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A geometrically based criterion to avoid infimum-gaps in Optimal Control

In optimal control theory the expression infimum gap means a strictly negative difference between the infimum value of a given minimum problem and the infimum value of a new problem obtained by the former by extending the original family V of controls to a larger family W. Now, for some classes of domain-extensions -- like convex relaxation or impulsive embedding of unbounded control problems -- the normality of an extended minimizer has been shown to be sufficient for the avoidance of an infimum gaps. A natural issue is then the search of a general hypothesis under which the criterium 'normality implies no gap' holds true. We prove that, far from being a peculiarity of those specific extensions and from requiring the convexity of the extended dynamics, this criterium is valid provided the original family V of controls is abundant in the extended family W. Abundance, which is stronger than the mere C^0-density of the original trajectories in the set of extended trajectories, is a dynamical-topological notion introduced by J. Warga, and is here utilized in a 'non-convex' version which, moreover, is adapted to differential manifolds. To get the main result, which is based on set separation arguments, we prove an open mapping result valid for Quasi-Differential-Quotient (QDQ) approximating cones, a notion of 'tangent cone' resulted as a peculiar specification of H. Sussmann's Approximate-Generalized-Differential-Quotients (AGDQ) approximating cone.


Introduction
One of the main reason for enlarging the domain of a minimum problem relies on the aim of establishing the existence of at least one solution. Actually, domain extension is a quite common and variously motivated practice, in particular in the Calculus of Variations and in Optimal Control. Of course, a crucial requisite of such a domain enlargement consists in the density of the original problem in the new one: the extended minimum should be approximable by processes of the original problem. However, because of the presence of a final target, even a dense extension of the domain may result in the occurrence of an infimum gap: namely, it can happen that the infimum value of the original problem is strictly greater than the infimum value of the extended problem. This might be undesirable in many respects, for instance in the convergence of numerical schemes as well as in the identification of the value function via Hamilton-Jacobi equations. This raises a natural question: how can one avoid this gap phenomenon? A sufficient condition for gap avoidance seems to emerge from investigations by J. Warga [42,43,44,45] and from some other more recent papers [28,31,32,33,34], dealing with some particular cases: this criterion is the so-called normality of minimizers. Therefore, the mentioned question can be turned into the following one: Q.Under which hypotheses on a general optimal control problem normality is sufficient for gap-avoidance?
In order be more precise, let us briefly sketch the abstract setting of our optimal control problem. The state variable y will range on a Riemannian manifold M, while the control maps v(·) will belong to an original family V ⊂ W := L 1 ([0, S], W) -where W is a subset of a metric space-or to a larger set W, which will be called the extended family of controls. Given an initial stateȳ ∈ M and a time interval [0, S], we will consider the control system (E) dy ds (s) = f (s, y(s), w(s)) y(0) =ȳ, and, for every w ∈ W, we will use y[w] : [0, S] → M the corresponding (supposedly unique) solution. The original optimal control problem is defined as where the cost function h : M → R is continuous, and T ⊂ M is a closed set called target.
Replacing the family of controls V by the larger set W, one obtains the extended optimal control problem: We will assume the existence of a local minimum for the extended problem, namely a controlŵ ∈ W such that, for some one says that the optimal control problem has an infimum gap at y [ŵ]. Obviously, via the usual reductions, one can formulate a notion of infimum gap for a general Bolza problem -where an integral cost is involved as well-. For problems defined on Euclidean spaces and such that the extended dynamics is convex, an insightfull investigation of the gap question and its relation with normality was carried out by J.Warga (see e.g. [43]). More recently two specific classes of domain extentions -still assuming the convexity of the extended dynamics-have been studied in [28,32,33,34]. As mentioned above, these investigations share the fact that a certain condition turns out to be necessary for the gap occurrence: (A) There is an infimum-gap only if the minimum of the extended problem is an abnormal extremal. 1 Since any version of the PMP states that 'an optimal process (ŷ,ŵ) := (y[ŵ],ŵ) for the extended problem (P ) W is an extremal', in order for (A) to have a precise meaning one has to specify which version of the Pontryagin Maximum Principle (PMP) one is considering. In turn, this is equivalent to specify which kind of approximating cones we are going to utilize for both the reachable set and the target T. Actually, for this goal we shall introduce a generalized differential called Quasi Differential Quotient (QDQ) (Def. 2.3) 2 and the 1 Equivalently: if the minimum is normal (=not abnormal) there is no gap. 2 A QDQ is a special case of Sussmann's Approximate Generalized Differential Quotient [40]. associated notion of QDQ approximating cone (Def. 2.5). Let us say immediately that this choice is perhaps the most important step for the validity of the main result. And while it is impossible at this stage to give an exhaustive description of what QDQ approximating cones are, let us point out that, on the one hand, they are sufficiently small for a fixed point theorem to hold true and, on the other hand, they are enough large to allow the utilization of the notion of abundance, which, as we shall see, proves crucial for normality to imply no gap.
This said, let us give the notion of extremal. For simplicity, we consider here only the case when the state ranges on a Euclidean space. Moreover, if C ⊂ R n is a cone, we use C ⊥ to denote the polar cone of C, namely the set of linear forms λ ∈ (R n ) * such that λ · c ≤ 0 for all c ∈ C. Definition 1.1 (Extremal). Consider a controlŵ ∈ W and the corresponding trajectorŷ y := y[ŵ]. Assume thatŷ(S) ∈ T, and let C be a QDQ approximating cone of the target T atŷ(S). We say that the process (ŷ,ŵ) is an extremal (with respect to h and C) if there exist an absolutely continuous (adjoint) path λ ∈ W 1,1 ([0, S]; (R n ) * ) and a cost multiplier λ c ∈ {0, 1} such that (λ, λ c ) = 0 and (iii) λ(S) ∈ −λ c ∇h(ŷ(S)) − C ⊥ .
Furthermore, we say that an extremal (ŷ,ŵ) is normal if for every choice of the pair (λ, λ c ) one has λ c = 1. We say that an extremal (ŷ,ŵ) is abnormal if it is not normal, namely, if exists a choice of (λ, λ c ) with λ c = 0.
As mentioned above, in [32,33,34], where the original set of controls V was embedded in the set W of relaxed controls, it has been shown the validity of criterion (A), that is: if the optimal process (ŷ,ŵ) is a normal extremal, then an infimum gap cannot occur atŷ. An akin result has been achieved in [28], where the system is control-affine and the original set V comprises unbounded controls ranging in a convex cone. In that case a space-time, impulsive, extension is considered, namely the larger set of trajectories corresponding to W comprises space-time paths which are allowed to evolve along fixed time directions. 3 It is worth noticing that in both the investigated cases the original set of trajectories is dense in the set of extended trajectories, when the latter is endowed with C 0 topology. So, one might conjecture that criterion (A) is still valid as soon as the trajectories corresponding to V are dense in the set of trajectories corresponding to W. In fact, this is not the case , as shown by the simple example in Appendix A.1.
Hence, a condition stronger than density is needed. For this goal we introduce Kaskoz' version of J. Warga's notion of V being abundant in W (Def. 4.1). This condition strengthens density by requiring that the trajectories of the extended system's convexification are uniformly approached by trajectories of the original system.
We will further extend the notion of abundant subfamily V ⊂ W to systems defined on manifolds and to fairly general classes of controls (which are merely required to belong to a metric space). Then, aiming to express normality of extended trajectories in geometric terms, we invoke local set separation of the target from the original reachable set.
A crucial result for the achievement of the main theorem consists in showing that, with this notion of abundance, every needle-variational cone C atŷ corresponding to the enlarged domain W is also a QDQ approximating cone to the original reachable set (Theorem 4.1).
The next step consists in showing that the local set separation of the target from the original reachable set implies the linear separability between a QDQ approximating cone to the target and the above mentioned needle-variational cone C (Theorem 5.1). This is exactly the point where the choice of QDQ approximating cones -rather then other more classical cones, e.g. Boltiansky cones-plays essential. By expressing this linear separation in terms of adjoint paths, one finally gets the main result of the paper (Corollary 5.1), where, under the abundance hypothesis, statement (A) is turned into an actual theorem.
In Section 7 we apply the main theorem to nonlinear systems whose dynamics are neither bounded nor convex. Finally, since normality cannot be verified a priori, it is important to find sufficient conditions on the data guaranteeing that all minimizers are normal. This is what is provided by Theorem 5.2, where a directly verifiable criterion for normality is proved to hold true in the general setting.

Linear spaces, manifolds
Let E be a real linear space, and let us use E * to denote the algebraic dual of E. If ·, · is a given scalar product on E, 4 we will use |·| to denote the norm associated with ·, · , namely, for every e ∈ E we set |e| = e, e . For every e ∈ E and every real number r ≥ 0 let us use e + B r to denote the closed ball of center e and radius r, namely e + B r = {e + f | |f | ≤ r}. When e = 0 we will write B r instead of 0 + B r If E 1 , E 2 are real linear spaces and L ∈ Lin(E 1 , E 2 ), we shall use L · e to denote the element of E 2 coinciding with the image of e ∈ E 1 . We will use the symbol · also to mean duality. Furthermore, if λ ∈ E * 2 and and L ∈ Lin(E 1 , E 2 ), sometimes we will use the notation λ · L to mean "the element of E * 1 coinciding the image of λ through the dual map of L." While it doesn't generate any confusion, this promiscuous use of the notation "·" makes the writing λ · L · e unambiguous, for one has (λ · L) · e = λ · (L · e) for all (e, λ) ∈ E 1 × E * 2 . By saying that M, ·, · is a Riemannian differentiable manifold we will mean that M is a differential manifold and ·, · is a Riemannan metric. For every x ∈ M and e, f ∈ T x M, e, f x will denote the corresponding scalar product of e, f , and |e| x := e, e x will be called the norm of e. We will often omit the subscript and we will write e, f and |e| instead of e, f x and |e| x .
We will use d to denote the distance induced on M by ·, · . We recall that, if x 1 , x 2 ∈ M, the distance d(x 1 , x 2 ) is defined as the minimum among the ·, · -lenghts of the absolutely continuous curves having x 1 , x 2 as end-points. For any x ∈ M and any r ≥ 0, we will use B[x, r] to denote the closed ball of radius r and center x, i.e. B[x, r] := {y ∈ M | d(x, y) ≤ r}.

Cones
If A ⊂ E is any subset, we use span + A to denote the smallest convex cone containing A. Let us introduce a notion of transversality for cones.
The idea of a non-trivial intersection between cones, which plays essential in setseparation results like Theorem 2.3 below, is made formal is made formal by the following notion of tranversality: Definition 1.2. Let E be a linear space and let K 1 , K 2 ⊆ E be convex cones. We say that 2. K 1 and K 2 are strongly transverse, if they are transverse and Transversality differs from strong transversality only when K 1 and K 2 are complementary subspaces: Proposition 1.1. Let E be a linear space, and let K 1 , K 2 ⊆ E be convex cones. Then K 1 , K 2 are transverse if and only if either K 1 , K 2 are strongly transverse or K 1 , K 2 are linear subspaces such that K 1 ⊕ K 2 = E. Definition 1.3. Let E be a finite-dimensional linear space, and let E * be its dual space. For any subset A ⊂ E, the (convex) cone (where the symbol · denotes duality) will be called the polar cone of A.
The transversality of two cones is equivalent to their linearly separability. More precisely: Proposition 1.2. Two convex cones K 1 and K 2 are not transverse if and only if K ⊥ 1 ∩ K ⊥ 2 \{0} = ∅ , namely there exists a linear form λ = 0 such that In this case one also says that K 1 and K 2 are linearly separable.

Scorza-Dragoni points
We say thats ∈ [a, b] is a Scorza-Dragoni point for ϕ if, for all y ∈ X, We shall use SD {ϕ} to denote the set of all the Scorza-Dragoni points for the function ϕ.
Notice in particular that, if s ∈ SD {ϕ}, one has lim x→y, δց0 for any y ∈ X. The importance of Scorza-Dragoni points relies on the fact they they form a full measure set [36]:

Quasi Differential Quotients
In order to state the set-separation theorem (Th. 2.3) we need the notion of Quasi Differential Quotients approximating cone to a set E at a point of its boundary. For this purpose let us introduce the notion of Quasi Differential Quotient, which in turn is a particular case of Sussmann's Approximate Generalized Differential Quotient [40]. The corresponding set-separation theorem is based on an Open Mapping we prove below. Let us recall the notion of Cellina continuously approximable set-valued function, which is the building block in the definition of Approximate Generalized Differential Quotient. Definition 2.1 (CCA). Let F : R N R n be a set-valued map. We say that F is a Cellina continuously approximable (CCA) set-valued map if, for any compact set K ⊂ R N , • the restriction of F on K has compact graph, that is, the set Gr(F |K ) := {(x, y) ∈ K × R n : y ∈ F (x)} is compact, and • there exists a sequence of single-valued, continuous maps f k : K → R n , k ∈ N, such that the following condition holds: for every open set Ω satisfying Gr(F |K ) ⊂ Ω, there exists k Ω such that Gr (f k ) : We will say that a function ρ : [0; +∞[→ [0; +∞] is a a pseudo-modulus if it is monotonically nondecreasing and lim s→0 + ρ(s) = ρ(0) = 0. We call modulus a pseudo-modulus taking values in [0, +∞[.
is a compact set, and Γ ⊂ R N is any subset. We say that Λ is an Approximate Generalized Differential Quotient (AGDQ) of F at (γ,ȳ) in the direction of Γ if there exists a pseudo-modulus ρ having the property that (*) for every δ > 0 such that ρ(δ) < +∞, there exists CCA set-valued map We will use a subclass of AGDQs, which we call the Quasi Differential Quotients. Their main property consists in the validity of an actual, not punctured, open mapping theorem (see Theorem 2.2below).
As pointed out in [40], this definition is intrinsic, that is, it is independent of the choice of the charts φ and ψ.

Open Mapping results
Theorem 2.1 (Directional Open Mapping). Let N, n be positive integers, and let Γ be a convex cone in R N . Let F : R N R n be a set-valued map, and let Λ be a AGDQ of F at (γ,ȳ) in the direction of Γ. Let us assume that there is an elementw ∈ R n such that w ∈ Int(L · Γ) for every L ∈ Λ. Then there exist a closed convex cone D ⊆ R n and positive constants α, β verifyingw ∈ Int(D) and (2.4) If one takesw = 0 in the statement of Theorem 2.1, the cone D necessarily coincides with the whole R n . As a consequence, one obtains the following 'punctured' Open Mapping Theorem.
Corollary 2.1 ('Punctured' Open Mapping). 6 Let N, n be positive integers, and let Γ be a convex cone in R N . Let F : R N R n be a set-valued map, and let Λ be an QDQ of F at (γ,ȳ) in the direction of Γ. Let us assume that Λ is surjective, by which we mean that L · Γ = R n for every L ∈ Λ. Then there are positive constants α, β verifyinḡ (2.5) If we replace AGDQ's with QDQ we get a real, non-punctured, open mapping result: Theorem 2.2 (Open Mapping). Let N, n be positive integers, and let Γ be a convex cone in R N . Let F : R N R n be a set-valued map, and let Λ be a GDQ of F at (γ,ȳ) in the direction of Γ. As above, let us assume that Λ is surjective, by which we mean that L · Γ = R n for every L ∈ Λ. Then the following statements (i), (ii) hold true: (i) there are positive constants α, β having the property that (2.6) (ii) there existsδ > 0 such that, for every δ ≤δ and every (L δ , h δ ) as in Definition 2.3, there exists γ δ ∈γ + Γ ∩ B δ such that In particular, by possibly reducing the size of α, one gets the open-mapping inclusions Proof. Without loss of generality, we can assume (γ,ȳ) = (0, 0) and, since a QDQ is an AGDQ, it is sufficient to prove only statement (ii). Namely, for every δ > 0 sufficiently small, we have to establish the existence of a γ δ ∈ B δ ∩ Γ such that Namely, L −1 r δ (γ) is the set of right inverse of L δ (γ). Let us first observe that, for every γ ∈ B δ ∩ Γ, and δ sufficiently small, L −1 r δ (γ) is non-empty. Indeed, it contains the Moore-Penrose pseudo-inverse where tr denotes transposition. Furthermore, it is trivial to verify that the set-valued map L −1 r δ is convex-valued. Finally, by possibly reducing the size ofδ, for every δ ∈ [0,δ], the set-valued map L −1 r δ has compact graph. Indeed, there exist a constant K > 0 such that Λ ρ(δ) is a compact subset made of linear operators whose right inverse are bounded (in the operator norm) by K. Moreover, let us consider a sequence (γ m ) m∈N ⊂ B δ ∩ Γ converging toγ ∈ B δ ∩ Γ, and, for every m ∈ N, let us choose M m ∈ L −1 r δ (γ m ). Hence, one has that L δ (γ m ) • M m = Id R n and, since the sequence (M m ) ranges in a compact set, there exists a subsequence (M m k ) converging to a linear operatorM . In particular, To prove that this map has non-empty values for every γ ∈ B δ ∩ Γ, it is sufficient to determine a linear mapping M ♭ : R n → R N and an element v ∈ Γ such that is surjective. Now, a geometrical intuition suggests that M ♭ might be obtained by adding a suitable linear operator to an element of L −1 r δ (γ), for instance the linear operator M ♯ . Actually, following [40], if ·, · is any scalar product on R n , we define the linear map M ♭ : R n → R N by setting, for every w ∈ R n , It is straightforward to verify that M ♭ verifies conditions (2.9), so that Ψ δ (γ) is not empty. Since for every δ the map h δ is continuous and h δ ≤ δρ(δ), by possibly reducing the size ofδ we conclude that, for every δ ∈ [0,δ], the set-valued map Ψ δ verifies Ψ δ (B δ ∩ Γ) ⊂ B δ ∩Γ and has non-empty, convex values, and a closed graph. Since the domain of Ψ δ is compact and convex, the set-valued map Ψ δ verifies the hypotheses of the Kakutani fixed point theorem, so that there exists γ δ ∈ B δ ∩ Γ such that γ δ ∈ Ψ δ (γ δ ). It follows that there is a matrix M ∈ L −1 r δ (γ δ ) such that 0 = γ δ + M · h δ (γ δ ). Therefore, one gets which concludes the proof.

QDQ approximating cones and set separation
Assume that M is a C 1 differentiable manifold , E ⊂ M, and z ∈ E. If X is a linear space, let us call convex multicone in X any family of convex cones of X.
Definition 2.5. An AGDQ approximating multicone [resp. a QDQ approximating multicone] to E at z is a convex multicone C ⊆ T z M such that there exist a non-negative integer N , a set-valued map F : R N M, a convex cone Γ ⊂ R N , and an AGDQ [resp. a QDQ] Λ of F at (0, z) in the direction of Γ such that F (Γ) ⊂ E and C = {L · Γ : L ∈ Λ}.
In the particular case when an AGDQ approximating multicone [ resp. a QDQ approximating multicone] is a singleton, namely Λ = {L} for some L ∈ Lin(R N , R n ), we say that C := L · Γ is an AGDQ approximating cone [ resp. a QDQ approximating cone] to E at z.

Let us introduce the notion of local set-separation:
Definition 2.6. Let X be a topological space, and let us consider two subsets A 1 , A 2 ⊂ X and a point z ∈ A 1 ∩ A 2 . We say that A 1 and A 2 are locally separated at z provided there exists a neighborhood V of z such that We are now ready to state our set-separation result, which connects set separation with the linear separability of QDQ approximating cones. Furthermore the result includes a special property in the case when the approximating cones are complementary linear subspaces.

Theorem 2.3 (Set separation)
. Let E 1 , E 2 be subsets of M, and let z ∈ E 1 ∩ E 2 Assume that C 1 , C 2 are AGDQ approximating cones of E 1 and E 2 , respectively, at z.
i) If C 1 and C 2 are strongly transverse, then the sets E 1 and E 2 are not locally separated.
ii) If moreover: Let us prove statement ii). Because of the local character of the statement, there is not loss of generality in considering only the Euclidean case when M = R n . For every i = 1, 2, let n i ≥ 0 be the dimensions of the subspace C i (so that n 1 + n 2 = n), and let N i ≥ 0 an integer such that Γ i ⊂ R Ni . By hypothesis, for every i = 1, 2 there exists a modulus ρ i : [0, +∞[ → [0, +∞[ having the property that, for every δ > 0, there exists a continuous map and observe that Furthermore, let us set ρ(δ) := ρ 1 (δ) + ρ 2 (δ) and let us define the continuous map Defining the linear map L ∈ Lin{R N1+N2 , R n } by setting Hence, Λ is a QDQ of F at (0, 0). Moreover, one has L · Γ = C 1 ⊕ C 2 = R n so that, by the Open Mapping result stated in Theorem 2.2 for k ∈ N sufficiently large, we get the existence of (γ 3 Gaps and set-separation

Original and extended controls
Let (M, ·, · ) be a Riemannian C 2 -differentiable manifold, let [0, S] be a time-interval and let W be a metric space which we call the set of control values. For every (s, w) ∈ [0, S]×W, let M ∋ y → (y, f (s, y, w)) ∈ T M be a vector field. We will consider two families of controls V, W := L 1 ([0, S], W), with V ⊂ W. We will call V and W the original family of controls and the extended family of controls, respectively. Let us choose an initial pointȳ ∈ M, and, for any control map w ∈ W, let us consider the Cauchy problem We shall assume the following regularity hypothesis: Hypothesis (SH) : for every (y, w) ∈ M × W.
In particular, for every w ∈ W there exists a unique trajectory y[w] of (E).
Let us fix a closed set T ⊆ M, which we will refer to as target.

Infimum gaps
Let us endow the set of controls W with the pseudo-distance d f defined as for all controls w 1 , w 2 ∈ W. The set will be called the original reachable set, and the set will be called the extended reachable set.
We will also consider local versions of the above reachable sets. Precisely, for a given extended process (ŷ,ŵ) and r ≥ 0, we set The occurrence of a local infimum gap is captured by the following definition: We say that (ŷ,ŵ) satisfies the infimum gap condition if, for any continuous cost function h : R n → R, there exists r > 0 such that one has Despite the name, the infimum gap condition (3.14) is clearly a fully dynamical property. Actually, it could be as well rephrased in terms of 'supremum gap' or even independently of any optimization procedure as shown in Lemma 3.1 below. Definition 3.3. Let (ŷ,ŵ) be a feasible extended process such thatŷ(S) ∈ R W \R V . We say that (ŷ,ŵ) is isolated from V if, for some r > 0 the sets Rŵ ,r V ∪ {ŷ(S)} and T are locally separated atŷ(S), namely, there exists a neighborhood N ⊂ M ofŷ(S) such that Lemma 3.1. Let (ŷ,ŵ) be an extended feasible process such thatŷ(S) ∈ R W \R V . Then the following conditions are equivalent: i) (ŷ,ŵ) satisfies (3.14) for a given continuous cost function h andr > 0; ii) the process (ŷ,ŵ) is isolated from V; iii) the process (ŷ,ŵ) satisfies the infimum gap condition. Furthermore the right handside of (3.14) is equal to +∞.
Proof. We give a proof just for the sake of completeness, all arguments being trivial. Let us start proving that i) implies ii). This means that one has to show that there existsr > 0 such that Rŵ ,r V ∩ T = ∅ ∀r <r. (3.15) Assume that (3.15) is false, which means that there exists a sequence r n ↓ 0 such that Rŵ ,rn V ∩ T = ∅ for all natural n. This implies that there exists a sequence (y k ) k∈N verifying y k ∈ Rŵ ,r k V ∩ T for every k ∈ N, so that y n →ŷ(S), which, in view of the continuity of h, contradicts i). Hence, (3.15) holds true, from which we get ii).
Let us now prove that ii) ⇒ iii). By hypothesis, there exists a neighborhood N of y(S) such that Rŵ ,r V ∪ {ŷ(S)} ∩ T ∩ N = {ŷ(S)}. Sinceŷ(S) ∈ R W \R V , by possibly reducing the size of r > 0 one obtains that Rŵ ,r V ∩ T = ∅, which obviously implies iii), with the right hand-side of (3.14) equal to +∞. Finally, the relation iii) ⇒ i) is trivial.

Abundance
Our main results -namely Theorem 5.1 and Corollaries 5.1 and 5.2-strongly rely on a property introduced by J. Warga and called "abundance". It consists in a particular pervasiveness of V in W, which happens to be stronger than density. In fact, because of the presence of a closed final constraint, the mere density of R V into R W is not enough in order to normality to be a sufficient condition for gaps' avoidance (see Subsection (A.1)). We will make use of a generalization of abundance provided in [21] and we will extend it to manifolds.
For every positive integer N , let Γ N be the convex hull of the union of the origin with the N -simplex, namely For every value of the parameter γ ∈ Γ N and every control w, w 1 , ..., w N ∈ W 1+N , let us use y γ w, w 1 , ..., w N to denote the corresponding solution of (4.16). 7 Notice, in particular, that y[w] = y γ w, w, ..., w for all w ∈ W and for all γ ∈ Γ N .  Proposition 4.1. [21] Assume that the subfamily V ⊂ W satisfies the concatenation property and is dense in W with respect to the pseudo-metric d f . Then V is an abundant subset of W.
The proof of this result for the special case when M = R n was given in ( [21], Theorem IV.3.9) by developing some arguments in [18]. The required, obvious, changes to prove the result on a Riemannian manifold reduce to a reformulation of estimate (4.17) in local coordinates, so we omit them.

Approximating the original reachable set by extended cones
Let us fix a a feasible extended process (ŷ,ŵ), and, for any s,š ∈ [0, S], s >š, let M (s,š) : Tŷ (š) M → Tŷ (s) M denote the differential of the diffeomorphism established by the differential equationẏ = f (s, y,ŵ) from a neighborhood ofŷ(š) to a neighborhood of y(s). As is known, s → M (s,š) is the solution of the variational Cauchy problem having the following coordinate representation: will be called extended variational cone corresponding to the feasible extended process (ŷ,ŵ).
The following result can be regarded as claiming a sort of infinitesimal thickness of V in W.
Proof of Theorem 4.1. We will prove this theorem assuming that M is an open subset of R n , so that we can identify Tŷ (S) M with R n . Clearly, this is not restrictive because of the local character of the result.
Proof. Let us begin proving the lemma in the case when N = 1 and 0 ≤ γ ≤ 1. One has

Hence, one has
which, by the basic theory of linear ODE's, implies that Therefore, the lemma is proved for N = 1 and for any 0 ≤ γ ≤ 1. The general case N ≥ 2 is easily obtained by a finite induction argument. The latter doesn't present any new difficulty with respect to the proof of the case N = 1. Moreover is almost verbatim the one utilized in the proof of the Pontryagin Maximum Principle when passing from single to multiple, finitely many needle variations (see e.g. [35], Theorem 4.2.1). For this reason we omit it.

The main result
Theorem 5.1 (A geometric principle for gaps). Let us assume that the family of controls V is abundant in W. Let (ŷ,ŵ) be a feasible extended process satisfying the infimum gap condition. Then any QDQ approximating cone C to T atŷ(S) is linearly separable from any extended variational cone C s1,...,sN w1,...,wN , i.e. there exists a non-zero linear form ξ ∈ T * y(S) M such that Let us give the definition of abnormal extremal, normal h-extremal, and h-abnormal extremal.
Furthermore, we say that an h-extremal (ŷ,ŵ) is normal if for every choice of the pair (λ, λ c ), one has λ c = 1. We say that an h-extremal (ŷ,ŵ) is abnormal if it is not normal, namely if exists a choice of (λ, λ c ) with λ c = 0.
Remark 5.1. Though these definitions have intrinsic meanings, we have chosen to adopt a notation reminiscent of coordinates. Of course, the adjoint equation (i) might be expressed -when coupled with the dynamics-as the Hamiltonian system where H : T * M → R is the maximized Hamiltonian defined by setting and X H is the Hamiltonian vector field, namely X H := J · DH, J being the symplectic matrix and D the differential operator with respect to x and λ.
Let us also point out that the dot · has obvious different meanings according to the context: for instance, in (i) of the definitions above, it denotes a linear operator on the cotangent space, while in (ii) it denotes the duality product.
Observe that every abnormal extremal is an abnormal h-extremal for any cost h differentiable atŷ(S), while every abnormal h-extremal is an abnormal extremal. We are now ready to state our main result on infimum gaps .

Corollary 5.1 (Normality No-Gap Criterion).
Let us assume that the family of controls V is abundant in W. If a feasible extended process (ŷ,ŵ) satisfies the infimum gap condition, then, for every QDQ approximating cone C to T atŷ(S), (ŷ,ŵ) is an abnormal extremal with respect to C.
When referred to a specific cost h, the contrapositive version of this theorem provides a sufficient condition for the absence of local infimum gaps. Precisely: Corollary 5.2 (A sufficient condition for avoiding infimum gaps). Let us assume that the family of controls V is abundant in W, and let (ŷ,ŵ) be a feasible extended process. Let h : M → R be a cost function, differentiable atŷ(S), and let (ŷ,ŵ) be a normal hextremal for some QDQ approximating cone C to T atŷ(S). Then there is no infimum gap at (ŷ,ŵ).
As we have mentioned in the Introduction, the relation between gap phenomena and abnormality has been quite investigated in two cases of embeddings: the embedding of bounded optimal control problems into their convex relaxation [32,33,34] and the embedding of unbounded (convex) control systems into their impulsive, space-time closure [28]. Since the original control families in such embeddings turn out to be abundant in their extensions, these kinds of results can be also obtained by Theorem 5.1. 12 In Section 7, we are going to present a new application to a dynamics which is neither convex nor bounded.

A verifiable sufficient condition for normality
In practical situations, it may be difficult or even impossible to directly verify the normality of an extremal, which, in view of Corollary 5.2, would guarantee the absence of gaps. This motivates Theorem 5.2 below, which provides a sufficient condition on the data of the problem in order for the extremals to be normal.
In the following definition we assume that a control system dy ds (s) = f (s, y(s), w(s)), w ∈ W, a.e. s ∈ [0, S], (5.31) as above is given, with an initial condition y(0) =ȳ, (5.32) and, still, we use R W to denote the reachable set fromȳ.

The case of a finite subset of controls
Let us consider a finite subset of control valuesW ⊆ W and let us set SinceW is finite, E(W) has measure equal to S. Therefore, by Lusin's theorem we can write where E 0 has zero measure and, for every j, the set E j is compact, and, for every w ∈W, the restrictions to E j of the map where property (P) F is as follows: 14 We recall that an element t ∈ B ⊂ R is a density point for B if (1) λ(S) =λ ∈ −C ⊥ (2) dλ ds = −λ · ∂f ∂y (s,ŷ(s),ŵ(s)), a.e s ∈ [0, S]; (3) λ(s) · f (s,ŷ(s),ŵ(s)) ≥ λ(s) · f (s,ŷ(s), w) for every (s, w) ∈ F .
Notice that, for every subset F ∈ D ×W, Λ(F,W) is compact and, moreover, By Theorem 5.1, Λ(F,W) = ∅ as soon as F is finite and can be written as Claim: One has Λ(F,W) = ∅ even when F is an arbitrary finite subset of D ×W, namely F can be written as Indeed, every s i belongs to a suitable D h , which can be labelled as D h(i) . Since D h(i) is made of density points, there exist sequences (s i,j ) such that The mapping s →λ j (s) satisfies the inequalitȳ for all j ∈ N, every i = 1, . . . , m and w ∈W. Since, for every i = 1, . . . m, the map s → r wi (s) := f (s,ŷ(s), w i ) − f (s,ŷ(s),ŵ(s)) is continuous on D h(i) , the function s → λ j (s) · r wi (s) is also continuous on D h(i) , so passing to the limit we can conclude that for every i = 1, ..., m and w ∈W. Since one also has 0 =λ =λ(S) ∈ −C ⊥ , the claim is proved.

The general case of an infinite control set
Up to now we have shown that, ifW is finite, and F ⊂ D ×W is finite -and we write card(F ) < ∞-, then Λ(F,W) is a nonempty compact set. We now conclude the proof throgh a standard non-empty intersection argument (see e.g. [39]). If we take a finite family F 1 , . . . , F r ⊂ D ×W such that card(F i ) < ∞ for every i = 1, . . . , r, one has Λ(F 1 ,W) ∩ · · · ∩ Λ(F r ,W) = Λ(F 1 ∪ · · · ∪ F r ,W) = ∅, (for card(F 1 ∪ · · · ∪ F r ) < ∞). Hence, is a family of compact subsets such that each finite intersection is nonempty. This implies that the (infinite) intersection of all Λ(F,W) such that cardF < ∞ is nonempty. Therefore has non-empty intersection. This can easily achieved by the same arguments as above. Indeed, Λ(Ŵ) is not empty and compact as soon asŴ is finite. Furthermore, for every In particular, the family Λ(Ŵ) : card(Ŵ) < ∞ is made of compact subsets and satisfies the finite intersection property, that is, the intersection of any finite finite subfamily Λ(Ŵ) : card(Ŵ) < ∞ is not empty. Therefore, it has non-empty intersection, namely This concludes the proof of Theorem 5.1.

An application to non-convex, unbounded, problems
Impulsive optimal control problems -where the dynamics is unbounded-have been extensively studied together with their applications [3,4,5,8,10,11,13,14,19,20,22,24,25,37,46]. The space-time representation (see (7.40) below) can be regarded as an extension of unbounded control systems. An important case is the one of a minimum problem with of control-affine dynamics: Here the set U where the controls u take values is unbounded. Furthermore, the state x range over a n-dimensional Riemannian manifold M of class C 2 , and the time-dependent vector fields f, g 1 , . . . , g m are of class C 1 in x, measurable in t, and uniformly bounded by a L 1 map. Moreover, the cost h : R × M × R → R is a continuous function, (t 1 ,x) ∈ R × M is a fixed initial condition, K is a non negative fixed constant, possibly equal to +∞, and the end-point constraintT ⊆ R × R n is a closed subset. Notice incidentally that the function η(t) coincides with the L 1 -norm of the control function u := (u 1 , u 2 , ..., u m ) on the interval [t 1 , t].
The gap-abnormality criterion for this kind of systems (where one considers the spacetime extension (7.40) below) has been already investigated in the case when the set of controls U is a convex cone [28]. Actually, thanks to Corollary 5.1 (see also [31]), the main result in [28] can be extended to the case in which the state ranges on a Riemannian manifold. However the generalization made possible by Corollary 5.1 allows one to go much further. Indeed, in what follows we are able to deduce from Corollary 5.1 that the gap-abnormality criterium holds true also in the situation when the control set U is unbounded but is neither convex nor a cone.
More precisely, we will consider the following two cases:  Remark 7.1. We will treat Case (i) in detail, describing the extension to the convex space-time system obtained by both convexification of the dynamics and the closure of suitably reparameterized processes. Instead, we will only suggest the needed changes to deal with Case (ii), where the only extension comes from reparameterization. However, Case (ii) is somehow more significative, in that it marks the most important improvement with respect to the former literature initiated by Warga's work. Indeed, in this case not only the original dynamics but also the extended dynamics is non-convex. This can be of interest in those application where the convexification of the dynamics is not needed (for instance because one gets existence of minima without invoking convexification).

Case (i) (Space-time convex extension)
In order to formulate this problem by means of the terminology adopted in Theorem 5.1, we need to embed our system into a suitably extended one. To this aim we need to perform both a 'compactification' (to manage unboundedness) and a 'convexification'. Let us begin by setting s. t., for a.e. s ∈ [0,Ŝ], (7.40) Accordingly, a pair (z 0 , z, ν , a, (w 0 1 , w 1 ), . . . , (w 0 n+3 , w n+3 ), d such that z 0 , z, ν is the solution of the above control system corresponding to the control a, (w 0 1 , w 1 ), . . . , (w 0 n+3 , w n+3 ), d is called a process of (P ) h W . The embedding of the problem (P ) into (P ) h W is as follows: fixŜ > 0, and, for every control u : [t 1 , t u 2 ] → U , consider the function σ u : [t 1 , t u 2 ] → [0,Ŝ] defined by (t + η(t)).
By a trivial use of the chain rule one gets the following result (see e.g. [3] for a similar embedding): Lemma 7.1. The embedding I is injective 16 . Moreover, the image space of the embedding I coincides with the set of all processes (z 0 , z, ν , a, (w 0 1 , w 1 ), . . . , (w 0 n+3 , w n+3 ), d such that a, (w 0 1 , w 1 ), . . . , (w 0 n+3 , w n+3 ), d ∈ V Thanks to Lemma 7.1 we can identify the original problem (P ) with the problem .

A Appendix
A.1 An example on why abundance is crucial The following example, which is due to H.J. Sussmann, 18 shows how the abundance hypothesis plays crucial for the validity of Theorem 5.
In particular this implies λ(s) ≡ 0 and λ c > 0. Therefore, if we set h(y) := y for every y ∈ R, the process (ŷ,ŵ) turns out to be a normal h-extremal. Therefore, in view of Then, for every δ sufficiently small, either there exists a γ δ ∈ [0, 1] such that 1 0 θ δ (γ δ )(s)ds = 1, or the map γ → 1 0 θ δ (γ)(s)ds is not continuous. Since the former case is ruled out by the fact that the map θ δ (·)(s) has to take values in V, the map γ → 1 0 θ δ (γ)(s)ds is not continuous, so providing a contradiction.