Transversality, the maximum principle, and the

If S is a subset of a finite-dimensional real vector space V , an ap-proximating cone (a.k.a. “Boltyanskii approximating cone”) to S at a point p if S ...

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Transversality, the maximum principle, and the approximation problem H´ ector J. Sussmann Department of Mathematics — Rutgers University Piscataway, NJ 08854, USA [email protected] Conference on Optimization, State Constraints and Geometric Control Dipartimento di Matematica “Tullio Levi-Civita” Universit` a degi Studi di Padova May 23-24, 2018 1

HAPPY BIRT HDAY F RAN CO !!!!!

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HAPPY BIRT HDAY GIOVAN N I !!!!!

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REMARKS FOR THE EXPERTS I All the ideas of this talk are reall contained, at least implicitly, in the original work of Pontryagin-Boltyanskii-GamkrelidzeMischenko. And they were understood quite explicitly by

Jack Warga.

The purposes of this talk are 1. to clarify these old ideas and explain them in simple modern language 2. to show, how properly formulated, these ideas can be extended further, in particular to the cases where the relevant maps are setvalued.

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REMARKS FOR THE EXPERTS II Most of the ideas discussed in this talk have been presented in previous lectures and papers. But the approach used in this talk is new. In particular, I have been able to do away with the distinction between “transversality” and “strong transversality”, thus making the new approach much simpler.

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REMARKS FOR THE EXPERTS III

The work discussed here is closely related to that of Michele Palladino and Franco Rampazzo on the gap problem.

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STRUCTURE OF THE TALK

1. Transversality

2. The maximum principle as a transversality theorem

3. Approximation of trajectories by trajectories for a smaller class of controls

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TRANSVERSALITY The idea of transversality is very simple. EXAMPLE: Let γ1 : [0, 1] 7→ R2, γ2 : [0, 1] 7→ R2, be two continuous curves in the plane. Then γ1, γ2 may:

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1. not intersect at all,

γ

γ

2

2

γ

1

γ

1

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2. intersect “tangentially without crossing”,

γ

2

γ

2

γ1

γ p

1

in which case it’s possible to make arbitrarily small perturbations of γ1, γ2 that will not intersect at all; 10

3. intersect transversally (i.e., cross),

γ

1

γ

γ

2

2

p γ

1

in which case all sufficiently small perturbations of γ1, γ2 will also intersect. 11

Transversality of the tangent approximations L1, L2 to γ1 and γ2 at an intersection point p is sufficient for the curves to intersect transversally:

γ2

γ1

L2

γ1

p L1

γ2

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but is not necessary:

γ

γ

1

p

2

γ

1

γ

2

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And the linear approximations that matter are approximations by tangent cones, not necessarily by tangent subspaces:

γ2

p γ1

γ2

γ

1

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The situation is totally different for two curves in three-space: If γ1 : [0, 1] 7→ R3, γ2 : [0, 1] → 7 R3 are two continuous curves in R3, then for every positive ε there exist curves ˜ γ1 : [0, 1] 7→ R3, ˜ γ2 : [0, 1] 7→ R3, such that i.

k˜ γ1 − γ1ksup < ε,

ii.

k˜ γ2 − γ2ksup < ε,

and iii.

˜ γ1 and ˜ γ2 do not meet at all,

that is, iii’.

˜ γ1(t1) 6= ˜ γ2(t2) for all (t1, t2) ∈ [0, 1] × [0, 1].

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We consider pointed continuous maps (PCMs), that is, triples

(S, f, p) where:

i. S is a topological space, ii. f is a continuous map from S to some other topological space T , iii. p is a point of S. If S, f, p, T are as above, then we say that f , or the PCM (S, f, p), have target T . 16

DEFINITION: Let (S1, f1, p1), (S2, f2, p2), be PCMs with target T . Assume that T is a metric space. We say that (S1, f1, p1) and (S2, f2, p2) meet transversally if

(TR) For every pair (N1, N2) consisting of neighborhoods Nj of pj in Sj , there exists a positive real number ε such that, if gj : Nj 7→ T are arbitrary continuous maps such that dist(gj (x), fj (x)) ≤ ε

whenever x ∈ Nj , j = 1, 2

(that is, g1, g2 are “ε-perturbations” of f1, f2 on N1, N2) it follows that g1 and g2 meet, that is, there exist qj ∈ Nj for which g1(q1) = g2(q2).

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An obvious

necessary condition for (S1, f1, p1) and (S2, f2, p2)

to meet transversally is

f1(p1) = f2(p2) . REMARK: Rather than assuming that T is a metric space, it would suffice to assume that T has a uniform structure, i.e., a structure that makes it possible to talk about two maps into T being “uniformly cloee”. For example, T could be a topological vector space. In that case, instead of talking about “ε-perturbations” of a map µ into T we would talk about “V -small perturbations”, where V is a neighborhood of 0 in T : a map ν : S 7→ T is a V -small perturbation of a map µ :7→ T if ν(s) − µ(s) ∈ V for all s ∈ S. If S is compact, then T can be an arbitrary toplogical space, because the space C 0(S, T ) of continuous maps has a natural topology (the compact-open topology).

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LINEAR PCMs A linear PCM is a PCM of the form (D, L, 0), where i. D is a convex cone, ii. L is a linear map with target a real vector space T . A REMARK ON THE DEFINITION OF “CONVEX CONE”: A convex cone is a nonempty subset D of a real vector space V , which is closed under addition and multipication by nonnegative scalars. In particular, 0 always belongs to D. The space V has a natural topology TV , namely, the one in which a subset Ω of V is open if and only if Ω ∩ W is open in W for every finite-dimensional subspace W of V . (Also: (i) TV is the inductive limit of the topologies of the finite-dimensional subspaces of V ; (ii) TV is the strongest topology that makes all the inclusion maps W ∋ w 7→ w ∈ V continuous, for all finite-dimensional subspaces W of V .) So D has a natural topology as well.

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TRANSVERSALITY OF LINEAR PCMs THEOREM Let T be a finite-dimensional real vector space, and let (D1, L1, 0), (D2, L2, 0), be linear PCMs with target T . Let Ci = LiDi for i = 1, 2. Then (D1, L1, 0) and (D2, L2, 0) meet transversally if and only if

C1 − C2 = T .

(1)

REMARK; The transversality condition (1) is equivalent to the following nonseparation condition: (NS) There does not exist a nonzero lineal functional λ : T 7→ R such that hλ, c1i ≤ 0 ≤ hλ, c2i

for all c1 ∈ C1 , c2 ∈ C2 .

(2)

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This theorem is

not true for infinite-dimensional targets.

EXAMPLE: Let T be an infinite-dimensional Hilbert space. Then, if D1 = T , L1 = idT , and D2 = {0}, L2 = 0, we have C1 = T , C2 = {0}, so the linear “transversality condition” C1 − C2 = T is satisfied. But (D1, L1, 0) does not meet (D2, L2, 0) transversally. Reason: Using a continuous retraction of the unit ball B of T onto the unit sphere ∂B (which exists if T is infinite-dimensional) one can construct, for any positive ε, a sequence B1, B2, . . . of pairwise disjoint balls in T that converge to zero, and retractions ρj : Bj 7→ ∂Bj , thus obtaining a continuous map Mε : T 7→ T which is an εperturbation of idT , and a sequence of points pj that are not in the image of Mε. And, for large enough j, these points are εperturbations of 0. 21

PROOF THAT THE CONDITION C1 − C2 = T IS NECESSARY FOR TRANSVERSALITY: Assume (D1, L1, 0) and (D2, L2, 0) meet transversally. Then in particular if v ∈ T is sufficiently small the cones C1 + v and C2 must intersect. So there exist c1 ∈ C1, c2 ∈ C2, such that c1 + v = c2. Then v = c1 − c2. So v ∈ C1 − C2. Hence the convex cone C1 − C2 contains a neighborhood of 0 in T . So C1 − C2 = T .

Q.E.D. 22

The proof that the condition C1 − C2 = T is sufficient for (D1, L1, 0) and (D2, L2, 0) to meet transversally is not very hard, but it needs some work. Furthermore, the proof yields a somewhat stronger conclusion:

If C1 − C2 = ˜ 1. subcones D (*) ˜ 1, L ˜1, 0) and (D

T , then there exist finitely spanned ˜ 2 of D1, D2 such that the PCMs D ˜ 2, L ˜2, 0) (where L ˜j is the restriction (D

˜ j ) meet transversally with a linear rate. of Lj to D

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˜ of a convex cone D is FINITELY SPANNED: A convex subcone D ˜ is finitely spanned it there exists a finite subset F of D such that D the convex cone spanned by F .

LINEAR RATE: There exists a positive constant K such that, for all sufficiently small positive δ, if Nj (δ) is the δ-neighborhood of 0 ˜j , then in C ˜1 , L ˜2 (#) If ε = Kδ, then, if g1, g2 are continuous ε-perturbations of L on N1(δ), N2(δ), then g1 and g2 meet (that is, there exist qj ∈ Nj (δ) such that g1(q1) = g2(q2)).

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PARTIAL LINEARIZATIONS Let (S, f, p) be a PCM with finite-dimensional   target T . A partial linearization of (S, f, p) is a pair (D, L, 0), µ , such that i. (D, L, 0) is a linear PCM, ii. µ is a continuous map from some neighborhood Dom(µ) of 0 in D into S, iii. µ(0) = p, def

iii. the map fµ = f ◦ µ satisfies fµ(x)) − fµ(0) = Lx + o(kxk)

as x → 0 , x ∈ Dom(µ) .

(3)

f (µ(x)) − f (p) = Lx + o(kxk)

as x → 0 , x ∈ Dom(µ) .

(4)

i.e.,

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PARTIAL LINEARIZATIONS

p

S

µ D L f−f(p)

T

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APPROXIMATING CONES If S is a subset of a finite-dimensional real vector space V , an approximating cone (a.k.a. “Boltyanskii approximating cone”) to S at a point p if S is a convex cone C in T which is the image of a partial linearization ((D, L, 0), µ) of the identity map idS .

L C p

S

µ D

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THEOREM. If (i.) (S1, f1, p1) and (S2, f2, p2) are PCMs with the same finite-dimensional target T , (ii.) (D1, L1, 0), (D2, L2, 0) are partial linearizations of (S1, f1, p1) and (S2, f2, p2), (iii.) (D1, L1, 0) and (D2, L2, 0) meet transversally, then (&) (S1, f1, p1) and (S2, f2, p2) meet transversally. PROOF: Trivial. Q.E.D.

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THE MAXIMUM PRINCIPLE (MP) I: THE DATA I. We are given I.a: I.b: I.c: I.d: I.e:

a state space Ω, a control set U , a time interval [a, b], a class of admissible controls U , consisting of functions η : [a, b] 7→ U , a controlled dynamical law, i.e., a differential equation x˙ = f (x, u, t) ,

I.f: I.g:

an initial condition, i.e., a point x∗ ∈ Ω, a reference trajectory-control pair, i.e., a pair (ξ∗, η∗), such that η∗ ∈ U , ξ∗ ∈ W 1,1([a, b]; Ω), and the curve ξ∗ satisfies ξ∗(a) = x∗ and ξ˙∗(t) = f (ξ∗(t), η∗(t), t)

I.h:

(5)

for a.e. t ∈ [a, b] ,

a subset S of Ω. 29

THE MAXIMUM PRINCIPLE (MP) II: THE MAPS One then defines: II.a: the (possibly set-valued) control-to-trajectory maps C−T Rx∗ : U 7→ W 1,1([a, b], Ω) , that assign to each control η ∈ U the trajectory (or the set of trajectories) ξ for η with initial condition ξ(a) = x, i.e., the solution (or set of solutions) ξ of ˙ ξ(t) = f (ξ(t), η(t), t) ξ(a) = x∗ .

a.e ,

II.b: the endpoint map E : W 1,1([a, b], Ω) 7→ Ω, defined by E(ξ) = ξ(b)

for , ξ ∈ W 1,1([a, b], Ω) .

II.c the (also possibly set-valued) control-to-terminal-point maps C−T P x∗ : U 7→ Ω defined by C−T P x∗ = E ◦ C−T Rx∗ . 30

THE MAXIMUM PRINCIPLE (MP) III: THE CONCLUSION The (MP) gives a sufficient condition for the following transversal intersection property:

(TI)

The PCM (U , C−T P x∗ , ξ∗(b)) meets (S, idS , ξ∗(b)) transversally.

Actually, (MP) gives a sufficient condition for a stronger property, namely, finite-dimensional transversal intersection

There exists a continuous map V from the unit (FDTI) ball Bn to U such that V (0) = ξ∗(b) such that (Bn, C−T P ◦V, 0) and (S, idS , ξ∗(b)) transversally. 31

THE MAXIMUM PRINCIPLE (MP) IV: THE SUFFICIENT CONDITION The sufficient condition has the form

There does not exist a nontrivial Hamiltonianmaximizing adjoint vector λ along (ξ∗, η∗) that satisfies the transversality condition (SC)

−λ(b) ∈ C ⊥ , where C is an approximating cone of S at ξ∗(b) and C ⊥ is the polar cone of C.

(Naturally, this requires having a topology on U . This will be discussed later.)

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EXAMPLE 1: LOCAL CONTROLLABILITY Assume S is just the single point ξ∗(b). (U , C−T P x∗ , ξ∗(b)) to ξ∗(b) implies:

(LC)

i.e.,

Then transversality of

The reachable set from ξ∗(a) over [a, b] contains a full neighborhood of ξ∗(b),

local controllability along ξ∗.

REASON: Every suffiently small perturbation of ξ∗(b) must meet the reachable set. That is, the reachable set must contain every point in some ε-neighborhood of ξ∗(b).

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EXAMPLE 2: OPTIMALITY R Assume we want to minimize an integral ab L(ξ(t), η(t), t)dt among all trajectory-control pairs (ξ, η) such that ξ(b) ∈ S.

In this case we apply the MP to the augmented control system ˙ ξ(t) = f (ξ(t), η(t), t) , ξ˙0(t) = L(ξ(t), η(t), t) , and the terminal set ˆ = (−∞, c∗] × S , S where c∗ is the cost along the reference trajectory, that is, c∗ =

Z b a

L(ξ∗(t), η∗(t), t)dt .

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\ Transversality of the control-to-terminal-point map C−T P x∗,0 of the \ ˆ implies that C−T agumented system to the set S P x∗,0 meets every ˆ In particular, sufficiently small perturbation of the identity map of S. we can consider the map (k, s) 7→ (k − ε, s) \ Then C−T P x∗,0 meets this map, so there \ exists a point (c, q) ∈ C−T P x∗,0(U ) such that c = k − ε, q = s for ˆ some (k, s) ∈ S. for a small positive ε.

Then k ≤ c∗, so c < c∗. So q is reachable from ξ∗(a) over [a, b] with cost c < c∗, and q ∈ S. So (ξ∗, η∗) is not optimal. Hence the MP gives a sufficient condition for non-optimality, which is of course equivalent to a necessary condition for optimality.

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The necessary condition for optimality has the form:

There exist a nontrivial Hamiltonianmaximizing adjoint vector (ˆ λ0, λ) along (NCO) (ξ0,∗, ξ∗, η∗) that satisfies the transversality condition −ˆ λ(b) ∈ C ⊥ , −ˆ λ0 ≥ 0 , If we write λ0 = −ˆ λ0 (so the control theory Hamiltonian H beecomes H = hλ, f (x, u, t)i − λ0L(x, u, t) (i.e., “H =momentum times velocity minus Lagrangian”, as in Physics), then the transversality condition for λ0 takes the familiar form λ0 ≥ 0 .

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So we see that in both cases, local controllabilty and optimal control, the transversality version of the MP implies, but is not equivalent to, the usual versions, namely, a. a sufficient condition for local controllability along a trajectory, b. a necessary condition for optimality. We now explore one of the stronger consequences of the MP.

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THE APPROXIMATION PROBLEM

PROBLEM: Given a dense subset U0 of the set U of admissible controls, we want to know which trajectories ξ, corresponding to controls η ∈ U , can be approximated by trajectories corresponding to a control in a subset U0 of U with the same endpoints as ξ.

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Obviously, for this question to make sense we need, to begin with, an f -adequate topology on U , i.e., a topology on such that

1. The “initial condition and control to trajecory map”, i.e. the map

(C)

Ω × [a, b] × U ∋ (x, t, η) 7→ ξx,t,η ∈ C 0([a, b], Ω) (6) (where ξx,η is the trajectory for the control η such that ξ(t) = x) is continuous. 2. The map U × U × [a, b] ∋ (η, ζ, t) 7→ ζ#tη ∈ U ,

(7)

where  

(ζ#tη)(s) = 

η(s) ζ(s)

if if

s
is continuous.

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With any such topology,

1. the “packets of needle variations” used is the classical proof of the Maximum Principle, which are maps Rm +(ε∗ ) ∋ (ε1 , . . . , εm ) 7→ V (ε1 , . . . , εm ) ∈ U n : ε + . . . + ε ≤ ε } ), are (where Rm (ε ) = { (ε , . . . , ε ) ∈ R ∗ m m ∗ 1 1 + + continuous,

2. the “parameter-to-trajectory” maps C−T R ◦ V are continuous,

3. the “parameter-to-terminal-point” maps C−T P ◦ V are continuous.

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A natural conjecture (F. Rampazzo, M. Palladino) is:

If U0 is dense in U in an appropriate way, then for every trajectory-control pair (ξ∗, η∗) that satisfies (SC), there exist controls ηn ∈ U0 (RP) that converge to η∗ aad are such that the correesponding trajectories ξn with initial condition ξn(a) = ξ∗(a) satisfy ξn(b) = ξ∗(b). There does not exist a nontrivial Hamiltonianmaximizing adjoint vector λ along (ξ∗, η∗) that (SC) satisfies the transversality condition −λ(b) ∈ C ⊥, where C is an approximating cone of S at ξ∗(b) and C ⊥ is the polar cone of C. 41

A natural conjecture is:

If U0 is dense in U in an appropriate way, then for every trajectory-control pair (ξ∗, η∗) that satisfies (SC), there exist controls ηn ∈ U0 (RP) that converge to η∗ aad are such that the correesponding trajectories ξn with initial condition ξn(a) = ξ∗(a) satisfy ξn(b) = ξ∗(b). There does not exist a nontrivial Hamiltonianmaximizing adjoint vector λ along (ξ∗, η∗) that (SC) satisfies the transversality condition −λ(b) ∈ C ⊥, where C is an approximating cone of S at ξ∗(b) and C ⊥ is the polar cone of C. 42

The conjecture is true, if the word “dense” is interpreted in an appropriate way” Just “dense” is not enough.

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EXAMPLE: Consider the system x˙ = u,

−1 ≤ u ≤ 1 ,

on R. Let U be the class of all measurable functions η : [0, 1] 7→ [−1, 1]. Then 0 is reachable from 0 over the interval [0, 1], using any control η : [0, 1] 7→ [−1, 1] that satisfies Z 1 0

η(t)dt = 0 .

(8)

Let U0 be the class of all η ∈ U that do not satisfy (8). Then U0 is obviously dense in U , but no trajectory for a control in U0 will go from 0 to 0 over [0, 1]. But: the trajectory-control pair (0, 0) satisfies (SC). (Reason: The Hamiltonian H(x, λ, u) is λu. For this to be maximimzed at u = 0 we need λ = 0. So (SC) holds.) CONCLUSION: we need something stronger that “density”. 44

In the previous example, the set U0 is the complement of B, where R1 B = { η ∈ U : 0 η(s)ds = 0 }. So U0 is the complement of the “bad” set B, where B is defined by R1 one condition, namely, 0 η(s)ds = 0. So B is a subset of U of codimension one. This means that in general a continuous curve γ : [0, 1] 7→ U cannot be approximated by curves γ0 : [0, 1] 7→ U0. In particular, it is easy to construct continuous curves γ : [−1, 1] 7→ U such that the control γ(s) steers 0 to s in time 1. (Just let γ(s) be the constant control with value s.) If we could approximate one of these curves by a U0-valued curve γ0, this would give us a continuous curve [−1, 1] ∋ s 7→ C−T P 0(γ0(s)) of terminal points and, by continuity, one of these terminal points would have to be 0. But we cannot approximate γ by U0-valued curves. 45

This suggests an idea: The conjecture should be true if U -valued curves, or, more generally, continuous maps from finite-dimensional balls Bν to U , can be approximated by U0-valued maps. So the extra hypothesis on U0 must be:

U0 is the complement of a subset of sufficiently (EH) high codimension. Let us make this precise.

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DEFINITION: Assume that 1. X is a topological space, 2. X0 is a subset of X, 3. ν is a nonnegative integer. We say that X0 is ν-dense in X if (*) C 0(Bν , X0) is dense in C 0(Bν , X), i.e., if (**) every continuous map θ : Bν 7→ X is a uniform limit of continuous maps θα : Bν 7→ X0. REMARK: If M is a smooth manifold and S is a smooth submanifold of M , then the complement of S is ν-dense if codim(S) > ν.

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EXAMPLES:

1. X0 is 0-dense iff X0 is dense.

2. X0 is 1-dense in X if every curve in X can be appproximated by curves in X0. 3. If X = R2, and X0 is the complement of a line, then X0 is dense but is not 1-dense. 4. On the other hand, if X = R2 and X0 is the complement of a point then X0 is 1-dense but not 2-dense.

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DEFINITION (Jack Warga): Assume that 1. X is a topological space, 2. X0 is a subset of X, We say that X0 is an abundant subset of X is X0 is νdense in X for every nonnegative integer ν. A sufficient condition for a subset X0 to be abundant in X is:

(R)

The identity map idX is a limit of continuous maps Φα : X 7→ X0, uniformly on compact sets.

The precise meaning of the convergence condition in (R), for a net {Φα}α∈A, is: For every compact subset K of X and every neighborhood Ω in X × X of the set {(x, x) : x ∈ K}, there exists α∗ such that {(x, Φα(x)) : x ∈ K} ⊆ Ω whenever α∗  α. 49

THIS WORKS

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EXAMPLE: RELAXED CONTROLS Assume that 1. U is a compact metrizable space. 2. U0 is the class of all measurable maps η : [a, b] 7→ U . 3. f (x, u, t) is (jointly) continuous with respect to x, u for each fiexed t, 4. f (x, u, t) is measurable with respect to t for each fixed x, u, 5. f satisfies Carath´ eodory-Lipschitz bounds kf (x, u, t)k ≤ CK (t) , kf (x, u, t) − f (y, u, t)k ≤ CK (t)kx − yk

(9) (10)

for all (x, y, u, t) ∈ K × K × U × [a, b], for every compact subset K of Ω, where CK ∈ L1([a, b], R) .

THE RELAXED CONTROL VALUES We let P(U ) be the set of all Borel probablity measures on U , and for µ ∈ P(U ), we define f (x, µ, t) =

Z

U

f (x, u, t)dµ(u) .

Then P(u) is a weak∗-closed, bounded subset of C 0(U, R)⊥, the dual of the Banach space C 0(U, R). So P(u), equipped with the weak∗ topology, is compact and metrizable.

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THE RELAXED CONTROLS We let U be the set of all bounded, measurable functions from [a, b] to P(U ). (The members of U are the “relaxed controls”.) Then U has a natural topology TU , namely, the weakest topology on U that makes all the maps U ∋ η 7→

Z bhZ a

U

i

ϕ(u, t)dη(t)(u) dt

continuous, for all functions ϕ ∈ L1([a, b], C 0(U, R)), i.e., all functions U × [a, b] ∋ (u, t) 7→ ϕ(u, t) ∈ R that are continuous with respect to u for each t, measurable with respect to t for each u, and such that Z b a

kϕ(·, t)ksupdt < ∞ .

(Basically, U is a subset of the unit ball of L∞([a, b], C 0(U, , R)⊥), the dual of L1([a, b], C 0(U, R)). And Tf is the weak∗ topology arising from this duality.) 52

The topology TU is an f -adequate control topology. And we get:

THEOREM: U0 is an abundant subset of U . SKETCH OF THE PROOF: Take your favorite way of approximating relaxed controls by ordinary ones, and verify that it yields a sequence {Φk }∞ k=1 of continuous maps Φk : U 7→ U0 such that Φk (η) → η uniformly as k → ∞.

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Then, using the Maximum Principle, we get:

THEOREM: If a trajectory-control pair (ξ∗, η∗) for a relaxed control η∗ is such that there does not exist a nontrivial Hamiltonian-maximizing adjoint vector that satisfies the transversality condition −λ(b) ∈ C ⊥ (where C is an approximating cone to S), then ξ∗ can be approximated by trajectories ηk ccrresponding to ordinary controls ηk in such a way that ξk (b) ∈ S.

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PROOF: The hypothesis implies that there exists a control vaiation V : Rν+(ε∗) 7→ U such that the map C−T P ◦ V meets S transversally. Let Vk be continuous maps from Rν+(ε∗) to U0 that converge to V . Then for sufficienctly large k the maps C−T P ◦ Vk are small perturbations of C−T P ◦ V , so they meet S.

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