t)(z −x), y0 + ξ · (z −x).

(5.13)From (5.12) and the assumption on y0 it follows thaty0 + ξ · (z −x) ≤M∀z ∈B(x, ρ). (5.14)Let y, y0 be as in (5.10), for some θi, ui, v. Using the concavity of α we then obtainy + A(t)(z −x) = A(t)z +n+1Xi=0θif(t, ui),y0 + ξ · (z −x) ≥α(t, x) +n+1Xi=0θiβ(t, ui) + ξ · (z −x)≥α(t, z) +n+1Xi=0θiβ(t, ui).This, together with (5.14), implies Φ(z) ∈F(t, z).Next, assume y0 ∈[M −1, M].

Then the definition (5.8) implies that the mapΨ(z) .=y + A(t)(z −x), y0is again a selection of F(t, ·).We now turn to the condition (C2). Fix (t, x) ∈Ω, ˜y = (y, y0) ∈F(t, x), ε > 0, andchoose ξ ∈∂xα(t, x).

Write ˜y as a convex combination˜y =νXj=1ϑj ˜yj,ϑj ∈(0, 1],νXj=1ϑj = 1,with ˜yj ∈extF(t, x), and define˜y′j = ˜yj + ε′(˜y −˜yj),choosing ε′ ∈(0, 1] such thath˜y′j, F(t, x)< ε∀j. (5.15)We now distinguish two cases.18

If y0 < M, then ˜y′j = (y′j, y′0,j) satisfies y′0,j < M for all j. Hence, for |z−x| sufficientlysmall, the mapsΦj(z) .=y′j + A(t)(z −x), y′0,j + ξ · (z −x)are affine selections of F. Moreover, (5.15) and the upper semicontinuity of h implyhΦ(z, z0), F(t, z)< εfor all z in a neighborhood of x.On the other hand, if y0 = M, then y0,j = y′0,j = M for all j.

Hence the mapsΦj(z) .=y′j + A(t)(z −x), Mare affine selections of F and satisfy (5.16) for |z −x| sufficiently small.Applying Theorem 2, we now obtain the existence of an optimal solution (x∗, x∗0) :[0, T] 7→IRn × IR to (5.4), (5.6), (5.11), with the additional property( ˙x∗, ˙x∗0)(t) ∈extF(t, x∗(t))for a.e.t ∈[0, T].The representation (5.10) and the selection theorem [11] now yield the existence of somemeasurable u∗: [0, T] 7→U such that(˙x∗(t) = A(t)x∗(t) + f(t, u∗(t)),x∗0(t) ∈α(t, x∗(t)) + β(t, u∗(t)),Mfor almost every t. Since the terminal value x∗0(T) is minimized, by (5.8) we must have˙x∗0(t) < M almost everywhere.Therefore, x∗is an optimal trajectory for the originalsystem (5.1), corresponding to the control u∗.6 - A counterexample.The following example shows how the bang-bang property may fail, if some of theassumptions in Theorem 2 or in Proposition 2 are not satisfied.More general resultsconcerning systems of this form can be found in [17, 18].On IR2, consider the control system˙x(t) = f(x(t)) + g(x(t))u(t),u(t) ∈[−1, 1],(6.1)19

withf(x1, x2) = (x1, x2),g(x1, x2) = (1, x1).For t ∈[0, 1], the trajectory t 7→(0, et), corresponding to the control u(t) ≡0, steers thesystem from p = (0, 1) to q = (0, e).Defining the auxiliary functionV (x1, x2) = x2 −x212 ,a straightforward computation yieldsddtV (x(t)) = V (x(t)) −x21(t)2for every solution of (6.1). This impliesV (x(t)) ≤etV (x(0)),with equality holding if and only if x1(s) = 0 for all s ∈[0, t].

In particular, the controlu ≡0 is the only one which steers the system from p to q in minimum time. Hence themultifunctionF(x1, x2) =(x1 + u, x2 + x1u);|u| ≤1does not have the bang-bang property.

Observe that(i) For each y = f(x) + g(x)ω ∈F(x), definingA =10ω1c =ω0,one checks that the condition (C1) in Theorem 2 holds. However, the condition (C2)here fails.

(ii) The multifunction extF(x) =f(x)+g(x), f(x)−g(x)satisfies both (C1) and (C2)in Theorem 2, but its values are not convex. (iii) Each set F(x) is a segment.

Moreover, F admits the representationF(x) =y;w · y ≤ψw(x) .= max|u|≤1 w ·f(x) + g(x)u .Since f, g are linear, each ψw is convex. However, this representation does not satisfyall assumptions in Proposition 2.20

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