Keywords: Commande optimale, principe de Pontryagine, conditions d'optimalité du second ordre, condition de croissance quadratique, minima faibles et forts

Analyse au second ordre de problèmes de commande optimale avec contraintes sur la commande et sur l'état initial-final

Résumé : Cet article présente une analyse des minima de Pontryagine satisfaisant une condition de croissance quadratique, pour des problèmes de commande optimale d'équations différentielles avec contraintes sur l'état initial-final et avec des contraintes sur la commande satisfaisant la condition d'indépendance linéaire positive uniforme.

Introduction

In this paper we discuss necessary or sufficient conditions for weak or bounded strong minima of optimal control problems with control constraints and constraint on the initial-final state. There is already an important literature on this subject.

Osmolovskii [START_REF] Osmolovskiȋ | Theory of higher order conditions in optimal control[END_REF][START_REF] Osmolovskiȋ | Quadratic extremality conditions for broken extremals in the general problem of the calculus of variations[END_REF][START_REF] Osmolovskiȋ | Second order sufficient optimality conditions for control problem with linearly independent gradients of control constraints[END_REF] analyzed second-order optimality conditions for such problems assuming linear independence of gradients of active constraints (LIG); see also Levitin,Milyutin and Osmolovskii [6, where these conditions were first formulated.

Malanowski [START_REF] Malanowski | Stability of solutions to convex problems of optimization[END_REF] obtained stability and sensitivity results in the case of convex cost and constraints (including state constraints) assuming the LIG hypothesis. More recently, Hermant and the first author [START_REF] Bonnans | Second-order analysis for optimal control problems with pure state constraints and mixed control-state constraints[END_REF] studied similar problems, without convexity assumption except for the (local) dependence of the Hamiltonian w.r.t. the control variable, and again with the LIG hypothesis.

The main novelty is that we do not assume any more the LIG hypothesis but instead a qualification condition that implies the uniform positive linear independence of gradients of active inequality constraints. Also, we do not assume the (local) convex dependence of the Hamiltonian w.r.t. the control variable, which makes the discussion of sufficient conditions more complex (since the Hessian of the Lagrangian of the problem is not a Legendre form, and so it is no more possible to pass to the limit in weakly convergent directions).

The paper is organized as follows. Section 2 sets the problem, recalls some basic concepts, and gives a decomposition principle, in the setting of abstract control constraints. Section 3 analyzes the multipliers associated with control constraints parameterized by finitely many inequalities. Section 4 proves, under a restoration property (of initial final state constraints), for which a verifiable sufficient condition is provided, that Pontryagines'principle makes the link between weak and bouded strong minima. Finally in section 5 we characterize the quadratic growth condition for weak minima.

Pontryagine minima 2.1 Pontryagine's principle

Let U := L ∞ (0, T ; R m ) and Y := W 1,∞ (0, T ; R n ) denote the control and state space. Set W := U × Y. When needed we denote w = (u, y), w = (ū, ȳ), etc. the elements of W. Similarly we denote when needed η = (y(0), y(T )), η = (ȳ(0), ȳ(T )), etc. the pair of initial-final states. The cost function is defined by

J(w) := T 0 ℓ(u(t), y(t))dt + φ(η), (1)
where ℓ : R m × R n → R (running cost) and φ : R n × R n → R (initial-final cost) are twice continuously differentiable (C 2 ) mappings. Consider the state equation ẏ(t) = f (u(t), y(t)) for a.a. t ∈ [0, T ]; [START_REF] Bonnans | Perturbation analysis of optimization problems[END_REF] where f : R m × R n → R n is a Lipschitz and C 2 mapping. We know that the state equation ( 2) has for any u ∈ U and given initial condition y(0) = y 0 a unique solution denoted y u,y0 ∈ Y. We consider problems having both control constraints u(t) ∈ U, for a.a. t ∈ (0, T ), [START_REF] Cominetti | Metric regularity, tangent sets and second order optimality conditions[END_REF] where U is a closed subset of R m , and initial-final state constraints of the form

Φ(η) ∈ K, with K := {0} R r 1 × R r2 -, (4)
and Φ : R 2n → R r , r = r 1 + r 2 , r 1 and r 2 are nonnegative integers. In other words, there are a finite number of equality and inequality constraints on the initial-final state:

Φ i (η) = 0, i = 1, . . . , r 1 , Φ i (η) ≤ 0, i = r 1 + 1, . . . , r.

Consider the following optimal control problem: Min w∈W J(w) subject to (2)-( 4). (P )

We call trajectory an element w of W that satisfies the state equation [START_REF] Bonnans | Perturbation analysis of optimization problems[END_REF]. If in addition the constraints (3)-( 4) hold, we say that w is a feasible point of problem (P ); the set of feasible points is denoted by F (P ). We briefly recall several concepts of solution. Denote by • s the norm of the space L s (0, T, R q ) for any q, and s ∈ [1, +∞]. A weak (resp. strong) solution of (P ) (or weak, strong minimum) is an element w ∈ F (P ) such that J( w) ≤ J(w) for all w ∈ F (P ) such that w -w W (resp. y -ȳ ∞ ) is small enough. Equivalently, w ∈ F (P ) is a weak (resp. strong) solution of (P ) if, for any sequence w k ∈ F (P ), such that w k → w in W (resp. y k → ȳ uniformly), we have that J( w) ≤ J(w k ) for large enough k.

Following [6, p.156] and [9, p. 291], we say that w ∈ F (P ) is a bounded strong solution (minimum) if for any bounded sequence w k ∈ F (P ), such that y k → ȳ uniformly, we have that J( w) ≤ J(w k ) when k is large enough. An element w of F (P ) is called Pontryagine solution (minimum) see [6, p.156] and [9, p. 2-3], if for any sequence w k ∈ F (P ), bounded in W, such that y k → ȳ uniformly and u k -ū 1 → 0, we have that J( w) ≤ J(w k ) when k is large enough.

Equivalently, w is a bounded strong (Pontryagine) solution if for any M > 0, there exists

ε M > 0 such that if w ∈ F (P ) is such that u ∞ ≤ M , y -ȳ ∞ ≤ ε M (and in addition u -ū 1 ≤ ε M in the case of a Pontryagine solution) we have that J( w) ≤ J(w).
Obviously any of the following concepts is implied by the previous one: strong, bounded strong, Pontryagine, weak solution. We define a strong, bounded strong, Pontryagine, weak perturbation of w ∈ F (P ) as a sequence w k of trajectories in W associated with the corresponding optimality concept, i.e., such that y k → ȳ uniformly, and in addition u k is bounded in L ∞ for the other types of perturbations, u k → u in L 1 (uniformly) for a Pontryagine (weak) perturbation. We say that the perturbation is feasible if the elements of the sequence belong to F (P ). We call δw k := w k -w a strong, bounded strong, Pontryagine, weak variation.

We say that a (strong, bounded strong, Pontryagine, weak) solution w satisfies the quadratic growth condition if there exists α > 0 (depending on M in INRIA the case of a bounded strong or Pontryagine solution) such that w is a solution of the same kind for the cost function

J α (u, y) := T 0 ℓ α (t, u(t), y(t))dt + φ(η), (6)
where

ℓ α (t, u, y) := ℓ(u, y) -1 2 α[|u -ū(t)| 2 + |y -ȳ(t)| 2 ]. (7)
Then we say that the (strong, bounded strong, Pontryagine, weak) quadratic growth condition is satisfied. So for instance the quadratic growth condition for a weak solution w (we speak then of weak quadratic growth) means that There exist α > 0, ε > 0 :

J(w) ≥ J( w) + 1 2 α w -w 2 2 , for all w ∈ F (P ), w -w ∞ < ε, (8)
and the bounded strong quadratic growth condition means that For any M > 0, there exist

α M > 0, ε M > 0 : J(w) ≥ J( w) + 1 2 α M w -w 2 2 for all w ∈ F (P ), y -ȳ ∞ < ε M , u ∞ ≤ M. (9)
We now recall the formulation of Pontryagine's principle at the point w ∈ F (P ). Let us denote by R q * the dual of R q (identified with the set of q dimensional horizontal vectors). We remind that K was defined in (4). The negative dual cone to K (set of vectors of R r * having a nonpositive duality product with each elements of

K) is K -= R r1 * × R r2 * + . We say that (θ, µ) ∈ K × K -is a complementary pair if µ i θ i = 0, for i = 1, . . . , r.
The normal cone to K at the point θ ∈ K is the set of elements of the negative dual cone that are complementary to θ. In particular, the expression of the normal cone to K at Φ(η) is

N K (Φ(η)) := {µ ∈ R r * ; µ i ≥ 0, µ i Φ i (η) = 0, i > r 1 }. ( 10
)
Let the end-point Lagrangian be defined by

Φ µ (y 0 , y T ) := φ(y 0 , y T ) + r i=1 µ i Φ i (y 0 , y T ). (11)
Consider the Hamiltonian function

H : R m × R n × R n * → R defined by H(u, y, p) := ℓ(u, y) + pf (u, y). (12)
Set P := W 1,∞ (0, T ; R n * ). For any µ ∈ R r * and p ∈ P, consider the following set of relations (reminding that η = (ȳ(0), ȳ(T ))):

(i) -ṗ(t) = H y ( w(t), p(t)), for a.a. t ∈ (0, T ); (ii) p(T ) = Φ µ yT (η); (iii) p(0) = -Φ µ y0 (η). ( 13
)
We call costate associated with µ at the point w ∈ F (P ), and denote by p µ , the unique solution in P of the backward equation ( 13)(i-ii). Existence and uniqueness of the costate follow from the fact that this is a Cauchy problem for a linear o.d.e. with measurable and bounded coefficients. Obviously the mapping µ → p µ is affine. We will obtain (13)(iii) as a necessary optimality condition.

Definition 2.1. Let w ∈ F (P ). We say that µ ∈ N K (Φ(η)) is a (regular) Pontryagine multiplier associated with w if the associated costate p µ satisfies (13)(iii), and is such that the following Hamiltonian inequality holds:

H(ū(t), ȳ(t), p µ (t)) ≤ H(v, ȳ(t), p µ (t)), for all v ∈ U, a.a. t ∈ (0, T ). ( 14
)
We denote by M P ( w) the set of Pontryagine multipliers associated with w; if this closed convex set is non empty, we say that w satisfies Pontryagine's principle (in qualified form), or that w is a Pontryagine extremal.

Remark 2.2. Let w be a Pontryagine extremal, and let µ ∈ M P ( w). We know (see e.g. [9, p. 24-25]) that there exists a constant c µ ∈ R such that

c µ := inf v∈U H(v, p(t), p µ (t)), for all t ∈ [0, T ]. ( 15
) By (13)(i-ii), the function µ → c µ is affine. Set h(v, µ, t) := H(v, ȳ(t), p µ (t)), t ∈ (0, T ). ( 16
)
By ( 14), we have that h(ū(t), µ, t) = c µ for a.a. t ∈ (0, T ). Define

U M := U ∩ B(0, M ), where M > ū ∞ . (17)
Let us show that there exists a representative ũ of ū such that

ũ(t) ∈ U M and h(ũ(t), µ, t) = c µ , for all t ∈ [0, T ]. ( 18
) Let t ∈ [0, T ]. If h(ū( t), µ, t) = c µ and ū( t) ∈ U M , let ũ( t) := ū( t).
Otherwise, since h(ū(t), µ, t) = c µ a.e., there exists a sequence t k ∈ [0, T ], t k → t such that ū(t k ) ∈ U M and h(ū(t k ), µ, t k ) = c µ . Extracting if necessary a subsequence, we may assume that ū(t k ) converges to some limit whose value will be ũ( t). Passing to the limit in the relation h(ū(t k ), µ, t k ) = c µ , we deduce that h(ũ( t), µ, t) = c µ so that h(ũ(t), µ, t) = c µ for all time. Clearly ũ( t) ∈ U M , and hence, (18) holds.

It is known that Pontryagine's principle, in a non qualified form, is satisfied by Pontryagine solutions of (P ), see [9, p. 24]. The qualified form is satisfied under some qualifications conditions to be presented in the next section.

Hamiltonian functions with a unique minimum

If A is a convex subset of a finite-dimensional space, we denote by ri(A) its relative interior, in the sense of convex analysis (the interior of A, in the topology induced by its affine hull). We check below that a relatively interior Pontryagine multiplier (i.e., some µ ∈ ri(M P ( w))) obtains an increase of Hamiltonian of the same growth rate as the maximum over bounded sets of Pontryagine multipliers: Lemma 2.3. Let w ∈ F (P ) satisfy Pontryagine's principle, and let M C ( w) be a nonempty, convex and compact subset of M P ( w). Then for any μ ∈ ri(M C ( w)), there exists β > 0 such that, for a.a. t, and any v ∈ U :

H(v, ȳ(t), p μ(t)) -H(ū(t), ȳ(t), p μ(t)) ≥ β max µ∈M C ( w) (H(v, ȳ(t), p µ (t)) -H(ū(t), ȳ(t), p µ (t))) ; ( 19
)
βµ i ≤ μi , for all i > r 1 . (20)

INRIA

Proof. Since μ ∈ ri(M C ( w)), and M C ( w) is compact, there exists ε 0 > 0 such that μ + ε 0 (μ -µ) ∈ M C ( w), for any µ ∈ M C ( w). The function µ → a(µ) := H(v, ȳ(t), p µ (t)) -H(ū(t), ȳ(t), p µ (t)) is affine. For given µ ∈ M C ( w), we have that µ ′ := μ + ε 0 (μ -µ) belongs to M C ( w). Since μ = 1 1+ε0 µ ′ + ε0 1+ε0 µ, µ ′ i ≥ 0 for i > r 1 , and a(•) is affine and nonnegative when v ∈ U , it follows that μi ≥ ε0 1+ε0 µ i for i > r 1 , and a(μ) ≥ ε0 1+ε0 a(µ). The conclusion follows with β = ε 0 /(1 + ε 0 ).

We next relate the convergence of cost, for bounded strong perturbations, to some integral of difference of Hamiltonian functions. For µ ∈ R r * , we define

J µ (w) := J(w) + µΦ(η) = T 0 ℓ(w(t))dt + Φ µ (η). ( 21
)
If w is a trajectory, then for any p ∈ P, we have that

J µ (w) = T 0 [H(w(t), p(t)) -p(t) ẏ(t)] dt + Φ µ (η). ( 22
)
Lemma 2.4. Let w ∈ W be a trajectory, let µ ∈ R r * , with associated costate p µ solution of (13)(i-ii), and let w be any trajectory. Denote η := (y(0), y(T )) and δη := η -η. Then (i) The following expansion holds:

J µ (w) -J µ ( w) = T 0 [H(w, p µ ) -H( w, p µ ) -H y ( w, p µ )δy]dt +(p µ (0) + Φ µ y0 (η))δy(0) + 1 2 (Φ µ ) ′′ (η)(δη) 2 + o(|δη| 2 ). (23)
(ii) Let w k be a bounded strong perturbation of w, and µ ∈ R r * . Then

J(w k ) -J( w) = T 0 [H(u k , ȳ, p µ ) -H(ū, ȳ, p µ )]dt + o(1). (24)
(iii) If w is a Pontryagine extremal, µ ∈ M P ( w), and w k is a feasible bounded strong perturbation of w, then lim inf k J(w k ) ≥ J( w), with equality iff the following holds:

lim inf k T 0 [H(u k , ȳ, p µ ) -H(ū, ȳ, p µ )]dt = 0. ( 25
)
In particular, J(w k ) → J( w) iff the integral in (25) converges to 0.

Proof. Denoting δy := y -ȳ, observe that for any µ ∈ R r * , using (22) with p = p µ , we have that

J µ (w) -J µ ( w) = T 0 [H(w, p µ ) -H( w, p µ ) -p µ δ ẏ]dt + Φ µ (η) -Φ µ (η). ( 26
)
Since by the integration by parts formula, we get

- T 0 p µ δ ẏdt = p µ (0)δy(0) -p µ (T )δy(T ) + T 0 ṗµ δydt, (27)
using the costate equation ( 13) and a second-order expansion of Φ µ (η), obtain (i). Since for a bounded strong perturbation we have that y k -ȳ ∞ → 0, we deduce from (23) that J µ (w k ) -J µ ( w) is equal to the r.h.s. of (24). Since η k → η, J µ (w k )-J µ ( w) = J(w k )-J( w)+o [START_REF] Bonnans | Second-order analysis for optimal control problems with pure state constraints and mixed control-state constraints[END_REF]. Relation (ii) follows. Combining with (14), we deduce (iii).

We next show that the uniqueness of the minimum of the Hamiltonian function for all times t implies that the control is continuous.

Given a Pontryagine extremal w, and M > ū ∞ , we say that µ ∈ M P ( w) satisfies the hypothesis of unique minimum of the Hamiltonian over U M if the associated costate p µ is such that, for all t ∈ [0, T ], the function h(•, µ, t) = H(•, ȳ(t), p µ (t)) has a unique minimum over U M .

Remark 2.5. If this hypothesis holds, then (i) by lemma 2.3, it holds for any element of ri(M P ( w)), and (ii) for given ε > 0 and M > 0, there exists

ε M > 0 such that For a.a. t ∈ (0, T ), whenever v ∈ U M and |v -ū(t)| ≥ ε : H(v, ȳ(t), p µ (t)) ≥ H(ū(t), ȳ(t), p µ (t)) + ε M . ( 28
)
Lemma 2.6. Let w be a Pontryagine extremal, and µ ∈ M P ( w) satisfy the hypothesis of unique minimum of the Hamiltonian over U M , with M > ū ∞ . Then (one representative of ) ū(t) is a continuous function of time, equal to this unique minimum.

Proof. Let t ∈ [0, T ]; then for t k → t in [0, T ] the function ũ constructed in remark 1.2 is such that h(ũ(t), µ, t) = h(ũ(t k ), µ, t k ) = c µ .
Passing to the limit obtain h(ũ(t), µ, t) = h(v, µ, t) = c µ for all limit point v of ũ(t k ) (they exist since ũ(t) ∈ U M for all t∈ [0, T ]). Since h(•, µ, t) has a unique minimum over U M we see that ũ(t) = v, which proves that ũ is continuous. The conclusion follows.

Consider the condition similar to (25), but with a limit instead of a lower limit:

lim k T 0 [H(u k , ȳ, p µ ) -H(ū, ȳ, p µ )]dt = 0. ( 29
)
Lemma 2.7. Let w be a Pontryagine extremal, and let µ ∈ M P ( w) be such that the Hamiltonian satisfies the hypothesis of unique minimum over U . If w k is a feasible bounded strong perturbation of w, then the four conditions below are equivalent:

(i) lim sup k J(w k ) ≤ J( w), (ii) lim k J(w k ) = J( w), (iii) (29) holds, (
iv) Any subsequence of u k has itself a subsequence converging to ū a.e.

Proof. The equivalence of (i),(ii) and (iii) follows from lemma 2.4. If (iii) holds, then by (28) u k → ū in measure (i.e., for all ε > 0, meas({t

∈ (0, T ); |u k (t) - ū(t)| > ε}) → 0)
, and hence (since this holds also for an arbitrary subsequence of u k ) condition (iv) holds. Finally assume that (iv) holds, but not (iii). Taking if necessary a subsequence we may assume that there exists ε > 0 such that

T 0 [H(u k , ȳ, p µ ) -H(ū, ȳ, p µ )]dt > ε.
But then, taking again a subsequence for which u k → ū a.e., we obtain a contradiction to Lebesgue's dominated convergence theorem.

INRIA

Corollary 2.8. Let w be a Pontryagine extremal. Assume that, for some µ ∈ M P ( w), the Hamiltonian satisfies the hypothesis of unique minimum over U . Then (i) any feasible bounded strong perturbation of w such that lim sup k J(w k ) ≤ J( w) is a Pontryagine perturbation, and (ii) w is a bounded strong minimum iff it is a Pontryagine minimum.

Proof. (i) Let w satisfy the hypotheses of the corollary. If w k is a feasible bounded strong perturbation of w such that lim sup k J(w k ) ≤ J( w), then, by the above lemma, we deduce that u k → ū a.e., and hence, by the dominated convergence theorem, u k -ū 1 → 0, so that w k is a Pontryagine perturbation. (ii) As already observed, a bounded strong minimum is a Pontryagine solution. So it remains to prove that if w is a Pontryagine solution, then it is a bounded strong solution. This follows from point (i).

Definition 2.9. Let w be a Pontryagine extremal. We say that the Hamiltonian function satisfies a local quadratic growth condition for µ ∈ M P ( w) if there exist α > 0 and ε > 0 such that For a.a.

t ∈ (0, T ), whenever v ∈ U , |v -ū(t)| ≤ ε : H(v, ȳ(t), p µ (t)) ≥ H(ū(t), ȳ(t), p µ (t)) + α|v -ū(t)| 2 . ( 30
)
Remark 2.10. In view of lemma 2.3, we have that, if M C ( w) is a nonempty, convex and compact subset of M P ( w), and there exist α > 0 and ε > 0 such that For a.a.

t ∈ (0, T ), whenever v ∈ U , |v -ū(t)| ≤ ε : max µ∈M C ( w) [H(v, ȳ(t), p µ (t)) -H(ū(t), ȳ(t), p µ (t))] ≥ α|v -ū(t)| 2 . ( 31
)
then the Hamiltonian function satisfies the local quadratic growth condition, for any µ ∈ ri(M C ( w)).

If µ ∈ M P ( w) is such that (28) and (30) hold, for M > ū ∞ , then ū has a representative ũ such that, for all t ∈ [0, T ], v → h(v, µ, t) has a unique minimum at ũ(t) over U M , and we have that

H(v, ȳ(t), p µ (t)) ≥ H(ũ(t), ȳ(t), p µ (t)) + min(α|v -ũ(t)| 2 , ε M ), whenever v ∈ U M , for all t ∈ [0, T ].
(32)

Second-order expansion of the (weighted) cost function

Having in mind Pontryagine perturbations, we now introduce the natural notion of linearization of the state equation in the framework of the study of Pontryagine minima. Given a Pontryagine perturbation w k of w ∈ W, satisfying the state equation ( 2), reminding that δy k := y k -ȳ is the variation of states, denote by δ L y k the solution of the Pontryagine linearization of the state equation at the point w = (ū, ȳ):

δ L ẏk = f y (ū, ȳ)δ L y k + f (u k , ȳ) -f (ū, ȳ); δ L y k (0) = y k (0) -ȳ(0). ( 33
)
As shows the following lemma, δ L y k gives a good approximation of δy k . We denote the remaining terms in a second-order Taylor expansion in L 1 as

R 1k := O( u k -ū 2 1 + |y k (0) -ȳ(0)| 2 ); r 1k = o( u k -ū 2 1 + |y k (0) -ȳ(0)| 2 ).
A lemma similar to the one below can be found in Milyutin and Osmolovskiȋ [9, p. 40-42, prop.8.1 to 8.3]. We give the (short) proof in order to make the paper self-contained. Lemma 2.11. Let w ∈ W satisfy the state equation (2), and w k be a Pontryagine perturbation of w. Then

δ L y -y k -ȳ ∞ = R 1k . ( 34
)
Proof. We may write

δ ẏk = f y (ū, ȳ)δy k + f (u k , ȳ) -f (ū, ȳ) + ∆ k (35)
where ∆ k satisfies, by the mean value theorem, for some θ : [0, T ] → (0, 1):

∆ k := f (u k , y k ) -f (u k , ȳ) -f y (ū, ȳ)δy k = [f y (u k , ȳ + θδy k ) -f y (ū, ȳ)]δy k = O((|u k -ū| + |δy k |)|δy k |), (36)
so that ∆ k 1 = R 1k . The conclusion follows then from Gronwall's lemma.

Lemma 2.12. Let w ∈ W be a trajectory, and w k be a Pontryagine perturbation of w. Then for any µ ∈ R r * , denoting by p µ the costate associated with µ at the point w (solution of (13)(i-ii)), we have that:

J µ (w k ) -J µ ( w) = T 0 [H(u k , ȳ, p µ ) -H(ū, ȳ, p µ ) + 1 2 H yy (ū, ȳ, p µ )(δ L y k ) 2 ]dt + T 0 [H y (u k , ȳ, p µ ) -H y (ū, ȳ, p µ )]δ L y k dt +(p µ (0) + Φ µ y0 (η))δy k (0) + 1 2 (Φ µ ) ′′ (η)(δ L y k (0), δ L y k (T )) 2 + r 1k . (37)
Proof. Expanding w.r.t. δy k := y k -ȳ the r.h.s. of the expression below:

H(u k , y k , p µ ) -H(ū, ȳ, p µ ) = [H(u k , y k , p µ ) -H(u k , ȳ, p µ )] +[H(u k , ȳ, p µ ) -H(ū, ȳ, p µ )], (38)
and since (all derivatives of f and ℓ being uniformly bounded and continuous over bounded sets)

H(u k , y k , p µ )-H(u k , ȳ, p µ ) = H y (u k , ȳ, p µ )δ k y + 1 2 H yy (u k , ȳ, p µ )δy 2 k +r 1k , (39)
and using the relation

T 0 H yy (u k , ȳ, p µ )δy 2 k dt = T 0 H yy (ū, ȳ, p µ )δy 2 k dt + r 1k , obtain with (23): J µ (w k ) -J µ ( w) = T 0 [H(u k , ȳ, p µ ) -H(ū, ȳ, p µ ) + 1 2 H yy (ū, ȳ, p µ )δy 2 k ]dt + T 0 [H y (u k , ȳ, p µ ) -H y (ū, ȳ, p µ )]δy k dt +(p µ (0) + Φ µ y0 (η))δy k (0) + 1 2 (Φ µ ) ′′ (η)(δy k (0), δy k (T )) 2 + r 1k .
(40) The conclusion follows with lemma 2.11.

INRIA

A decomposition principle

Now given w ∈ W and a Pontryagine perturbation w k , both satisfying the state equation ( 2), we start to use second-order expansions of the cost function with the remainder

r 2k := o( δu k 2 2 + |y k (0) -ȳ(0)| 2 ). Note that the norm ( u 2 2 + |y(0)| 2 ) 1/2
is equivalent to w 2 over the set of trajectories. Consider a sequence of "measurable partitions" of [0, T ], i.e.,

[0, T ] = A k ∪ B k ; meas(A k ∩ B k ) = 0, (41)
such that meas(B k ) → 0, and a Pontryagine perturbation w k of w ∈ W. Denote the restriction of variations of control variables to the set A k by

δ A u k = 1 A k (u k -ū); u A,k := ū + δ A u k , (42)
and similarly for δ B u k and u B,k . The associated states are defined as solution of the Cauchy problems

ẏA,k = f (u A,k , y A,k ); y A,k (0) = y k (0), ( 43
) ẏB,k = f (u B,k , y B,k ); y B,k (0) = ȳ(0), (44)
that is, the variation in the initial condition is "absorbed" by the "A" part. We set

δ A y k = y A,k -ȳ; δ B y k = y B,k -ȳ. ( 45
)
Theorem 2.13 (Decomposition principle). Let w be a trajectory in W, w k be a Pontryagine perturbation, and (A k , B k ) be a measurable partition of (0, T ), such that meas(B k ) → 0. Then, for all µ ∈ R r * , we have a decomposition principle

J µ (w k ) = J µ (w A,k ) + (J µ (w B,k ) -J µ ( w)) + r 2k , (46)
and, p µ being the costate associated with µ at the point w, J µ (w B,k ) has the following expansion:

J µ (w B,k ) = J µ ( w) + B k [H(u k , ȳ, p µ ) -H(ū, ȳ, p µ )]dt + O( δ B u k 2 1
). ( 47)

If in addition δ A u k ∞ → 0, then setting r A2k := o( δ A u k 2 2 + |y k (0) -ȳ(0)| 2 ), we have that J µ (w A,k ) = J µ ( w) + δ A J µ k + r A2k , (48)
with

δ A J µ k := T 0 [ 1 2 H ww ( w, p µ )(δ A w k ) 2 + H u ( w, p µ )δ A u k ]dt +(p µ (0) + Φ µ y0 (η))δy k (0) + 1 2 (Φ µ ) ′′ (η)(δ A y k (0), δ A y k (T )) 2 .
(49)

Proof. The Pontryagine linearization (33) being the sum of the ones for perturbations u A,k and u B,k , it follows from lemma 2.11 that

δy k -δ A y k -δ B y k ∞ = R 1,k , and we have (i) δ A y k ∞ = O( δ A u k 1 + |y k (0) -ȳ(0)|); (ii) δ B y k ∞ = O( δ B u k 1 ). ( 50
)
Relation (47) follows then from (37) and (50)(ii). Now by the Cauchy-Schwarz inequality:

δ B u k 1 ≤ meas(B k ) 1/2 δ B u k 2 = o( δ B u k 2 ), (51) so that δ B y k ∞ = o( δ B u k 2 )
, and using r 1k = O(r 2k ), obtain with (37):

J µ (w k ) -J µ ( w) = T 0 [H(u k , ȳ, p µ ) -H( w, p µ ) + 1 2 H yy ( w, p µ )δ A y 2 k +(H y (u k , ȳ, p µ ) -H y ( w, p µ ))δ A y k ]dt +(p µ (0) + Φ µ y0 (η))δy k (0) + 1 2 (Φ µ ) ′′ (η)(δ A y k (0), δ A y k (T )) 2 + r 2k . (52)
Using ( 50) and(51), get

B k [H y (u k , ȳ, p µ ) -H y (ū, ȳ, p µ )]δ A y k dt = O( δ B u k 1 δ A y k ∞ ) = r 2k . ( 53
)
We deduce with (52)-(53) that

J µ (w k ) -J µ ( w) = δA J µ k + B k [H(u k , ȳ, p µ ) -H(ū, ȳ, p µ )]dt + r 2k , (54)
where

δA J µ k := A k [H(u k , ȳ, p µ ) -H( w, p µ )]dt + 1 2 T 0 H yy ( w, p µ )δ A y 2 k dt + A k (H y (u k , ȳ, p µ ) -H y ( w, p µ ))δ A y k dt +(p µ (0) + Φ µ y0 (η))δy k (0) + 1 2 (Φ µ ) ′′ (η)(δ A y k (0), δ A y k (T )) 2 . (55)
Specializing to the case when δ B u k = 0, we obtain

J µ (w A,k ) = J µ ( w) + δA J µ k + r A2k , (56)
which combined with (47), ( 51) and (54), proves (46). Finally when δ A u k ∞ → 0, it follows from a Taylor expansion that δA J µ k = δ A J µ k + r A2k , proving (48). The conclusion follows.

Inequality control constraints

We assume in this section that the control constraints are parameterized by finitely many inequalities:

U := {u ∈ R m ; g(u) ≤ 0}, ( 57
)
where g : R m → R q is a C 2 mapping. In other words, the control constraints are defined by

g i (u(t)) ≤ 0, for a.a. t ∈ (0, T ), i = 1, . . . , q. ( 58
)
We consider the "abstract" formulation where the state is a function of initial state and control. So we may write the state as y u,y0 (t), and define G :

U ×R n → R r by G i (u, y 0 ) := Φ i (y 0 , y u,y0 (T )), i = 1, . . . , r. (59)

INRIA

By G 1:r1 (u, y 0 ) we denote the (vertical) vector of components 1 to r 1 of G(u, y 0 ). We say that the following qualification condition [START_REF] Robinson | First order conditions for general nonlinear optimization[END_REF] (a natural infinite dimension generalization of the Mangasarian-Fromovitz condition [START_REF] Mangasarian | The Fritz-John necessary optimality conditions in the presence of equality and inequality constraints[END_REF]; see also [2, Section 2.3.4])) holds at w ∈ F (P ) if

G ′ 1:r1 (ū, ȳ0 ) is onto, There exists β > 0 and (v, z0 ) ∈ Ker G ′ 1:r1 (ū, ȳ0 ); g(ū(t)) + g ′ (ū(t))v(t) ≤ -β, for a.a. t ∈ [0, T ] G ′ i (ū, ȳ0 )(v, z0 ) ≤ -β, for all i > r 1 such that G i (ū, ȳ0 ) = 0.
(60) We also define for future reference Ĵ(u, y 0 ) := J(u, y u,y0 ); Ĵµ (u, y 0 ) := Ĵ(u, y 0 ) + µG(u, y 0 ),

as well as the augmented Hamiltonian function by

H a (u, y, p, λ) := H(u, y, p) + λg(u) = ℓ(u, y) + pf (u, y) + λg(u), (62)
where u ∈ R m , y ∈ R n , p ∈ R n * , and λ ∈ R q * . Given w = (ū, ȳ) ∈ F (P ), we recall that the set of normal directions to K at the point Φ(η) was defined in [START_REF] Osmolovskiȋ | Theory of higher order conditions in optimal control[END_REF]. The costate p µ ∈ P associated with µ ∈ N K (Φ(η)) was defined as the solution of (13)(i-ii). For w ∈ F (P ), µ ∈ N K (Φ(η)) such that (13)(iii) holds and t ∈ [0, T ], define

Λ t ( w, µ) := {λ ∈ R q * + ∩ g(ū(t)) ⊥ ; H a u ( w(t), p µ (t), λ) = 0}, LM L ( w) := {(λ, µ); µ ∈ N K (Φ(η)); (13) holds ; λ ∈ L ∞ (0, T, Λ t ( w, µ))}.
We call LM L ( w) (the superscript L is reminiscent of Lagrange) the set of first-order multipliers, denote by M L ( w) its projection on the second component, and say that ( w, p, λ, µ) is a first-order extremal if w ∈ F (P ), (λ, µ) ∈ LM L ( w), and p is the associated costate. Remember that J µ (w) was defined in (21). The Lagrangian for the abstract formulation is

L(u, y 0 , λ, µ) := Ĵµ (u, y 0 ) + λ, g(u) , (63)
the last duality product being in the space L ∞ (0, T, R q ). Theorem 3.1. Let w be a weak solution of (P ), satisfying the qualification hypothesis (60). Then (i) the sets LM L ( w) and M L ( w) are nonempty and bounded, and (ii) M P ( w) is a (possibly empty) subset of M L ( w).

Proof. (i) An abstract formulation of problem (P ) is

Min

u,y0 Ĵ (u, y 0 ); g(u) ≤ 0; G(u, y 0 ) ∈ K. ( 64
)
Let K g := L ∞ (0, T, R q -), with associated normal cone at the point g(ū) denoted N Kg (g(ū)). The corresponding set of Lagrange multipliers at point (ū, ȳ0 ) is defined as

(LM L ) ♯ ( w) := {(λ, µ) ∈ N Kg (g(ū)) × N K (G(ū, ȳ0 )); D (u,y0) L(ū, ȳ0 , λ, µ) = 0}.
(65) The qualification hypothesis (60) being a particular case of Robinson's qualification condition [START_REF] Robinson | First order conditions for general nonlinear optimization[END_REF], we know that (LM L ) ♯ ( w) is nonempty and bounded in L ∞ (0, T, R q ) * × R r * . It remains to prove that any (λ, µ) ∈ (LM L ) ♯ ( w) is such that λ can be identified with some λ in L ∞ (0, T, R q * ), the norm in L ∞ (0, T, R q * ) of λ being uniform over all (λ, µ) ∈ (LM L ) ♯ ( w). More precisely, we have to check the existence of c > 0 such that, | λ, a | ≤ c a L 1 (0,T,R q ) , for all a ∈ L ∞ (0, T, R q ). If this holds then, as L ∞ (0, T, R q ) is a dense subset of L 1 (0, T, R q ), λ has a unique extension λ in the dual space of L 1 (0, T, R q ), i.e., L ∞ (0, T, R q * ).

Since the norm of a ∈ L ∞ (0, T, R q ) is the sum of the norms of its positive and negative parts, it suffices to check this inequality when a ≥ 0, i.e., since λ ≥ 0, λ, a ≤ c a L 1 (0,T,R q ) . We can write a(t) = α(t)ā(t), with α(t) = |a(t)| and |ā(t)| = 1. Set h := -(g(ū) + g ′ (ū)v). By (60), β ≤ h i (t), i = 1, . . . , q, for a.a. t. Since a i (t) ≤ α(t), i = 1, . . . , q, for a.a. t, we have that βa(t) ≤ α(t)h(t), and so β λ, a = λ, βa ≤ λ, αh .

Since λ ≥ 0, a ≥ 0 and g(ū) ≤ 0, we have that:

0 ≥ λ, αg(ū) ≥ λ, a ∞ g(ū) = a ∞ λ, g(ū) = 0, (67)
the last equality being the complementarity condition between elements of a convex cone and elements of the corresponding normal cone. It follows that λ, αg(ū) = 0. Combining with (66), and using (65), obtain

β λ, a ≤ -λ, αg ′ (ū)v = -λ, g ′ (ū)αv = D u Ĵµ (ū, ȳ0 )(αv) ≤ D u Ĵµ (ū, ȳ0 ) ∞ α 1 v ∞ = D u Ĵµ (ū, ȳ0 ) ∞ v ∞ a 1 , (68)
which, since µ remains in a bounded set, gives the desied estimate. Point (i) follows.

(ii) Let µ ∈ M P ( w). Then ū is solution of the problem Min u∈L ∞ (0,T,R m ) T 0 H(u(t), ȳ(t), p µ (t))dt; g(u(t)) ≤ 0, for a.a. t ∈ (0, T ).

(69) In view of the qualification condition, there exists some λ such that (λ, µ) ∈ (LM L )( w). The conclusion follows.

Quadratic growth with initial-final state constraints

When dealing with initial-final state constraints we need to combine the previous decomposition principle with a certain restoration hypothesis, for which we will give sufficient conditions. Given a Lagrange extremal w, and an arbitrary μ ∈ ri(M L ( w)) (the relative interior of M L ( w)), denote the set of strictly (non strictly) complementary active constraints by

I + := {1, . . . , r 1 } ∪ {r 1 < i ≤ r; μi > 0}, I 0 := {r 1 < i ≤ r; Φ i (η) = 0} \ I + . (70)
Similarly to (20), all µ ∈ ri(M L ( w)) have the same set of positive components, as can be easily checked, so that the definition does not depend on the choice INRIA of the particular μ. Define

K + := {θ ∈ IR r ; θ i = 0, i ∈ I + , θ i ≤ 0, i ∈ I 0 }. (71)
Note that, if μ ∈ ri(M L ( w)), then K + = K ∩ μ⊥ . The function

d(η) := i∈I0 Φ i (η) + + i∈I+ |Φ i (η)| (72)
is the distance of the initial-final state constraint to the set K + , in the L 1 (R r ) norm (the unique projection of θ ∈ R r in this norm being θ ′ defined by θ ′ i = 0 if i ∈ I + , and θ ′ i = min(θ i , 0) otherwise). We have that

d(η) = O(-μΦ(η)) whenever w ∈ F (P ), (73)
since then Φ 1:r1 (η) = 0 and Φ r1+1:r (η) ≤ 0. Call Pontryagine norm the following one:

w P := u 1 + y ∞ . (74)
For given ε D > 0, and u ∈ U, define the set times of ε D -deviation of ū as

B εD (u) := t ∈ (0, T ); |u(t) -ū(t)| ≥ u -ū 1 ε D . ( 75
)
We have the following relation. Proof. The conclusion follows from the relations below:

meas(B εD (u)) = T 0 1 {|u(t)-ū(t)|≥ u-ū 1 ε D } dt ≤ ε D u -ū 1 T 0 |u(t)-ū(t)|dt = ε D .
Definition 4.2. Let w be a Pontryagine extremal. We say that the restoration property (for the initial-final state constraints) is satisfied at w ∈ F (P ) for μ ∈ M P ( w) in the Pontryagine sense if there exists ε P > 0 and ε B > 0 such that, for any trajectory w such that w -w P ≤ ε P and u(t) ∈ U a.e., and measurable set B ⊂ (0, T ) such that meas(B) ≤ ε B over which u and ū coincide, there exists w ′ ∈ F (P ) such that u ′ = ū on B and

w ′ -w ∞ = O(d(η)); J(w ′ ) = J μ(w) + O( w -w P d(η)). (76)
Let us give a sufficient condition for the restoration property. Denote the kernel of derivatives of almost active control constraints (relative to w ∈ W), parameterized by ε R > 0 (this notation is a reminder of "restoration");

U εR := v ∈ U; g ′ i (ū(t))v(t) = 0 whenever g i (ū(t)) ≥ -ε R , i = 1, . . . , q, for a.a. t ∈ (0, T ) . ( 77
)
We will call special qualification condition the Mangasarian-Fromovitz qualification condition [START_REF] Mangasarian | The Fritz-John necessary optimality conditions in the presence of equality and inequality constraints[END_REF] for constraints on initial-final state in K + , over the Banach space E εR := U εR × R n . Setting ē := (ū, ȳ(0)), it can be formulated as:

   (i)
There exists e j ⊂ E εR , j = 1, . . . , |I + |, such that {G ′ I+ (ē)e j } j=1,...,|I+| is an independent family, (ii) There exists e 0 ∈ E εR such that G ′ I+ (ē)e 0 = 0; G ′ I0 (ē)e 0 < 0.

(78)

RR n 6707

Proof. It suffices to prove (86)(i). We see that this is just the analysis of the square linear system (85). We know that the matrix inversion is locally Lipschitz over the set of invertible square matrices of a given dimension. When meas(B) = 0, the solution is ᾱ = 0; otherwise the perturbation of the matrix and of the r.h.s. is, by ( 81) and ( 83), of order O(meas(B)). The result follows.

Lemma 4.5. Let w ∈ F (P ) satisfy the special qualification condition (78), for some ε R > 0. Let M > ū ∞ . Then, if ε B > 0 and ε P > 0 are small enough, for any measurable B ⊂ (0, T ) such that meas(B) ≤ ε B , any trajectory w such that u ∞ ≤ M and w -w P ≤ ε P , there exists a trajectory w ′′ ∈ W (that does not in general satisfy the control constraints) such that

w ′′ -w ∞ = O(d(η)); G(u ′′ , y ′′ 0 ) ∈ K + ; u ′′ -u ∈ E εR (B). ( 87
)
Proof. Denote by F B the finite dimensional space spanned by e 0 B , . . . , e |I+| B , endowed with the norm of U ×R n . Consider the mapping T from U ×R n into itself, that with a given e := (u, y 0 ) ∈ U × R n associates e + δe, where δe = (δu, δy 0 ) is a vector of F B satisfying the following conditions:

(i) G I+ (e) + G ′ I+ (ē)δe = 0; (ii) G I0 (e) + G ′ I0 (ē)δe ≤ 0, ( 88
)
so we can write δe = |I+| j=0 θ j e j B . Coefficients θ j , for j = 1 to |I + |, are uniquely determined by (88)(i), and for θ 0 we choose the smallest possible nonnegative value. In view of lemma 4.4, we have that, for some

c 1 > 0, if ε B is small enough: δe ∞ ≤ c 1 |G I+ (e)| + |G I0 (e) + | . ( 89
)
Over the set V 1,M := {w ∈ W; w -w P ≤ 1; u ∞ ≤ M }, the mapping e → G ′ (e) is, when E εR is endowed with the L 1 norm, Lipschitz from E εR into L(E εR , R r ), so that for some c 2 > 0:

|G(e + δe) -G(e) -G ′ (e)δe| ≤ c 2 δe 2 P . ( 90
)
In view of ( 88) and (89), and since (making the abuse of notation of denoting also by P the induced norm for G ′ (e)) G ′ (ē) -G ′ (e) P → 0 when u ∞ ≤ M and w -w P → 0, it follows that when w -w P is small enough, say w -w P ≤ ε P 1 we have that

|G I+ (e + δe)| + |G I0 (e + δe) + | ≤ 1 2 (|G I+ (e)| + |G I0 (e) + |). ( 91
)
Consider the sequence defined by e k+1 := T (e k ), for k ∈ IN . It follows from ( 89) and (91) that, if the trajectory w corresponding to e 0 satisfies the hypotheses of the lemma for small enough ε P , then the sequence w k is well-defined, remains in V 1,M , and converges to some e ′′ such that e ′′ -e 0 ≤ 2c 1 |G I+ (e)| + |G I0 (e) + | . The corresponding associated state y ′′ is such that w ′′ := (u ′′ , y ′′ ) satisfies (87).

Given w ∈ W satisfying the state equation ( 2), we consider the following measure of constraint defect:

D(w) := g(u) + ∞ + r1 i=1 |Φ i (η)| + r i=r1+1 Φ i (η) + .
(92) B = B εD , where B εD is the deviation set introduced in (75). Then, by lemma 4.1, meas B εD ≤ ε D and by definition of the deviation set we have that

|u(t) -ū(t)| ≤ ε D -1 w -w P , for a.a. t ∈ (0, T ) \ B εD . (97)
By lemma 4.5, there exists a trajectory w ′′ ∈ W satisfying (87), and then using (87) and (97) we get that, for some c 1 > 0 and for a.a. t ∈ (0, T ) \ B εD :

g i (u ′′ (t)) = g i (u(t)) + g ′ i (u(t))(u ′′ (t) -u(t)) + O(d(η) 2 ), ≤ g i (u(t)) + g ′ i (ū(t))(u ′′ (t) -u(t)) + c 1 w -w P d(η) + d(η) 2 . ( 98
) If t ∈ B εD then g i (u ′′ (t)) = g i (u(t)) ≤ 0. Otherwise, if g i (ū(t)) ≥ -ε R (we remind that ε R was introduced in (77)), since g i (u(t)) ≤ 0 and g i (ū(t))(u ′′ (t) - u(t)) = 0, we get g i (u ′′ (t)) ≤ c 1 w -w P d(η) + d(η) 2 . ( 99
)
If g i (ū(t)) < -ε R and t / ∈ B εD , then by (97), whenever ε P is small enough, we have g i (u(t)) < -1 2 ε R . Using lemma 4.4, (98) and the estimate d(η) = O( w -w P ), we obtain:

g i (u ′′ (t)) ≤ -1 2 ε R + O(d(η)) ≤ -1 2 ε R + O(ε P ) ≤ 0, ( 100
)
so that finally with (99), and using again d(η) = O( w -w P ):

g(u ′′ (t)) ≤ c 2 w -w P d(η), a.e. on (0, T ). ( 101
)
Let µ be the element of the singleton M P ( w). We next apply lemma 2.12 at point w, denoting therefore pµ the costate evaluated at the point w (and not w). Note that o(•) and O(•) in the statement of this theorem come from Taylor expansions on bounded sets, and hence, are uniform over the reference point. Since w ′′ -w ∞ = O(d(η)), we deduce that, denoting by p µ the costate evaluated at the point w and associated with µ:

J µ (w ′′ ) -J µ (w) = T 0 H u (w, pµ )(u ′′ -u)dt +(p µ (0) + (Φ µ y0 )(η))(y ′′ (0) -y(0)) + O(d(η) 2 ). ( 102
)
Since w -w P ≤ ε P , using

pµ (0) + Φ µ y0 (η) -p µ (0) + Φ µ y0 (η) = O( w -w P ), H u (w, pµ ) -H u ( w, p µ ) 1 = O( w -w P ) (103)
as well as the relations p µ (0) + Φ µ y0 (η) = 0 and w ′′ -w ∞ = O(d(η)), we can write

J µ (w ′′ ) -J µ (w) = T 0 H u ( w, p µ )(u ′′ -u)dt + O( w -w P d(η)). ( 104
)
Since u ′′ -u ∈ U εR , we deduce from the first-order optimality conditions that

H u ( w(t), p µ (t))(u ′′ (t) -u(t)) = -λ t g ′ (ū(t))(u ′′ (t) -u(t)) = 0. (105)
With ( 104)-(105), we get J µ (w ′′ ) -J µ (w) = O( w -w P d(η)). Now by the definition (87) of w ′′ , we have that J(w ′′ ) = J µ (w ′′ ). By lemma 4.6, (87) and (101), there exists w ′ ∈ F (P ), such that w ′ -w ′′ W = O( w -w P d(η)) and u ′ (t) = u ′′ (t) = ū(t) a.e. on B εD . Consequently, J(w ′ ) -J(w ′′ ) = O( ww P d(η)); therefore (76) holds. The conclusion follows.

As shows the following theorem, the quadratic growth condition for the Hamiltonian makes a bridge between the notions of weak and bounded strong quadratic growth of the cost functional J. Theorem 4.8. a) Let w ∈ F (P ) satisfy the qualification condition (60). Then the bounded strong quadratic growth condition (9) at w implies the three following conditions: (i) the weak minimum quadratic growth condition (8), (ii) Pontryagine's principle with hypothesis of unique minimum of the Hamiltonian over U for some µ ∈ M P ( w), and (iii) the local quadratic growth condition for Hamiltonian (30), for some µ ∈ M P ( w). b) Conversely, if (i)-(iii) holds as well as the restoration property (76), then the bounded strong quadratic growth condition holds at w. Proof. a) Let w satisfy the bounded strong quadratic growth condition. Then of course the condition of weak quadratic growth holds, and in addition, for any M > ū ∞ , Pontryagine's principle holds for the problem of minimizing J(w) -α M w -w 2 2 over the set

W M := {w ∈ F (P ); |u(t)| ≤ M a.e.}. (106)
from which the local quadratic growth condition for Hamiltonian function (30) follows, as well as the hypothesis of unique minimum of the Hamiltonian over U for some µ ∈ M P ( w). b) Assume that w satisfies conditions (i)-(iii) of the theorem, as well as the restoration property (76). If it does not satisfy the bounded strong quadratic growth condition, then there exists a sequence of bounded strong perturbation

w k such that J(w k ) ≤ J( w) + o( w k -w 2 2 ). (107)
By Corollary 2.8, w k is a Pontryagine perturbation of w. Let M > sup k u k ∞ . We know that (32) holds for some constants α > 0 and ε M > 0. Let ε > 0 be such that αε 2 < ε M , and set

B k := {t ∈ (0, T ); |u k (t) -ū(t)| > ε}; A k := (0, T ) \ B k . (108)
In view of lemma 2.3 we may assume that conditions (ii) and (iii) hold for the same µ ∈ ri(M P ( w)). By (30), since meas(B k ) → 0 by lemma 4.1, we can write, ← by theorem 2.13 and using (32):

J µ (w k ) = J µ (w A,k ) + B k [H(u k , ȳ, p µ ) -H( w, p µ )]dt + r 2k ≥ J µ (w A,k ) + αε 2 meas(B k ) + r 2k . (109)
Applying the restoration property (76) to w A,k , get the existence of

w ′ A,k ∈ F (P ) such that    (i) w ′ A,k -w A,k = O(d(η A,k )); (ii) J(w ′ A,k ) = J µ (w A,k ) + O( w A,k -w P d(η A,k )). (iii) u ′ A,k (t) = u A,k (t) = ū(t) a.e. on B k . (110)
Combining (110)(ii) with (109), we obtain

J µ (w k ) ≥ J(w ′ A,k ) + αε 2 meas(B k ) + O( w A,k -w P d(η A,k )) + r 2k . (111) INRIA On the other hand, since y B,k -ȳ ∞ = O(meas(B k )), we have that Φ(η k ) = Φ(η A,k ) + O(meas(B k )). (112)
We deduce that

d(η A,k ) = d(η k ) + O(meas(B k )). (113)
Also, since w k ∈ F (P ), and hence, Φ 1:r1 (η k ) = 0 and Φ r1+1:r (η k ) ≤ 0, combining (111) and ( 113), and finally with (73), get

J(w k ) ≥ J(w ′ A,k ) + αε 2 meas(B k ) -µΦ(η k ) + O( w A,k -w P d(η A,k )) + r 2k ≥ J(w ′ A,k ) + 1 2 αε 2 meas(B k ) -µΦ(η k ) + O( w A,k -w P d(η k )) + r 2k ≥ J(w ′ A,k ) + 1 2 αε 2 meas(B k ) -1 2 µΦ(η k ) + r 2k . (114) Since w ′
A,k is feasible, by (110)(i), we have that

w ′ A,k -w = O u ′ A,k -ū ∞ + |y ′ A,k,0 -ȳ0 | = O ( u A,k -ū ∞ + |y A,k,0 -ȳ0 | + d(η A,k )) = O (ε) + o(1), (115)
then for small enough ε > 0, since w is a weak minimum, we have that J(w ′ A,k ) ≥ J( w), and so, using 1 2 µΦ(η k ) ≤ 0, (114) implies

J(w k ) ≥ J( w) + 1 2 αε 2 meas(B k ) + r 2k . (116)
In view of (107), we must have meas(B k ) = r 2k . By lemma 4.6 and (113), there exists

w ′ k ∈ F (P ) such that w ′ k -w A,k ∞ = r 2k , and therefore w ′ k -w 2 2 = w A,k -w 2
2 + r 2k so that, using theorem 2.13 for the first equality, when ε > 0 is small enough,

J(w k ) = J(w A,k ) + r 2k = J(w ′ k ) + r 2k ≥ J( w) + 1 2 α w A,k -w 2 2 + r 2k , = J( w) + 1 2 α w k -w 2 2 + r 2k , (117)
contradicting (107).

5 Second-order necessary or sufficient conditions

Critical directions

Since the qualification hypothesis (60) is a particular case of Robinson's qualification condition, the second-order optimality condition due to Cominetti [START_REF] Cominetti | Metric regularity, tangent sets and second order optimality conditions[END_REF] (see also [START_REF] Bonnans | Perturbation analysis of optimization problems[END_REF]Thm. 3.45]) holds. We denote ȳ(0) as ȳ0 , and recall that Ĵ(u, y 0 ) := J(u, y u,y0 ), where y u,y0 is the solution of the state equation ( 2), with initial condition y(0) = y 0 , and that G was defined in (59). In order to state it, reminding the notations in (61) and (63), define the set of active inequalities at time t and for the initial-final state constraints:

I t := {1 ≤ i ≤ q; g i (ū(t)) = 0}; I F := {r 1 + 1 ≤ j ≤ r; G j (ū, ȳ0 ) = 0}. (118)
Set U 2 := L 2 (0, T, R m ). The linear mappings Ĵ′ (ū, ȳ0 ) and G ′ (ū, ȳ0 ) have a unique extension over U 2 × R n that will be denoted in the same way. We define the set of extended tangent directions to the control and initial-final state constraints (they are extended in the sense that we take L 2 spaces instead of L ∞ ):

T g (ū) := {v ∈ U 2 ; g ′ It (ū(t))v(t) ≤ 0, a.a. t ∈ (0, T )}, (119)

T Φ (ū, ȳ0 ) := {(v, z 0 ) ∈ U 2 × R n ; G ′ 1:r1 (ū, ȳ0 )(v, z 0 ) = 0; G ′ IF (ū, ȳ0 )(v, z 0 ) ≤ 0}, ( 120
) the set of extended critical directions:

C 2 (ū, ȳ0 ) := {(v, z 0 ) ∈ T Φ (ū, ȳ0 ); v ∈ T g (ū); Ĵ′ (ū, ȳ0 )(v, z 0 ) ≤ 0}. ( 121
)
The set of critical directions (in the original space) is

C ∞ (ū, ȳ0 ) := {(v, z 0 ) ∈ C 2 (ū, ȳ0 ); v ∈ L ∞ (0, T, R m )}. ( 122
)
Finally T 2 -(g(ū), g ′ (ū)v) stands for the second-order tangent set to L ∞ (0, T, R q -) at the point g(ū), in the direction g ′ (ū)v, i.e., for s > 0:

T 2 -(g(ū), g ′ (ū)v) = {v ∈ L ∞ (0, T, R q ); supess t (g(ū) + sg ′ (ū)v + 1 2 s 2 v) ≤ o(s 2 )}.
(123) By Cominetti [START_REF] Cominetti | Metric regularity, tangent sets and second order optimality conditions[END_REF], we have that (as usual, σ K (•) denotes the support function to a set K, i.e. the supremum of duality products with elements of K, and the corresponding term in (124) is called "sigma-term"): Theorem 5.1. Let w = (ū, ȳ) be a weak solution of (P ), satisfying the qualification hypothesis (60). Then max (λ,µ)∈LM L ( w)

L (u,y0) 2 (ū, ȳ0 , λ, µ)(v, z 0 ) 2 -σ(λ, T 2 -(g(ū), g ′ (ū)v) ≥ 0, for all (v, z 0 ) ∈ C ∞ (ū, ȳ0 ). ( 124
)
Note that the second-order tangent set in L ∞ has no practical characterization (see however [START_REF] Cominetti | Tangent sets to unilateral convex sets[END_REF]). We do not detail the proof of theorem 5.1 since it is a standard application of [START_REF] Cominetti | Metric regularity, tangent sets and second order optimality conditions[END_REF] (note that since the constraint on the initial-final state consist in a finite number of inequalities they have no contribution to the "sigma term") and since we next prove a stronger result. Let us denote Ω(v, z 0 ) := max

(λ,µ)∈LM L ( w) L (u,y0) 2 (ū, ȳ0 , λ, µ)(v, z 0 ) 2 . ( 125
)
Note that the maximum is indeed attained since the function to be maximized is continuous for the weak * topology, the set of multipliers is bounded in the L ∞ norm, and any bounded sequence in an L ∞ space has a weak * converging subsequence. Since the sigma term is nonpositive (e.g. [2, Equation (3.110)]), and C ∞ (ū, ȳ0 ) ⊂ C 2 (ū, ȳ0 ), a sufficient condition for (124) is

Ω(v, z 0 ) ≥ 0, for all (v, z 0 ) ∈ C 2 (ū, ȳ0 ). ( 126
)
Note that the above condition makes sense since, for any (λ, µ) ∈ LM L ( w), the quadratic form (v, z 0 ) → L (u,y0) 2 (ū, ȳ0 , λ, µ)(v, z 0 ) 2 , defined over U × R n , has a unique extension over U 2 × R n .

Theorem 5.2. Let w = (ū, ȳ) be a weak solution of (P ), satisfying the qualification hypotheses (60) and (78). Then condition (126) holds.

INRIA

Proof. Define the set of times with small negative constraints as

I ε = {t ∈ (0, T ); -ε ≤ g i (ū(t)) < 0, for some 1 ≤ i ≤ q}. (127)
Let (v, z 0 ) ∈ C 2 (ū, ȳ0 ) be an extended critical direction. For given ε > 0, consider the perturbed direction (v ′ ε , z 0 ) ∈ C ∞ (ū, ȳ0 ), defined by

v ′ ε (t) = 0 if either t ∈ I ε , or |v(t)| > ε -1 ; v ′ ε (t) = v(t) otherwise. ( 128
)
By (78), where vectors e k were defined, there exists (v ε , z 0ε ) ∈ C ∞ (ū, ȳ0 ) of the form

(v ε , z 0ε ) = (v ′ ε , z 0 ) + |I+| k=0 α ε,k e k (129) such that v ε -v ′ ε ∈ U εR , |α ε | → 0 when ε ↓ 0, and so (v ε , z 0ε ) -(v ′ ε , z 0 ) ∞ → 0. When ε < ε R , if -ε ≤ g i (ū(t))
, we have that v ′ ε (t) = 0 and g ′ i (ū(t))e k (t) = 0. It follows that g(ū) + ρg ′ (ū)v ε ≤ 0 when ρ > 0 is small enough. In that case we know that σ(λ, T 2 -(g(ū), g ′ (ū)v ε ) = 0. Therefore, by theorem 5.1, Ω(v ε , z 0ε ) ≥ 0, so that there exists λ ε ∈ L ∞ (0, T, R q ) such that (λ ε , µ) ∈ LM L ( w) and

L (u,y0) 2 (ū, ȳ0 , λ ε , µ)(v ε , z 0ε ) 2 ≥ 0. ( 130
)
Since the set of Lagrange multipliers is bounded, λ ε has weak * limit points when ε ↓ 0. As (v ε , z 0ε ) → (v, z 0 ) in the L 2 norm (so that the term in product of λ ε strongly converges in the L 1 norm), we may pass to the limit in this inequality when ε → 0. The conclusion follows.

Remark 5.3. The method of proof is a variant of the one used for "extended polyedricity", see [2, Section 3.2.3]. The basic concept there is the one of radial critical directions, i.e., critical directions v for which there exists κ > 0 such that (in our notations) g(u) + κg ′ (u)v ≤ 0. Here the L ∞ smoothness of the multiplier compensates the fact that extended critical directions belong to L 2 spaces.

We next discuss relations with weak quadratic growth, defined in (8).

Corollary 5.4. Let w be a weak solution of (P ), satisfying the qualification hypotheses (60) and (78) and the weak quadratic growth condition (8) with parameter α.

Then

Ω(v, z 0 ) ≥ α( v 2 2 + |z 0 | 2 ), for all (v, z 0 ) ∈ C 2 (ū, ȳ0 ). ( 131
)
This is a consequence of the second-order necessary condition for the problem of minimizing over F (P ) the perturbed cost function (the proof follows the one of theorem 5.2)

Ĵα (u, y 0 ) := Ĵ(u, y 0 ) -1 2 α u -ū 2 2 + |y 0 -ȳ0 | 2 . ( 132
)

Characterization of the weak quadratic growth condition

Let w ∈ F (P ) satisfy the special qualification hypothesis (78). We know that the projection M L ( w) of the set LM L ( w) on the second component is a singleton {µ}. Let p denote the associated costate. We consider the following two conditions: uniform local quadratic growth of Hamiltonian functions along the trajectory (analogous to (30)) i.e.,

       There exists c H > 0, ε ∞ > 0; for a.a. t ∈ (0, T ), H(u, ȳ(t), p(t)) ≥ H(ū(t), ȳ(t), p(t)) + c H |u -ū(t)| 2 , for all u ∈ R m ; g(u) ≤ 0; |u -ū(t)| ≤ ε ∞ ,
where p is the costate associated with μ,

and uniform quadratic growth along critical directions (131).

Theorem 5.5. Let w ∈ F (P ) satisfy the qualification hypotheses (60) and (78).

Then w satisfies the weak quadratic growth condition iff (131) and (133) hold.

Proof. Since the weak quadratic growth condition means that w is, for some α > 0, a weak solution of the perturbed problem Min w∈W J(w) -1 2 α w -w 2 2 , and the latter has the same first-order multipliers, conditions (133) and (131) (for the latter, in view of corollary 5.4) are necessary conditions for weak quadratic growth.

Let us show the converse by contradiction. So let us assume that, while w satisfies (131) and ( 133), there exist a sequence w k ∈ F (P ) such that w k → w in W, w k = w for all k, and

J(w k ) ≤ J( w) + o( w k -w 2 2 ). (134)
Define the local critical cone C t at time t ∈ [0, T ] as (I t was defined in (118)):

C t := {v ∈ R m ; H u (ū(t), ȳ(t), p(t))v ≤ 0; g ′ i (ū(t))v ≤ 0, i ∈ I t }. (135)
Note that, by (121):

C 2 (ū, ȳ0 ) = {(v, z 0 ) ∈ T Φ (ū, ȳ0 ); v(t) ∈ C t for a.a. t; μG ′ (ū, ȳ0 )(v, z 0 ) = 0}. ( 136
) Since C t is a polyhedron, by Hoffman's lemma [START_REF] Hoffman | On approximate solutions of systems of linear inequalities[END_REF], there exists a constant κ

t such that dist(v, C t ) ≤ κ t [H u ( w(t), p(t))v] + + i∈It [g ′ i (ū(t))v] + . (137)
This constant κ t can be estimated as a finite maximum of those for projections over vector subspaces corresponding to a subset of active inequalities (for the problem of projection over the local critical cone), and therefore is a measurable function of time. Take a sequence of positive numbers ε k → 0 (we will be more precise later) and consider the measurable partition

B k := {t ∈ (0, T ); κ t ≥ 1/ε k }; A k := (0, T ) \ B k . ( 138
) Denote u ′ k := ū + (u k -ū)1 A k ; u ′′ k := ū + (u k -ū)1 B k . (139)
Denote also

γ ′ k := u ′ k -ū 2 2 + |y k0 -ȳ0 | 2 ; γ ′′ k := u ′′ k -ū 2 2 ; γ k := γ ′ k + γ ′′ k . ( 140
) INRIA Note that γ k = u k -ū 2 2 + |y k0 -ȳ0 | 2 .
Let y ′ k be the state associated with control u ′ k and initial state y k0 . Since ε k → 0, we have that meas(B k ) → 0. The decomposition principle (theorem 2.13) implies

J μ(w k ) = J μ(w ′ k ) + B k [H(u ′′ k , ȳ, p) -H(ū, ȳ, p)]dt + o(γ k ). (141)
In view of the uniform local quadratic growth condition for Hamiltonian functions (133), this implies

J μ(w k ) ≥ J μ(w ′ k ) + c H γ ′′ k + o(γ k ). ( 142
) Let (λ, μ) ∈ LM L ( w). Adding 0 ≥ T 0 λ(t)g(u ′ k (t)
)dt to (37), obtain by a second-order expansion that

J μ(w ′ k ) ≥ J μ(w ′ k ) + λ, g(u ′ k ) = J( w) + O(γ ′ k ). (143)
If (for a subsequence) γ ′ k = o(γ k ), deduce then with (142) that

J(w k ) ≥ J μ(w k ) ≥ J( w) + c H γ ′′ k + o(γ k ) = J( w) + c H γ k + o(γ k ), (144)
which gives the desired contradiction to (134). So in the sequel (and this will be used at the end of the proof) we may assume that

γ k = O(γ ′ k ). (145)
In view of (134), we obtain

J( ŵk ) -1 2 μΦ(η k ) ≤ J( w) + o(γ k ). ( 152
)
RR n 6707

We will end the proof by obtaining a contradiction to (152). Set γk := ûkū 2 2 + |ŷ k0 -ȳ0 | 2 . By (146)(i), we have that

γ1/2 k = (γ ′ k ) 1/2 + O(d(η ′ k )) = O((γ ′ k ) 1/2 ) = O((γ k ) 1/2 ). (153)
Since (a + b) 2 ≥ 1 2 a 2 -b 2 (a, b ∈ R), by the first equality above, we have that for some c > 0 not depending on k:

γk ≥ 1 2 γ ′ k -cd(η ′ k ) 2 . ( 154
)
This relation will be used later. Combining (37) and ( 153), obtain J( ŵk ) ≥ J μ( ŵk ) = J( w) +

vk 1 = v k -(û k -ū) 1 = O γ k ε k . ( 160
)
Since a projection is nonexpansive, we also have

(û k -u k )1 A k ∞ = (û k -u ′ k )1 A k ∞ = O(d(η ′ k )) = O(γ 1/2 k ) = O(ε 2 k ), (162) so that in particular ûk -ū ∞ ≤ ûk -u k ∞ + u k -ū ∞ = O(ε 2 k ). ( 163
)

INRIA

Combining with (160), get 164) By ( 157), ( 161) and (163), we have that vk ∞ ≤ 2 ûk -ū ∞ = O(ε 2 k ). Therefore we obtain in a similar way

T 0 |û k (t) -ū(t)||v k (t)|dt ≤ ûk -ū ∞ ε k ε k vk 1 = O(ε k γ k ) = o(γ k ). (
vk 2 2 ≤ vk ∞ vk 1 = O ûk -ū ∞ ε k ε k vk 1 = o(γ k ), ( 165
)
v k 2 2 = ûk -ū + vk 2 2 = ûk -ū 2 2 + o(γ k ). ( 166
)
Since ŵk ∈ F (P ), setting êk := (û k , ŷk0 ), and using (153), we have that (I F was defined in (118))

G ′ i (ē)(û k -ū, ŷk0 -ȳ0 ) = G i (ê k ) -G i (ē) + O(γ k ) = O(γ k ), 1 ≤ i ≤ r 1 , ≤ O(γ k ), i ∈ I F .
(167) In view of (160), using γ k /ε k = o(γ 1/2 k ), we have that

G ′ i (ē)(v k , ŷk0 -ȳ0 ) = G ′ i (ē)(û k -ū, ŷk0 -ȳ0 ) + o((γ k ) 1/2 ) = o(γ 1/2 k ), 1 ≤ i ≤ r 1 , G ′ i (ē)(v k , ŷk0 -ȳ0 ) = G ′ i (ē)(û k -ū, ŷk0 -ȳ0 ) + o((γ k ) 1/2 ) ≤ o(γ 1/2 k ), i ∈ I F .
(168) Also, when (λ, μ) ∈ LM L ( w), using (152), ( 153) and (158), for some c > 0: μG ′ (ē)(û k -ū, ŷk0 -ȳ0 ) = -Ĵ′ (ū, ȳ0 )(û k -ū, ŷk0 -ȳ0 ) -λ, g ′ (ū)(û k -ū) ≥ -cγ k .

(169) Therefore, using again (160) and γ k /ε k = o(γ

1/2 k ) we get -μG ′ (ē)(v k , ŷk0 -ȳ0 ) ≤ o(γ 1/2 k ). ( 170
)
It follows with (168) that

G ′ i (ē)(v k , ŷk0 -ȳ0 ) = o((γ ′ k ) 1/2
), for all i ∈ I + .

Therefore, by the special qualification condition (78), there exists a critical direction (v ′ k , z ′ k ) such that, setting z k0 := ŷk0 -ȳ0

v ′ k -v k ∞ + |z ′ k0 -z k0 | = o((γ k ) 1/2 ). ( 172
)
We recall that Ω(v, z 0 ) was defined in (125). Note that Ω(v k , z k0 ) := max (173) By (131), we obtain that

Ω(v ′ k , z ′ k0 ) ≥ α v ′ k 2 2 + |z ′ k0 | 2 , (174)
RR n 6707

Lemma 4 . 1 .

 41 For any u ∈ U, we have that meas(B εD (u)) ≤ ε D .

By lemma 4 . 7 ,

 47 the restoration property (definition 4.2) is satisfied. So there exists ŵk ∈ F (P ) such that (η ′ k being the initial-final state associated with w ′ k ):   (i) ŵk -w ′ k ∞ = O(d(η ′ k )); (ii) J( ŵk ) = J μ(w ′ k ) + O ( w ′ k -w P d(η ′ k )) . (iii) ûk (t) = ū(t), for a.a. t ∈ B k . (146)Combining (142) and (146)(ii), obtainJ μ(w k ) ≥ J( ŵk ) + c H γ ′′ k + O ( w ′ k -w P d(η ′ k )) + o(γ k ). (147)Since meas(B k ) → 0, and hence,u ′′ k -ū 1 = o( u ′′ k -ū 2 ), we have that |Φ(η ′ k ) -Φ(η k )| = O( u ′′ k -ū 1 ) = o((γ ′′ k ) 1/2 ). (148) Consequently d(η ′ k ) = d(η k ) + o((γ ′′ k ) 1/2 ). (149) Since w ′ k -w P = O((γ ′ k ) 1/2), we deduce from (147) thatJ μ(w k ) ≥ J( ŵk ) + c H γ ′′ k + O (γ k ) 1/2 d(η k ) + o(γ k ).(150)Using (73), we deduce that, for large enough k, J(w k ) ≥ J( ŵk ) + c H γ ′′ k -1 2 μΦ(η k ) + o(γ k ).

(i)

  |v k (t)| ≤ |û k (t) -ū(t)|, for a.a. t; (ii) v k ∞ = O( ûk -ū ∞ ). (161) We now fix ε k := ( u k -ū ∞ + |y k0 -ȳ0 |)1/2 . Using (146)(i), (149), and γ and denoting by 1 A k (t) the characteristic function of the set A k , obtain

  ( w, p, λ)(v k , z k ) 2 dt + (Φ μ) ′′ (η)(z k0 , z k (T )) 2 .

  H u ( w(t), p(t))(û k (t) -ū(t))dt + O(γ k ). (155)In view of (152), we deduce that Note that v k is measurable and that, in view of the restoration property, ûk = ū a.e. on B k , and hence, v k = vk (t) = 0 a.e. on B k . If i ∈ I t , since g i (ū(t)) = 0 and g i (û k (t)) ≤ 0, we have a.e.

	(159)
	so that by (153) and (156)-(159):

T 0 T 0 H u ( w(t), p(t))(û k (t) -ū(t))dt ≤ O(γ k ). (

156

)

Consider the projection, for each time t, of the displacement ûk (t) -ū(t) over the local critical cone, and its difference with the projected direction:

v k (t) := P Ct (û k (t) -ū(t)); vk := v k -(û k -ū). (

157

)

g ′ i (ū(t))(û k (t)-ū(t)) ≤ g i (û k (t))+O(|û k (t)-ū(t)| 2 ) ≤ O(|û k (t)-ū(t)| 2 ). (

158

)

By the definition of sets A k and B k , a.e. on [0, T ], we have that

|v k (t)| ≤ 1 ε k [H u ( w(t), p(t))(û k (t) -ū(t))] + + i∈It [g ′ i (ū(t))(û k (t) -ū(t))] + ,

RR n 6707

We first note that this condition implies the uniqueness of the "µ part" of the multiplier. Lemma 4.3. Let w ∈ F (P ) satisfy the special qualification condition (78). Let (λ i , µ i ) ∈ LM L ( w), for i = 1, 2. Then µ 1 = µ 2 . Proof. Let (λ i , µ i ) ∈ LM L ( w), for i = 1, 2. Set λ := λ 2 -λ 1 and µ := µ 2 -µ 1 . Let e = (v, z 0 ) ∈ E εR . Then λ(t)g ′ (ū(t))v(t) = 0 for a.a. t, and therefore, taking the difference of equations of stationarity of Lagrangians (last relation in (65)), µG ′ (ū, ȳ0 )(v, z 0 ) = 0. Since by (78) G ′ I+ (ē) is onto from E εR onto R |I+| , it follows that µ = 0, as was to be proved.

When dealing with a Pontryagine perturbation w k = (u k , y k ), in the process of restoration, it is useful to freeze control variations set of small measure where |u k -ū| is large. So, given an arbitrary measurable subset B (of small measure) of [0, T ], we need the following notation:

Denote by e j = (u j , z j 0 ) the components of vectors e j in (78), j = 0, . . . , |I + |, and define e j B = (u j B , z j 0B ) in E εR (B), for j = 1, . . . , |I + |, by

Since the functions u j , j = 1, . . . , |I + |, are essentially bounded, we have that

For j = 0, we proceed in two steps. First define ê0 B = (û 0 B , z 0 0B ) in the same way, i.e.,

Now define

where the coefficient ᾱ is solution of

Lemma 4.4. Let w ∈ F (P ) satisfy the special qualification condition (78).

Then there exists ε B > 0 such that, for any measurable subset B of [0, T ], if meas(B) ≤ ε B , then (85) has a unique solution, and {e 0 B , . . . , e

INRIA

It is well-known that Robinson's qualification condition (60) implies the following local Hoffman bound: there exists c > 0 such that, if w ∈ W satisfies the state equation ( 2), and is close enough to w, then there exists ŵ ∈ F (P ) such that ŵ -w W ≤ cD(w).

We now check that this inequality still holds in some cases if w ∈ W is close enough to w in the Pontryagine norm.

Let e 0 := (zv, z0 ) and β > 0 be the direction and constant stated in (60). By arguments similar to those in the proof of lemma 4.4 we obtain the existence of a direction e 0

Let T be the mapping from U × R n into itself, that with a given (u, y 0 ) ∈ U × R n associates (u, y 0 ) + δe, where δe = (δu, δy 0 ) is a vector of Span{e 0 B , . . . , e r1 B } of minimum norm satisfying the following conditions:

(96) We obtain by arguments similar to those at the end of the proof of lemma 4.5 that the sequence computed by the mapping T , with initial point (u, y 0 ) such that u ∞ ≤ M and w -w P is small enough converges to a point (û, ŷ0 ) such that the corresponding associated state ŷ satisfies the conclusion of the lemma. 172)

By ( 164)-(166), and since LM L ( w) is bounded, we have

and hence, in view of ( 175) and ( 164)-(166) again:

For a given λ such that (λ, μ) ∈ LM L ( w), adding to (37) (written for the sequence ŵk ) the inequality 0 ≥ T 0 λ(t)(g(û k (t))-g(ū(t))) dt and using H a u ( w, p, λ) = 0, obtain

so that, maximizing w.r.t. the bounded set LM L ( w), we obtain in view of ( 177)

Combining with (145) and (154), we get for some α ′ > 0:

By (149), d(η ′ k ) 2 ≤ 2d(η k ) 2 + o(γ k ) and so using also (145)

Using ( 73) and (145), we see that this gives a contradiction to (152), as was to be shown.


MSc Catarina Veiga

PhD Guillaume Janssens

PhD; Ching-Ling Teng

PhD; Thomas Baudier

PhD Lucian Hotoiu

PhD Jamie R Mcclelland

PhD Gary Royle

PhD Liyong Lin

PhD Lingshu Yin

PhD James Metz

PhD Timothy D Solberg

PhD Zelig Tochner

II Charles B Simone

MD James Mcdonough

PhD, Boon-Keng Kevin Teo

PhD ; Lingshu Yin

PhD ; Boon-Keng

First clinical investigation of CBCT and deformable registration for adaptive proton therapy of lung cancer

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

Lung cancer is the leading cause of cancer-related death in the United States and worldwide. Modern radiotherapy techniques allow dose escalation [START_REF] Belderbos | Final results of a phase I/II dose escalation trial in non-small-cell lung cancer using three-dimensional conformal radiotherapy[END_REF][START_REF] Machtay | Higher biologically effective dose of radiotherapy is associated with improved outcomes for locally advanced non-small cell lung carcinoma treated with chemoradiation: An analysis of the radiation therapy oncology group[END_REF] and reduced injuries to normal tissues [START_REF] Rosenzweig | Results of a phase I dose-escalation study using threedimensional conformal radiotherapy in the treatment of inoperable nonsmall cell lung carcinoma[END_REF][START_REF] Bradley | Toxicity and outcome results of RTOG 9311: A phase I-II dose-escalation study using three-dimensional conformal radiotherapy in patients with inoperable non-small-cell lung carcinoma[END_REF][START_REF] Bradley | A phase I/II radiation dose escalation study with concurrent chemotherapy for patients with inoperable stages I to III non-small-cell lung cancer: Phase I results of RTOG 0117[END_REF][START_REF] Levin | Proton beam therapy[END_REF] . Proton therapy offers better dose localization than that achieved by conventional photon therapy. [7][START_REF] Roelofs | Results of a multicentric in silico clinical trial (ROCOCO) comparing radiotherapy with photons and protons for non-small cell lung cancer[END_REF][START_REF] Simone Cb 2nd | Comparison of intensity-modulated radiotherapy, adaptive radiotherapy proton radiotherapy, and adaptive proton radiotherapy for treatment of locally advanced head and neck cancer[END_REF][START_REF] Chang | Significant reduction of normal tissue dose by proton radiotherapy compared with three-dimensional conformal or intensity-modulated radiation therapy in stage I or stage III non-small-cell lung cancer[END_REF][START_REF] Kesarwala | Intensity-modulated proton therapy for elective nodal irradiation and involved-field radiation in the definitive treatment of locally advanced non-small-cell lung cancer: a dosimetric study[END_REF] Proton dose distribution, however, is highly sensitive to changes in patient geometry, especially in the lungs. [START_REF] Berman | Proton beam therapy for non-small cell lung cancer: Current clinical evidence and future directions[END_REF] For example, interfractional tumor enlargement or development of atelectasis increase density along the beam path and shorten beam penetration. The under-ranging can potentially reduce target coverage. Conversely, tumor regression reduces density along the beam path and increases beam penetration. The over-ranging may result in unplanned dose to otherwise spared organs distal to the tumor volume. Therefore, accurate patient positioning and regular evaluation computed tomography (CT) scans are critical components of proton therapy. [START_REF] Wink | Particle therapy for non-small cell lung tumors: Where do we stand? A systematic review of the literature[END_REF] Replanning will be required if the new dose distribution based on evaluation CTs compromises target coverage and/or exceeds tissue tolerance.

Volumetric imaging afforded by on-board cone-beam CT (CBCT) is an alternative to routine CT imaging and may play an important role in adaptive radiation therapy (ART). The advantages of onboard CBCT are threefold: (1) it offers highly accurate patient positioning in three dimensions; [START_REF] Yang | Evaluation of on-board kV cone beam CT (CBCT)-based dose calculation[END_REF][START_REF] Grills | Image-guided radiotherapy via daily online cone-beam CT substantially reduces margin requirements for stereotactic lung radiotherapy[END_REF] (2) it enables daily monitoring of the patient in the treatment position; and (3) and it facilitates rapid assessment of the "dose of the day". [START_REF] Landry | Phantom based evaluation of CT to CBCT image registration for proton therapy dose recalculation[END_REF][17][START_REF] Veiga | Cone-beam computed tomography and deformable registration-based "dose of the day" calculations for adaptive proton therapy[END_REF] We propose an ART workflow using on-board CBCT where replanning is triggered after three decision-points (Fig. 1). First, a fast range-corrected dose distribution based on water equivalent thickness (WET) is calculated on a virtual CT (vCT) derived from the CBCT. [START_REF] Peroni | Automatic segmentation and online virtual CT in headand-neck adaptive radiation therapy[END_REF] When significant dosimetric changes are observed, treatment may continue if normal tissue dose limits are not exceeded and after consultation with a physician. However, an offline review is triggered for a full dose recalculation on the vCT. If the dosimetric impact is still evaluated as significant, a rescan CT (rCT) is scheduled. If dosimetric changes are confirmed on the rCT, a replan is triggered. We retrospectively evaluated this workflow for twenty consecutive lung cancer patients summarizing common radiationinduced changes in the lung and critically assessed the workflow. The rCT was used as the goldstandard to gauge the accuracy of the vCT. To our knowledge, this study is the first clinical investigation of CBCT in adaptive lung proton therapy.

METHODS AND MATERIALS

Patient selection and data acquisition

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Data from twenty consecutive patients treated for lung malignancies were included in this retrospective study. All patients underwent passive scattering proton therapy (PSPT) using two treatment fields with a median dose of 66.3Gy[CGE] (range:40-66.6Gy [CGE]) in a median of 1.8Gy/fraction (range:1.5-4Gy/fraction). The patient cohort included a variety of tumor sizes, locations and anatomical changes that occurred throughout the treatment course (Table 1). The imaging protocol consisted of a 4D PET/CT for treatment planning, CBCT and rescan 4D CT acquired in treatment position for verification during the course of treatment. The proton-gantry mounted CBCT system (Ion Beam Applications SA, Ottignies-Louvain-la-Neuve, BE) has a sourceto-axis distance of 288.4cm, detector-to-axis distance of 58.6cm and a maximum field-of-view (FoV) of 34cm; the images were acquired in half-scan mode at 110kVp and 1142mAs. The CBCTs were reconstructed with a resolution of 1.33×1.33×2.5mm [START_REF] Rosenzweig | Results of a phase I dose-escalation study using threedimensional conformal radiotherapy in the treatment of inoperable nonsmall cell lung carcinoma[END_REF] using the open-source Reconstruction Toolkit (RTK). [START_REF] Rit | The Reconstruction Toolkit (RTK), an open-source cone-beam CT reconstruction toolkit based on the Insight Toolkit (ITK)[END_REF] The average phase of the 4D CT was used for dose calculations. One pair of CBCT and rCT at mid-treatment was selected for evaluation for each patient. The chosen rCT and CBCT scans were acquired on the same day for 17 patients; for 3 they were acquired within 2 days of each other.

Virtual CT and image correction

The CBCT was rigidly aligned to the planning CT (pCT) and the diffeomorphic Morphons deformable image registration (DIR) algorithm available in the open-source REGGUI package was subsequently applied to register the pCT to the CBCT. [START_REF] Janssens | Diffeomorphic registration of images with variable contrast enhancement[END_REF] This method was previously validated for head and neck cancer patients. [START_REF] Landry | Phantom based evaluation of CT to CBCT image registration for proton therapy dose recalculation[END_REF]17 Some anatomical changes in the thorax cannot be modeled by deformation alone. The situations are diverse, but include changes within the lung (such as atelectasis and pleural effusion) and different tumor responses to treatment (such as regression of infiltrating tumors and erosion [START_REF] Sonke | Adaptive radiotherapy for lung cancer[END_REF] ). Therefore, a semi-automatic correction step was applied to the vCT for these anatomical modifications. This step consists of a watershed-cuts algorithm [START_REF] Bernard | Incremental classification of objects in scenes: Application to the delineation of images[END_REF][START_REF] Cousty | Watershed cuts: minimum spanning forests and the drop of water principle[END_REF] combined with an exclusive OR logical operator and a classifier [START_REF] Freund | Large margin classification using the Perceptron algorithm[END_REF][START_REF] Gallant | Perceptron-based Learning Algorithms[END_REF] to identify regions showing significant intensity mismatch between vCT and CBCT.

When appropriate, the vCT intensities were replaced by the bulk value of lung or tissue based on thresholding of the CBCT intensities. This algorithm was validated in-house [START_REF] Xxx | The accuracy of deformable registration for adaptive lung proton therapy[END_REF] ; technical details of the implementation can be found as supplementary material.

Clinical indicators: WET and dose warping

A complementary set of clinical indicators, based on WET and dose, that support the replanning decision-making, were used to estimate the impact of anatomical changes on the treatment objectives.

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Variation in WET on the distal surface of the target is a good surrogate for potential under/overranges. Changes in range were estimated by computing the difference between the WET from the pCT and vCT (i.e. WET pCT >WET vCT corresponds to over-ranging and WET pCT <WET vCT to under-ranging).

In PSPT compensator smearing is used to ensure target coverage in presence of errors in patient positioning and motion. [START_REF] Urie | Compensating for heterogeneities in proton radiation therapy[END_REF] Therefore, potential under-ranges are partially taken into account by the compensator; thus, for under-ranges a morphological dilation using the same radius as the compensator is applied on the pCT-based WET map before computing the difference with the vCTbased WET map to identify under-ranges not accounted for in smearing. Quantitative measurements of the 2D WET difference maps at the PTV distal surface (per beam) for the pCT and vCT were calculated as clinical indicators, which include the percentage of pixels with under/over-ranges larger than 3mm (WET under>3mm /WET over>3mm ) and the 95% percentile of the under/over-range distribution (WET under-95% /WET over-95% ).

Complementary to the WET analysis, it is necessary to evaluate the dosimetric impact of under/overranging, particularly for OAR. Online dose review requires a fast dose recalculation tool; thus we validated in-house a fast range-corrected dose approximation method 29 by warping the original dose onto the vCT based on changes in the 3D WET maps between the vCT and pCT. Doses were recalculated using Eclipse version 11.0 (Varian Medical Systems, Palo Alto, CA). The following abbreviations are used to identify the different doses: pCT dose (D pCT ), vCT warped dose (D vCT-WET ), vCT recalculated dose (D vCT ) and rCT recalculated dose (D rCT ). DVHs and dosimetric statistics representative of target coverage and OAR tolerances used in-house to trigger replanning were chosen as dose indicators. These were for PTV and iCTV the V 95% and V 99% , respectively, with a threshold of 3% change in the rCT. For OARs the dose tolerances used were: for heart, D max =72Gy, V 45Gy <35% and V 30Gy <50%; for esophagus, D max =70Gy and V 55Gy <30%; for cord (and cord+5mm), D max =50Gy(65Gy); and for brachialplexus, D max =66Gy. The iCTV/PTV contours were rigidly propagated while the OARs were propagated using DIR.

RESULTS

Case studies of different anatomical changes

Lung changes

Atelectasis is the collapse of lung that is sometimes reversible. PT#1 developed partial atelectasis at the upper left lobe during week two (Fig. 2) resulting in increased WET along the beam paths and subsequent under-ranging (WET under-95% =10.4/12.3mm for LPO 1 /LPO 2 field). Tumor coverage was compromised and a higher dose was delivered to the esophagus (D max from 50Gy to 71/71/68Gy for When tumors regress, the previously blocked airway can reopen and reinflate the collapsed lung (PT#3, Fig. 3). Lung reinflation reduced the WET along the beam path, and caused beam overranging (WET over-95% =41.4/44.1mm for RPO/PA field). The change in dose distribution compromised tumor coverage (iCTV ∆V 99% =-27/-27/-13% for D vCT-WET /D vCT /D rCT ), which triggered replanning. The predicted loss of coverage was higher in the vCT than in rCT, which can be attributed to the partial truncation of the CBCT at the beam entrance (see section 3.1.3).

3.1.2: Tumor changes

Different tumor response scenarios were identified and detailed below.

Infiltrating tumors

For PT#14, the GTV decreased from 4.1 to 2.7cm in diameter after four weeks of treatment (Fig. 3).

The uncorrected vCT resulted in DIR errors of the lung tissue between the tumor and chest wall. After applying the correction algorithm, the clinical indicators were nearly identical between vCT and rCT (iCTV ∆V 99% =-5/-7/-6% for D vCT-WET /D vCT /D rCT ).

Tumor regression

When tumors regress, the topological changes may not be handled by DIR alone. In PT#2 a 22.3mm cavity appeared within the original tumor volume (Fig. 2). Its size and location were accurately identified and accounted for by the vCT correction step. The reduction in WET along the beam path resulted in beam over-ranging to the heart (WET over-95% =24.6/25.2mm for RPO 1 /RPO 2 field, Figs. 4 and5). Dosimetric indicators between vCT and rCT were similar (Fig. 4 proton range, i.e., WET over>3mm =27.0/40.7% for vCT/rCT (RPO 1 field). Regardless of the differences in WET between the vCT and rCT, identical reduction in dose coverage was detected in the DVHs.

Moderate shrinkage/enlargement

Moderate tumor regression was here defined as a visually apparent change in tumor volume less than 25% of its original GTV. An example is PT#11, who has focal shrinkage resulting in modest beam over-ranging (WET over-95% =8.5/7.5mm for AP/RPO field), and an increase in dose to the cord (D max from 30 to 36/37/37Gy for D vCT-WET /D vCT /D rCT ). PT#16 was the only case of tumor enlargement during radiation treatment; the diameter increased ~5mm along the beam path. Because of the complex organ geometry at the mediastinum, both beam under/over-ranging were observed (i.e., WET under-95% =1.1mm and WET over-95% =3.6mm for RPO field), resulting in increased dose to the cord (D max from 35 to 46/47/40Gy for D vCT-WET /D vCT /D rCT ), and a right shift of iCTV/PTV DVH curves. In these two cases the vCT and rCT offered similar clinical indicators with DIR alone.

3.1.3: Other factors affecting clinical indicators

In addition to DIR errors, setup variations and differences in the respiratory pattern between CBCT and rCT scans can result in differences between D vCT-WET /D vCT and D rCT . This implies that different clinical indicators are being extracted, and can, therefore, create false positive/negative triggers for offline review. Our online workflow was less robust for patients in whom the magnitude of dose differences arising from setup variations was comparable to those arising from internal anatomical change. For PT#13, tumor shrinkage was well recovered by DIR; however the position of the main bronchi was shifted superiorly in rCT in comparison to the CBCT (Fig. 3). The movement of the main airway cause different predictions of target coverage (iCTV ∆V 99% =+3/+1/-4% for D vCT- WET /D vCT /D rCT ). WET difference maps were also affected: the magnitude of WET over-95% was small but the 2D WET maps gave different indicators (WET over>3mm was 18.8/46.6% for vCT/rCT for PT#19 RPO field, see Fig. 5). Clinically, none of these patients required replanning as the dosimetric changes were generally small and the effects averaged out during the course of treatment.

Setup errors should not be confused with systematic drift of tumor position through the treatment course. For PT#5, the primary tumor shifted in the inferior direction, and this was consistent between the CBCT and rCT. DIR accurately described the change in tumor position, but the modest change in WET had minimal effect on target coverage or dose to OAR.

Due to FoV limitations, a minority of CBCTs did not encompass the entire exterior of the patient body at the beam entrance (Fig. 2 and3). For PT#4, the CBCT truncation resulted in overestimation

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of over-ranging of the RPO field (WET over-95% =19.1/11.0mm for rCT/vCT), but the overall changes in range and dosimetry were small (Fig. 4). Nevertheless, uncorrected truncation may in some cases lead to inaccurate clinical indicators (PT#3: iCTV ∆V 99% =-27/-27/-13% and, PT#17: iCTV ∆V 99% =0/0/-5% for D vCT-WET /D vCT /D rCT ).

General results

Eighteen of the twenty patients exhibited tumor regression resulting in over-ranging of the proton beams; two showed considerable under-ranging. These were cases of atelectasis (PT#1) and tumor enlargement (PT#16). For all patients, the average absolute difference in WET over-95% and WET over>3mm between vCT and rCT were 3.4±2.7mm and 12±12%, respectively. Figure 5 shows examples of WET and WET difference maps. WET difference maps identified the same regions of under/over-ranging for all patients with large anatomical changes. This was true even for PT#8 and PT#20, in spite of the full magnitude of the over-ranging not being fully recovered due to limitations of the corrected vCT to reproduce complex shrinkage and/or density changes (Fig. 5). In cases of smaller changes or setup variations, the WET difference maps were less uniform. In general, values of WET over-95% needed to exceed 15mm before significant dosimetric changes could be detected. The most common issues that could lead to a replan were loss of tumor volume coverage, increase in maximum dose to the cord, and over-ranging of dose into the heart.

DISCUSSION

In proton therapy, accurate Hounsfield units are a requirement to make clinical decisions for ART.

CBCT plays an important role in image-guided therapy, and vCTs is one step in that direction and may play a complementary role to rCT.

In terms of WET information, although the vCT may not reproduce identical WET maps, it identifies the same trends as the rCT regarding the effect of the WET changes. 90% of the fields with WET under- 95% /WET over-95% larger than 10mm were properly identified as such from the vCT. The dose warping method reproduced similar clinical indicators for patients with considerable changes that may trigger a replan. The most common issue was loss of target coverage of the iCTV. For PT#1, PT#2 and PT#14, the impact to OARs was detected (esophagus, heart/cord and cord respectively). For PT#8, the changes in OAR dose were not properly detected, while for PT#20 an increase in cord dose was incorrectly detected. When smaller changes occurred, differences in OAR dose were also detected (PT#11, PT#12 and PT#16), but some false positives/negatives occurred for loss of target coverage (PT#13, PT#15 and PT#17). Variations in setup can result in overestimation (PT#9) and underestimation (PT#19) of over-ranging, but with minimal dosimetric impact. In general, OAR doses

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were maintained within tolerance; however, special care should be given to fields that point towards an OAR, such as lateral oblique fields that may range out at the heart (PT#2, PT#7 and PT#12).

An important conclusion taken from this retrospective investigation was the necessity to evaluate multiple parameters during the ART decision-making process: changes in WET, qualitative review of images and dose distributions, DVHs and corresponding dose-statistics. Flags raised by a single indicator should be backed by additional evidence. For example, iCTV-V 99% statistic was quite sensitive even when the DVHs did not reflect major changes. In cases where the anatomical changes are small the decision to replan should not be based on individual scans, but rather on continued monitoring. Smaller changes can in fact be comparable to setup errors and may average out.

The limitations of the proposed workflow were identified through an evaluation using a diverse cohort of patients. DIR has inherent uncertainties and associated errors, and traditional algorithms are not adequate when tissue appears/disappears (such as atelectasis). If unaccounted for, such situations result in significant errors in WET/range estimation. The correction step works well for gross registration errors, but cannot recover complex changes in tumor topology. When applying the method prospectively and if the correction step is not adequate, manual adjustments to the vCT may be necessary during the second decision-making point (i.e., offline review of vCT). Using CBCT directly for dose recalculation is a viable alternative to remove the errors associated with DIR. Despite the vast work on directly using CBCT in conventional photon therapy [START_REF] Yang | Evaluation of on-board kV cone beam CT (CBCT)-based dose calculation[END_REF][START_REF] Guan | Dose calculation accuracy using cone-beam CT (CBCT) for pelvic adaptive radiotherapy[END_REF] and/or to improve CBCT image quality [START_REF] Li | Feasibility of improving cone-beam CT number consistency using a scatter correction algorithm[END_REF][START_REF] Park | Proton dose calculation on scatter-corrected CBCT image: feasibility study for adaptive proton therapy[END_REF] , its usability is still limited in proton therapy [START_REF] Veiga | Cone-beam computed tomography and deformable registration-based "dose of the day" calculations for adaptive proton therapy[END_REF][START_REF] Kurz | Comparing cone-beam CT intensity correction methods for dose recalculation in adaptive intensity-modulated photon and proton therapy for head and neck[END_REF] and therefore the corrected vCT is a good interim solution. For smaller registration errors, we identified two common patterns of failure: first, the interface of lung-tissue-bone at the posterior rib wall due scatter artifacts, and second, the positioning of the scapula (which can often move in and out of the path of lateral oblique fields).

From a clinical perspective, two scenarios are possible with DIR errors. The first is a false positive trigger, i.e., the dose calculated on the vCT indicated a change in dosimetry when there is none. The outcome is an unnecessary CT scan to confirm the findings. The other is a false negative trigger, i.e., the dose calculated on the vCT failed to detect the change in dosimetry. While this scenario poses a bigger risk it is unlikely to occur. Higher DIR errors are associated with larger anatomical changes, and in such cases variations in dosimetry are usually still predicted even if with a different magnitude.

The use of CBCT instead of rCT has its own associated challenges. First, subtle changes in density of lung tissues between planning and verification (PT#20) were undetectable on CBCT but apparent on rCT. Second, the limited FoV and artifacts caused by the couch may result in incomplete information of the external contour. This was problematic for some lateral oblique fields (10% of fields); regardless, it can be avoided by closely matching the patient geometry near the beam entrance during

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image acquisition. This is not a limitation of the proposed workflow for CBCT systems with larger FoVs.

We established that a vCT is comparable to rCT; the next step is to test the workflow on a larger cohort and transition the workflow to clinical routine. The clinical indicators investigated here were empirically sensible; yet further investigations are required for defining the appropriate action threshold for replanning. On the technical side, improvement of the workflow to minimize its current limitations is a priority, which includes investigating lung-specialized DIR algorithms, [START_REF] Al-Mayah | Toward efficient biomechanical-based deformable image registration of lungs for image-guided radiotherapy[END_REF] automatic segmentation validation, [START_REF] Peroni | Automatic segmentation and online virtual CT in headand-neck adaptive radiation therapy[END_REF] improvement of CBCT image quality [START_REF] Park | Proton dose calculation on scatter-corrected CBCT image: feasibility study for adaptive proton therapy[END_REF] and integration with TPS or using more accurate dose calculations. [START_REF] Jia | GPU-based fast Monte Carlo dose calculation for proton therapy[END_REF]

CONCLUSIONS

We retrospectively evaluated a novel workflow to quantitatively assess WET and dose distributions using CBCT for proton therapy. This workflow was shown to provide similar clinical indicators as rCT on patients with considerable interfractional anatomical changes.

FIGURE LEGENDS

Table 1-Patient characteristics.

Figure 1-Workflow for ART. A corrected vCT was created using pCT-to-CBCT DIR, and the variation in WET between vCT and pCT was used to range-correct the planned dose. This process can be performed online to trigger an in-depth offline review of the vCT (and rCT) if deemed necessary. PT#2 and PT#4. For PT#1 the appearance of atelectasis increased the WET, resulting in underranging and loss of iCTV coverage. For PT#2 the shrinkage of the GTV resulted in decreased WET, and thus in over-ranging and increase in dose delivered to the heart and cord. For PT#4 the changes in WET were small in spite of CBCT truncation, resulting in similar dose distributions and DVH curves.

  WET /D vCT /D rCT ) which triggered immediate replanning. The vCT predicted similar dosimetric indicators as the rCT.

  ), i.e., decreased iCTV coverage (iCTV ∆V 99% =-7/-6/-8% at D vCT-WET /D vCT /D rCT ), and increased dose to the cord (D max from 45 to 52/52/49Gy for D vCT-WET /D vCT /D rCT ) and the heart (V 45Gy from 25% to 31/31/35% for D vCT- WET /D vCT /D rCT ). Changes in tumor density PT#20 had both regression and changes in tumor density; the average intensity of the GTV decreased from 30 to -110HU between pCT/vCT and rCT, corresponding to a local WET variation of approximately 7mm. The vCT retained the HUs from the pCT and underestimated the change in M A N U S C R I P T A C C E P T E D ACCEPTED MANUSCRIPT

Figure 2 -

 2 Figure2-Images used and generated by the workflow for PT#1 (atelectasis), PT#2 (large tumor shrinkage) and PT#4 (small tumor shrinkage). A consistent soft tissue window level was used for CT datasets (except the CBCT). For PT#1 and PT#2, the vCT needed the correction step (region defined by the red contour). For PT#4, DIR alone recovered the changes well; however, CBCT truncation affected the similarity between vCT and rCT.

Figure 3 -

 3 Figure3-Images used and generated by the workflow for PT#3 (lung reinflation), PT#14 (regression of infiltrating tumor) and PT#13 (shrinkage and changes in breathing pattern). A consistent soft tissue window level was used for CT datasets (except the CBCT). For PT#3 and PT#14, correction of the vCT was necessary (region defined by the red contour). For PT#13, DIR was able to recover the tumor shrinkage; however, visible differences in setup occur between vCT and rCT, particularly in the position of the main airways (yellow arrow).

Figure 4 -

 4 Figure 4-(A) Color overlay of the CTs and corresponding dose distributions and (B) DVHs for PT#1,

Figure 5 -

 5 Figure 5-WET and WET difference maps for PT#2 (RPO 1 field), PT#8 (PA field) and PT#19 (RPO field).

  map planning CT WET map rescan CT WET map map of over-ranges (vCT) map of over-ranges (rCT) PT#19 -RPO feld virtual CT WET map planning CT WET map rescan CT WET map map of over-ranges (vCT) map of over-ranges (rCT)

Table 1 -

 1 Patient characteristics F rCT /F t =treatment fraction at which rCT was acquired/total number of fractions c SI=superior-inferior; LR=left-right; AP=anterior-posterior d Large tumor shrinkage = visually apparent change in tumor volume greater than 25% of its original GTV.

	PT #	Age (Y)	G	TNM	Fields (gantry angle/name)	a V iCTV (cm 3 )	F rCT /F t b	Characteristics at verification scan	Tumour motion: SI/LR/AP c (mm) pCT rCT
	1	85	M	T3N2M0	175(LPO 1 ) & 135(LPO 2 )	130	9/37	Atelectasis	6/6/4	6/7/5
	2	71	F	T3N2M0	210(RPO 1 ) & 220(RPO 2 )	410	20/37	Large tumor shrinkage d	8/5/5	10/4/4
	3	68	F	T4N2M0	220(RPO) & 180(PA)	280	4/30	Lung reinflation	10/4/10	12/3/2
	4	69	M	T4N2M0	180(PA) & 220(RPO)	280	21/37	Small tumor shrinkage/setup error	5/1/3	5/1/3
	5	72	M	T3N3M0	180(PA) & 10(RAO)	260	8/27	Tumor position drift	9/2/4	8/3/2
	6	81	M	T4N0M0	180(PA) & 155(LPO)	320	9/37	Small tumor shrinkage/setup error	3/1/2	2/1/1
	7	77	M	T2aN2M0	180(PA) & 200(RPO)	180	10/37	Small tumor shrinkage/setup error	7/3/2	5/2/2
	8	62	F	T4N2M0	270(ASO) & 180(PA)	340	4/34	Large regression of infiltrating tumor	2/3/6	4/2/5
	9	64	M	T2aN0M0	180(PA) & 155(LPO)	130	22/37	Small tumor shrinkage/setup error	4/2/2	4/1/3
	10	65	M	T4N1M0	180(PA) & 150(LPO)	180	10/25	Small tumor shrinkage/setup error	5/3/4	8/3/4
	11	31	M	T2aN2M0	0(AP) & 205(RPO)	200	20/37	Moderate tumor shrinkage e	7/2/5	7/4/4
	12	76	F	T2bN0M0	180(PA) & 210(RPO)	150	20/37	Moderate tumor shrinkage	2/0/0	2/0/1
	13	71	M	T3N2M0	180(PA) & 145(LPO)	190	21/25	Moderate tumor shrinkage	6/3/3	6/3/6
	14	65	F	T4N0M0	180(PA) & 150(LPO)	500	20/37	Large regression of infiltrating tumor	2/2/2	1/0/0
	15	57	F	T3N2M0	180(PA) & 224(RPO)	340	10/30	Moderate tumor shrinkage	3/2/0	3/3/1
	16	58	F	T4N0M0	205(RPO) & 20(LAO)	580	15/33	Moderate tumor enlargement	6/6/4	2/1/1
	17	62	F	T1aN2M1b	173(LPO) & 150(LPO)	100	5/37	Moderate tumor shrinkage	2/0/2	3/1/4
	18	67	M	T3N1M0	0(AP) & 335(RAO)	330	20/30	Small shrinkage/setup error	3/3/3	5/3/3
	19	69	M	T3N0M1b	180(PA) & 205(RPO)	310	7/16	Small shrinkage/setup error	2/2/2	1/1/1
	20	78	M	T2aN0M1b	195(RPO) & 210(RPO)	140	9/15	Large tumor shrinkage; density changes	5/2/2	5/4/3

a V iCTV =volume of the iCTV b e Moderate tumor shrinkage = visually apparent change in tumor volume less than 25% of its original GTV.

Acknowledgments: C. V. was funded by Fundação para a Ciência e a Tecnologia (FCT) grant SFRH/BD/76169/2011, co-financed by FSE, POPH/QREN and EU. J.M. was supported by the EPSRC Intelligent Imaging Program Grant EP/H046410/1. This work was supported by the US Army Medical Research and Materiel Command under Contract Agreement no DAMD17-W81XWH-04-2-0022. Opinions, interpretations, conclusions and recommendations are those of the author and are not necessarily endorsed by the US Army.