Análisis de los datos Detección de errores

The goal of this section is to prove Theorem 2.19 on the smoothness of the value function. All the relevant definitions can be found in Chapter 2. As a first step, we generalize the classical definition of conjugate points to our setting.

Definition _{A.1. Let γ : [0, T ] → M be a strictly normal trajectory, such that x}0 = γ(0) and

γ(t) =Ex0(t, λ0). We say that γ(t) is conjugate with x0 along γ if λ0 is a critical point for Ex0,t.

Observe that the relation “being conjugate with” is not reflexive in general. Indeed, even if γ(t) is conjugate with x0, there might not even exist an admissible curve starting from γ(t) and ending at x0.

We stress that, if γ is also abnormal, any γ(t) is a critical value of the sub-Riemannian exponential map. Indeed, this is a consequence of the inclusion Im Dλ0Ex0,t ⊂ Im DuEx0,t 6= Tx0M for abnormal

trajectories; being strongly normal is a necessary condition for the absence of critical values along a normal trajectory. Actually, a converse of this statement is true.

Proposition_{A.2. Let γ : [0, T ] → M be a strongly normal trajectory. Then, there exists an ε > 0} such that γ(t) is not conjugate with γ(0) along γ for all t∈ (0, ε).

The proof of Proposition A.2 in the sub-Riemannian setting can be found in [ABB12] and can be adapted to a general affine optimal control system. See also [AS04] for a more general approach.

We are now ready to prove Theorem 2.19 about smoothness of the value function which, for the reader’s convenience, we restate here. Recall that M′ _{⊂ M is the relatively compact subset chosen for}

the definition of the value function.

Theorem. Let γ : [0, T ] → M′ _{be a strongly normal trajectory. Then there exists an ε > 0 and an}

open neighbourhood U ⊂ (0, ε) × M′_{× M}′ _{such that:}

(i) (t, γ(0), γ(t)) ∈ U for all t ∈ (0, ε),

(ii) For any (t, x, y) ∈ U there exists a unique (normal) minimizer of the cost functional Jt, among

all the admissible curves that connect x with y in time t, contained in M′_,

(iii) The value function (t, x, y) 7→ St(x, y) is smooth on U.

Proof. We first prove the theorem in the case M′ _{= M compact. We need the following sufficient}

condition for optimality of normal trajectory. Let a ∈ C∞_{(M). The graph of its differential is a smooth}

submanifold L0 = {d. xa| x ∈ M} ⊂ T∗M , dimL0 = dim M. Translations of L0 by the flow of the

Hamiltonian field Lτ= eτ ~H(L0) are also smooth submanifolds of the same dimension.

Lemma _{A.3 (see [AS04, Theorem 17.1]). Assume that the restriction π : L}τ → M is a diffeomor-

phism for any τ_{∈ [0, ε]. Then, for any λ}0∈ L0, the normal trajectory

γ(τ ) = π_{◦ e}τ ~H(λ0), τ∈ [0, ε],

is a strict minimum of the cost functional Jεamong all admissible trajectories connecting γ(0) with γ(ε)

in time ε.

Lemma A.3 is a sufficient condition for the optimality of a single normal trajectory. By building a suitable family of smooth functions a ∈ C∞_{(M), one can prove that, for any sufficiently small compact}

set K ⊂ T∗_{M , we can find a ε = ε(K) > 0 sufficiently small such that, for any λ}

0 ∈ K, and for any

t_{≤ ε, the normal trajectory}

γ(τ ) = π_{◦ e}τ ~H(λ0), τ∈ [0, t], t≤ ε

is a strict minimum of the cost functional Jt among all admissible curves connecting γ(0) with γ(t) in

time t.

We sketch the explicit construction of such a family. Let K ⊂ T∗_{M sufficiently small such that}

by a choice of coordinates x on O ⊂ M. Then, consider the function a : K × O → R, defined in coordinates by a(p0, x0; y) = p∗0y. Extend such a function to a : K × M → R. For any λ0 ∈ K, denote

by a(λ0) = a(λ

0; ·) ∈ C∞(M). Indeed, for x0 = π(λ0), we have λ0 = dx0a

(λ0). In other words we can

recover any initial covector in K by taking the differential at x0of an appropriate element of the family.

Therefore, let L(λ0)

0

.

= {dxa(λ0)| x ∈ M}, and Lτ(λ0) = e. τ ~H(L(λ₀0)). M is compact, then there exists

ε(K) = sup_{{τ ≥ 0| π : L}(λ0)

s → M is a diffeomorphism for all s ∈ [0, τ], λ0∈ K} > 0.

Let us go back to the proof. Set x0 = γ(0), and let γ(t) = Ex0(t, λ0). By Proposition A.2, we can

assume that γ(t) is not conjugate with γ(0) along γ for all t ∈ (0, ε). In particular, Dλ0Ex0,thas maximal

rank for all t ∈ (0, ε). Without loss of generality, assume that ~H is complete. Then, consider the map φ : R+

× T∗_{M → R}+

× M × M, defined by

φ(t, λ) = (t, π(λ),Eπ(λ)(t, λ)).

The differential of φ, computed at (t, λ0), is

D(t,λ0)φ =  1 00 I 00 ∗ ∗ Dλ0Ex0,t   , _{∀ t ∈ (0, ε),}

which has maximal rank. Therefore, by the inverse function theorem, for each t ∈ (0, ε), there exist an interval It and open sets Wt, Ut, Vtsuch that

t_{∈ I}t⊂ (0, ε), λ0∈ Wt⊂ T∗M, γ(0)∈ Ut⊂ M, γ(t)∈ Vt⊂ M,

and such that the restriction

φ : It× Wt→ It× Ut× Vt

is a smooth diffeomorphism. In particular, for any (τ, x, y) ∈ It× Ut× Vt there exists an unique initial

covector λ0(τ, x, y) = φ. −1(τ, x, y) such that the corresponding normal trajectory starts from x and

arrives at y in time τ, i.e. Ex(τ, λ0(τ, x, y)) = y. Moreover, we can choose Wt⊂ K. Then such a normal

trajectory is also a strict minimizer of Jτ among all the admissible curves connecting x with y in time

τ . In particular, it is unique.

As a consequence of the smoothness of the local inverse, the value function (t, x, y) 7→ St(x, y) is

smooth on each open set It× Ut× Vt. Indeed, for any (τ, x, y) ∈ It× Ut× Vt, St(x, y) is equal to the cost

Jτ of the unique (normal) minimizer connecting x with y in time τ, namely

Sτ(x, y) = Z τ 0 L(_Ex0(s, λ0(τ, x, y)), ¯u(e s ~H_(λ 0(τ, x, y))))ds, (τ, x, y) ∈ It× Ut× Vt,

where ¯u : T∗_{M → R}k _{is the smooth map which recovers the control associated with the lift on T}∗_M

of the trajectory (see Theorem 2.17). Therefore the value function is smooth on It× Ut× Vt, as a

composition of smooth functions. We conclude the proof by defining the open set U=. [

t∈(0,ε)

It× Ut× Vt⊂ (0, ε) × M × M,

which is indeed open and contains (t, γ(0), γ(t)) for all t ∈ (0, ε).

In the general case the proof follows the same lines, although the optimality of small segments of geodesics is only among all the trajectories not leaving M′_{. If we choose a different relatively compact}

M′′⊂ M, we find a common ε such that the restriction to the interval [0, ε] of all the normal geodesics with initial covector in K is a strict minimum of the cost function among all the admissible trajectories not leaving M′′_{∪ M}′_{. Therefore, the value functions associated with the two different choices of the}

relatively compact subset agree on the intersection of the associated domains U.

APPENDIX B