Subriemannian minimizers H. J. Sussmann Department of Mathematics - - PowerPoint PPT Presentation
Subriemannian minimizers H. J. Sussmann Department of Mathematics Rutgers University Piscataway, NJ 08854, USA sussmann@math.rutgers.edu Conference on Nonlinear Partial Differential Equations and Applications In honor of Jean-Michel
Department of Mathematics — Rutgers University Piscataway, NJ 08854, USA sussmann@math.rutgers.edu
Conference on Nonlinear Partial Differential Equations and Applications In honor of Jean-Michel Coron’s 60th birthday Paris, June 20-24, 2016
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A vector distribution on a smooth manifold M is a vector subbundle E of the tangent bundle TM of M. (That is, E assigns to each x ∈ M a linear subspace E(x) of the tangent space TxM, in such a way that the dimension of E(x) is the same for all x ∈ M. The fiber dimension of E is the dimension of the spaces E(x). A section of a vector distribution E on M over an open subset Ω of M is a vector field V on Ω such that V (x) ∈ E(x) for each x ∈ Ω. If κ = ∞ or κ = ω, and M is of class Cκ, we use Γκ(E, Ω) to denote the set of all sections of E over Ω that are of class Cκ. ———————
In this lecture,
b.“smooth” means “of class C∞”,
bundles of M. If x ∈ M, then TxM, T ∗
xM are, respectively, the tangent and cotangent spaces
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The vector distribution E is smooth if E is a smooth submanifold
(Equivalently, E is smooth iff for every x ∈ M and every v ∈ E(x) there exists a smooth section V on some neighborhood Ω
If M is real analytic, then the vector distribution E is real analytic if E is a real analytic submanifold of TM. (Equivalently, E is real analytic iff for every x ∈ M and every v ∈ E(x) there exists a real analytic section V on some neighborhood Ω of x such that V (x) = v.) A TRIVIAL WELL-KNOWN FACT: If E is a smooth (or real ana- lytic) distribution on M with fiber dimension d, then for every x ∈ M there exists a local basis of smooth (or real analytic) sections of E near x, that is, a d-tuple (X1, . . . , Xd) of smooth (or real analytic) sections of E defined on an open neighborhood Ω of x such that (X1(y), . . . , Xd(y)) is a basis of E(y) for every y ∈ Ω.
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A global section of E is a section of E over M. A WELL-KNOWN FACT: If E is a smooth distribution on M and dim M = n, then E has 2n+1 smooth global sections X1, . . . , X2n+1 such that the vectors X1(x), . . . , X2n+1(x) linearly span E(x) for each x ∈ M. (Proof: Use Whitney’s mbedding theorem.) ANOTHER WELL-KNOWN FACT: If E is a real analytic distribu- tion on the real analytic manifold M and dim M = n, then E has 4n + 2 real analytic global sections X1, . . . , X4n+2 such that the vec- tors X1(x), . . . , X4n+2(x) linearly span E(x) for each x ∈ M. (Proof:
Use the Morrey-Grauert embedding theorem.)
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A smooth Riemannian metric on a smooth distribution E on a smooth manifold M is a map M ∋ x → Gx such that (i) each Gx is a strictly positive definite symmetric bilinear form on E(x), (ii) whenever V, W are smooth sections on E over an open subset Ω of M, it follows that the function Ω ∋ x → Gx(V (x), W(x)) is smooth. If M and E are real analytic then the metric G is real analytic if (ii’) whenever V, W are real analytic sections on E over an open subset Ω of M, it follows that the function Ω ∋ x → Gx(V (x), W(x)) is real analytic.
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A WELL-KNOWN FACT: If E is a smooth (or real analytic) dis- tribution on the smooth (or real analytic) manifold M and G is a smooth (or real analytic) Riemannian metric on E, then G is the restriction to E of a smooth (or real analytic) Riemannian metric on M (that is, a Riemannian metric on the full tangent bundle TM). A TRIVIAL WELL-KNOWN FACT: If E is a smooth (or real ana- lytic) distribution on M with fiber dimension d, and G is a smooth (or real analytic) Riemannian metric on E, then for every x ∈ M there exists an orthonormal local basis of smooth (or real analytic) sections of E near x, that is, a d-tuple (X1, . . . , Xd) of smooth (or real analytic) sections of E defined on an open neighborhood Ω of x such that, for every y ∈ Ω, (X1(y), . . . , Xd(y)) is a basis of E(y) for which Gy(Xi(y), Xj(y)) = δij .
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If L is any linear space of smooth vector field on a manifold M, and x ∈ M, we define L(x) def = {V (x) : V ∈ L} . We say that L has full rank at x if L(x) = TxM. If S is any set of smooth vector fields on M, then L[S] will denote the Lie algebra of vector fields generated by S, that is, the smallest Lie algebra (over R) of vector fields that contains S. A smooth distribution E of fiber dimension d on a manifold M is bracket-generating if one of the following equivalent conditions hold: (1) L[Γ∞(E, M)] has full rank at every x ∈ M . (2) For every x ∈ M, if (X1, . . . , Xd) is any basis of sections of E defined on a neighborhood Ω of x, then L[X1, . . . , Xd] has full rank at x.
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A smooth subriemannian manifold is a triple (M, E, G) such that (1) M is a smooth manifold, (2) E is a smooth vector distribution on M, (3) G is a smooth Riemannian metric on E. Naturally, we call (M, E, G) real analytic if M, E and G are real analytic.
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If (M, E, G) is a subriemannian manifold, an admissible arc is an absolutely continuous arc ξ : [a, b] → M, defined on some compact interval [a, b], such that ˙ ξ(t) ∈ E(ξ(t)) for almost every t ∈ [a, b] . If we define the G-length vG of a tangent vector v ∈ TxM by vG =
if v ∈ E(x) , vG = +∞ if v / ∈ E(x) , and let the G-length of an arbitrary absolutely continuous arc ξ : [a, b] → M be the (finite or infinite) number ξG
def
=
b
a ˙
ξ(t)Gdt , then it is easy to see that an absolutely continuous arc ξ : [a, b] → M is admissible if and
ξG < ∞ .
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If M = (M, E, G) is a subriemannian manifold, and ξ : [a, b] → M is an absolutely continuous arc in M, we use ∂ξ to denote the ordered pair (ξ(a), ξ(b)), and refer to ∂ξ as the endpoint value, or boundary value, of ξ. We then define the distance dM(x, y) between two points x, y of M to be the infimum of ξG, taken over all absolutely continuous arcs ξ in M such that ∂ξ = (x, y). A WELL-KNOWN FACT: If M is connected, then (M, dM) is a metric space, whose topology is the same as the manifold topology
bracket-generating condition and Chow’s Theorem imply that dM(x, y) < ∞ for all x, y.)
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An absolutely continuous arc ξ : [a, b] → M is a length minimizer if ξG = dM(∂ξ). An admissible arc ξ : [a, b] → M is parametrized by arc-length, or a PAL arc, if ˙ ξ(t)G = 1 for almost all t ∈ [a, b]. TRIVIAL FACT: Every admissible arc can be reparametrized to be- come a PAL arc. (Reason: Just use arc-length as the new time parameter.)
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REMARK: In the real analytic case, for the purpose of the results considered here, we may always assume, without loss of generality, that the bracket-generating condition holds even if it originally does not hold. The reason is: through every point of M there passes a unique maxi- mal integral submanifold of the Lie algebra of vector fields generated by the real analytic global sections of E. And, in addition: all the results of this talk are about regularity of distance-minimizing arcs, and every such arc is entirely contained in an integral manifold, so in order to study such arcs we can restrict
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To study the smoothness problem, it suffices to work with sufficiently short arcs, so we are allowed to assume
It follows that the admissible arcs are exactly the arcs ξ that are trajectories of the control system ˙ x =
d
uifi(x) . (1)
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Precisely, an arc ξ : [a, b] → M is admissible if and only if it is absolutely continuous and satisfies ˙ ξ(t) =
d
ηi(t)fi(ξ(t)) for a.e. t , (2) for some integrable function η = (η1, . . . , ηd) : [a, b] → Rd. Furthermore, ξ is PAL if and only if d
i=1 ηi(t)2 = 1 for a.e. t.
And, for a PAL arc ξ : [a, b] → M, the length of ξ is exactly b − a. So the PAL minimizers are the minimum time arcs for the control system (1) with contol constraint d
i=1 ui(t)2 = 1.
For convenience, we use instead the control constraint
d
ui(t)2 ≤ 1 . (3) (This does not change the minimizers.)
So, from now on we study the minimum-time trajec-
d
d
i=1 u2 i ≤ 1 .
trajectory for the system ˙ x =
d
uifi(x) , x ∈ M ,
d
u2
i ≤ 1 ,
(5) and η = (η1, . . . , ηd) is the corresponding control, then
d
d
i ≤ 1 .
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xM, and p = 0.
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xM .
n
n
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d
i=1 uifi(x)
d
i=1 uifi(x) ∈ TxM.
d
xM, p = 0.
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d
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d
j=1 u2 j ≤ 1}.)
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Therefore, if (ξ, η) is an optimal trajectory-control pair, there exist an adjoint vector π and a constant π0 such that π0 = Hη(t)(Ξ(t)) = max{Hu(Ξ(t)) : u ∈ Bd} for a.e. t . Since u → Hu(X) is a linear function d
j=1 ujaj(X) for each X, the
maximum of this function on the unit ball Bd is attained for uj = aj(X)
d
k=1 ak(X)2
if
d
ak(X)2 = 0 , and u an arbitrary member of Bd if d
k=1 ak(X)2 = 0.
Furthermore, the maximum value of Hu(X) is
d
k=1 ak(X)2.
Since aj(X) = p, fj(x) if X = (x, p), we obtain
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d
j=1π(t), fj(ξ(t))2 is constant
d
k=1π(t), fj(ξ(t))2 = 1
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A Hamiltonian-maximizing adjoint vector π along a trajectory-control pair (ξ, η) is normal if π0 > 0, and abnormal if π0 = 0. A minimizer (ξ, η) is normal if it has a normal Hamiltonian-maximizing adjoint vector, and abnormal if it has an abnormal Hamiltonian- maximizing adjoint vector.
(Notice that a minimizer can be both normal and abnormal.)
The minimizer (ξ, η) is strictly abnormal if it is abnormal and not normal (that is, if all the Hamiltonian-maximizing adjoint vectors along (ξ, η) are abnormal). Then the following follows from the PMP:
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This is because if (ξ, η) is a normal minimizer, and π is a normal Hamiltonian-maximizing adjoint vector along (ξ, η), then the pair (ξ, π) is (in coordinates) a solution of the system of ordinary differ- ential equations ˙ ξ = 1 π0
d
π, fj(ξ)fj(ξ) , ˙ π = − 1 π0
d
π, fj(ξ)∂fj ∂x (ξ) , from it follows easily that ξ and π are smooth functions. And then the control η is also smooth, because ηj(t) = 1 π0 π(t), fj(ξ(t)) .
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REMARK: The previous result contains, in particular, the theorem
That’s because in the Riemannian case every minimizer is normal. Indeed, if d = n = dim M, then the sum d
j=1π(t), fj(ξ(t))2 cannot
vanish, because if it did then all the numbers π(t), fj(ξ(t)) would vanish, so π(t) would be the zero covector, because the vectors fj(ξ(t)) span the tangent space Tξ(t)M. But π(t) = 0, because (π(t), ξ(t)) ∈ T #M.
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Clearly, now, the problem of the smoothness of optimal trajectories reduces to that of the smoothness of strictly abnormal extremals. For this problem, there is another useful necessary condition called the Goh condition:
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Is it possible that the Goh condition can be generalized to the fol- lowing statement? If (ξ, η) is a strictly abnormal minimizer, then there exists a Hamiltonian-maximizing adjoint vector π along (ξ, η) such that π(t), [[fj1, [fj2, . . . , [fjk−1, fjk] . . .]](ξ(t)) = 0 for all t, all k, and all (j1, . . . , jk). The answer is NO, because if such a condition was true, it would follow from the bracket-generating condition that strictly abnormal minimizers do not exist. But:
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The proof is by induction on the fiber dimension d. The case d = 1 is trivial. So we assume that the theorem is true for fiber dimensions ≤ d − 1, and prove it’s true for fiber dimension d.
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d
i=1 ui
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d
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Let P be a stratification of T #M having the properties of the lemma. Suppose (ξ, η) is a PAL trajectory-control pair which is a minimizer, and is defined on an interval [a, b]. It suffices to prove that there exists τ such that a < τ < b and ξ is real analytic on a neighborhood ]τ − ε, τ + ε[ of τ. We may assume that (ξ, η) is strictly abnormal. Let π be a Hamiltonian-maximizing adjoint vector, and let Ξ = (ξ, π) be the corresponding Hamiltonian lift. Then Ξ is entirely contained in the abnormal set A. So each point Ξ(t) of Ξ belongs to a unique stratum S(t) of P, and S(t) ⊆ A. Let δ(t) = dim S(t).
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Pick ¯ t such that δ(¯ t) = max{δ(t) : a < t < b}. Let S = S(¯ t). We claim that Ξ(t) ∈ S for t in some neighborhood of ¯ t. To see this suppose otherwise. Then there exists a sequence {tk}∞
k=1,
converging to ¯ t, such that S(tk) = S for every k. Since P is locally finite, the set of all the S(tk) is finite, so we may assume, after passing to a subsequence, that all the S(tk) are one and the same stratum ˇ S. Naturally, ˇ S = S. Since the point Ξ(¯ t) is in S, so Ξ(¯ t) / ∈ ˜ S, but Ξ(¯ t) is a limit of points of ˇ S, it follows that Ξ(¯ t) ∈ (Clos ˜ S)\˜ S. So S ⊆ (Clos ˇ S)\ˇ S, and this implies that dim S < dim ˇ
Hence δ(¯ t) < δ(tk) contradicting our choice of ¯ t as a time when δ has its maximum value.
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We now know that, for some positive ε,
Since S is a nice stratum, the spaces US(X), for all X ∈ S, have the same dimension ¯ d, and there exist real analytic maps ej : S → Rd, for j = 1, . . . , ¯ d, such that e(X)def = (e1(X), . . . , e¯
d(X))
is a basis of US(X) for each X. Furthemore, we can apply the Gram-Schmidt orthogonalization procedure and assune that e(X) is an orthonormal basis of US(X).
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Let ΞI be the restriction of Ξ to I. Then ΞI is a trajectory of the system ˙ X =
d
uj Fj(X) , but the vectors ˙ ΞI(t), for t ∈ I, are tangent to S. Therefore the control η is such that η(t) ∈ US(ΞI(t)) for each t ln I, and then η(t) is a linear combination η(t) =
¯ d
ck(t)ek(t) , with coefficients ck(t) such that ¯
d k=1 ck(t)2 = 1.
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So ΞI is a PAL trajectory of the system ˙ X =
¯ d
vkGk(X) , (7) with state space S and control constraint ¯
d k=1 v2 k ≤ 1.
(Here Gk = ek · F.) Furthermore, ΞI is a time-optimal trajectory of (7). (This is easy.) We now prove the
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To prove this, assume ¯ d = d. (¯ d cannot be larger than d, because ¯ d = dim US(X), and US(X) is a subspace of Rd.) This would mean that US(X) = Rd for every X ∈ S. Then every vector field Fj would be tangent to S. Then every iterated Lie bracket [ Fi1, [ Fi2, . . . , [ Fiℓ−1, Fiℓ] . . .]] would be tangent to S as well. Since the momentum functions Fi vanish on S (because S ⊆ A), it follows that the directional deriva- tives [ Fi1, [ Fi2, . . . , [ Fiℓ−1, Fiℓ] . . .]]Fj vanish on S. That is, all the iterated Poisson brackets Ψ = {{Fi1, {Fi2, . . . , {Fiℓ−1, Fiℓ} . . .}}, Fj} vanish.
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But Ψ(x, p) = p, [[fi1, [fi2, . . . , [fiℓ−1, fiℓ] . . .]], fj](x) . Hence p vanishes against all the iterated brackets of the fk. By the bracket-generating assumption, it follows that p = 0, contradicting the fact that S ⊆ T #M. So ΞI is a PAL solution of a minimum-time optimal control problem exactly like our original problem, except that the fiber dimension of the new problem is smaller. Hence, by the inductive hypothesis, there is a point τ of I such that ΞI is real analytic near ¯ τ. And then of course, ξ is real analytic near ¯ τ.
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Can one do something similar in the smooth (i.e., C∞) case?. Yes, but only (so far) for d = 2.
REMARK: This argument does not seem to work for d > 2.
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PROOF: Write our system (locally) as ˙ x = u1f1(x) + u2f2(x) , x ∈ U , u2
1 + u2 2 ≤ 1 ,
where U is an open subset of Rn. Let B be the set of all iterated Lie brackets B of the form B = [fim, [fim−1, [· · · , [f2, f1] · · ·]]] where each ij is 1 or 2. (The number m is the degree of B.) It is easy to see, using the Jacobi identity, that every iterated of f1 and f2 is a linear combination of the brackets in B. (For example, [[f1, f2], [f1, [f1, f2]]] = [f1, [f2, [f1, [f1, f2]]]] − [f2, [f1, [f1, f1, f2]]] , and so on.)
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To each vector field X on Rn, associate the momentum function µX : Rn × Rn → R , given by µX(p, x) = p† · X(x) . (Here p is a covector, and Rn × Rn is the cotangent bundle of the state space.) Then there is a simple formula for differentiating a momentum func- tion µX along Hamiltonian lifts of trajectories. Roughly: ˙ µX = η1µ[f1,X] + η2µ[f2,X] .
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More precisely, if ξ : [a, b] → Rn is a trajectory, corresponding to a control [a, b] ∋ t → η(t) = (η1(t), η2(t)) ∈ R2 , and Ξ : [a, b] → Rn × Rn , is a Hamiltonian lift of ξ, given by Ξ(t) = (π(t), ξ(t)) , where ˙ π(t) = −η1(t)π(t)† · ∂f1 ∂x (ξ(t)) − η2(t)π(t)† · ∂f2 ∂x (ξ(t)) we have d dt
for every smooth vector field X
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If (ξ, η) is a PAL minimizer, then it is either a normal extremal, in which case it is smooth, or a strictly abnormal one, in which case there exists a Hamiltonian lift Ξ = (π, ξ) such that µf1(Ξ(t)) ≡ µf2(Ξ(t)) ≡ µ[f1,f2](Ξ(t)) ≡ 0 . For every t ∈ [a, b], there exists a B ∈ B such that µB(Ξ(t)) = π(t)† · B(ξ(t)) = 0 . (Reason: if π(t)† · B(ξ(t)) = 0 for every B ∈ B, then π(t)†v = 0 for every v ∈ Rn, so π(t) = 0.) Given t, let m∗(t) be the smallest m such that µB(Ξ(t)) = 0 for some B ∈ B of degree m.
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Pick a t0 ∈ [a, b] such that m∗(t0) ≤ m∗(t) for every t ∈ [a, b] . Let ˆ m = m∗(t0). Let B be a bracket in B of degree ˆ m such that µB(t0) = 0. Write B = [fi ˆ
m, C], where C ∈ B is of degree ˆ
m − 1. Then µC(Ξ(t)) = 0 for all t ∈ [a, b]. Assume i ˆ
m = 1. (The case when i ˆ m = 2 is identical.)
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Then d dt
Therefore η1(t)µ[f1,C](Ξ(t)) + η2(t)µ[f2,C](Ξ(t)) ≡ 0 , that is η1(t)µB(Ξ(t)) + η2(t)µ[f2,C](Ξ(t)) ≡ 0 , Let ε be a positive number such that µB(Ξ(t)) = 0 for t ∈ (t0ε, t0 + ε) . Let I = (t0ε, t0 + ε) . Since η1(t)2 + η2(t)2 = 1, the vector η(t) is determined uniquely, up to a sign, for t ∈ I.
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More precisely: let Ω be the set of points (p, x) ∈ Rn × Rn such that µB(p, x) = 0. Then Ξ(t) ∈ Ω for all t ∈ I. On Ω, define “feedback controls” ϕj : Ω :→ R by letting ϕ1(p, x) = − µ[f2,C](p, x)
ϕ2(p, x) = µB(p, x)
(These are smooth functions on Ω because µB and µ[f2,C] are smooth functions and µB is = 0 on Ω.) Then define a “feedback Hamiltonian vector field” V on Ω by letting V (p, x) = ϕ1(p, x)F1(p, x) + ϕ2(p, x)F2(p, x) , where F1, F2 are the Hamiltonian lifts of f1 and f2.
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Then our Hamiltonian lift Ξ satisfies ˙ Ξ(t) = ±(V (Ξ(t))) for a.e. t ∈ I . So Ξ is “almost” a trajectory of a smooth vector field. So ξ is smooth, if we can make sure that the sign in the previous equation does not change. Furthermore, the sign cannot change, because if it did our trajectory would not be optimal. (Easy proof.) Hence Ξ is smooth on I, and so is ξ. So we have proved that ξ is smooth on a nonempty open subinterval
This result can be applied to the restriction of ξ to any nontrivial subinterval of [a, b], and our conclusion follows.
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One can also prove that for every optimal trajectory the control is
sense:
j=0 of measurable sets
j=1 of smooth feedback vector
∞
The proof is essentially as before:
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µB(Ξ(t)) = 0 but µC(Ξ(t)) = 0 for all C ∈ B of smaller degree.
before.
Lebesgue points of the control η. (This is a huge subset of AB. In particular, AB\SB is a null set.)
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sj in a sequence (sj)∞
j=1 of points that converge to t and are
distinct from t.
µC(Ξ(t)) as the limit as j → ∞
µC(Ξ(sj)) − µC(Ξ(t)) sj − t .
η1(t)µB(Ξ(t)) + η2(t)µ[f2,C](Ξ(t)) = 0 .
Ξ(t) = ± VB(Ξ(t)) .
B∈B SB has measure zero. So we are
done.
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What is to be done next? If you want to prove that all minimizers are smooth, then you must continue the analysis that was started in work by Leonardi and Monti, and more recent work by Hakavuori and Le Donne, who proved that PAL subriemannian minimizers cannot have corners. If, on the other hand, you want to prove that it is not true that all minimizers are smooth, then you have to find an example of a nonsmooth minimizer. This will be, of course, a strictly abnormal extremal. Finding nonsmooth abnormal extremals is easy. And finding lots of strictly abnormal minimizers is also easy. (See HJS “A cornucopia of abnormal minimizers”.)
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The hard part is to prove that a given extremal is actually a mini- mizer. For normal extremals there are methods, especially the construction
The methods of the “cornucopia” paper do work for lots of fam- ilies of abnormal extremals, but I do not know how to construct nonsmooth abnormal extremals to which those methods apply.
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