No Infimum Gap and Normality in Optimal Impulsive Control Under State Constraints

In this paper we consider an impulsive extension of an optimal control problem with unbounded controls, subject to endpoint and state constraints. We show that the existence of an extended-sense minimizer that is a normal extremal for a constrained Maximum Principle ensures that there is no gap between the infima of the original problem and of its extension. Furthermore, we translate such relation into verifiable sufficient conditions for normality in the form of constraint and endpoint qualifications. Links between existence of an infimum gap and normality in impulsive control have previously been explored for problems without state constraints. This paper establishes such links in the presence of state constraints and of an additional ordinary control, for locally Lipschitz continuous data.


Introduction
In Optimal Control Theory it is quite common practice to extend the domain of a minimum problem to ensure the existence of the minimum or to identify optimality conditions. In doing this, it is of course desirable to avoid the so-called infimum gap phenomenon, i.e. that the minimum of the extended problem is different from the minimum of the original problem. This is relevant not only for theoretical reasons of 'well-posedness' of the extension, but also for the actual usefulness of the extended problem in order to identify, for instance, necessary optimality conditions or a non degenerate Hamilton-Jacobi-Bellman equation for the original minimum problem. For the classical extension of a minimum problem by convex relaxation -where the original velocity set of the trajectories is replaced by its convexification-it has emerged that a sufficient condition to avoid the infimum gap is the normality of an extended sense minimizer, namely, that all sets of multipliers verifying a Maximum Principle have cost multiplier, λ in the following, different from zero [29,30,36,[38][39][40]. In [26] the 'normality test' has been proved to be sufficient to guarantee the absence of an infimum gap also for the impulsive extension of an optimal control problem with unbounded controls. Very recently, in [27,28] this link between normality and no-infimum-gap has been established for the extension of an abstract minimum control problem, which includes both relaxation and impulsive extension. All the above results in the case of the impulsive extension concern problems without state constraints, with C 1 data, and no ordinary controls in the dynamics. However, state constraints, together with nonsmoothness of the data and additional ordinary controls, arise very frequently in the applications of impulsive optimal control (see e.g. [7,16,19,22,34] and the references therein).
This paper provides 'normality type' sufficient conditions to avoid a gap between the infima of the following optimization problem (P) and the extended optimization problem (P e ) below: minimize a.e. t ∈ [t 1 , t 2 ], du dt (t) ∈ C a.e. t ∈ [t 1 , t 2 ], h 1 (t, x(t)) ≤ 0, . . . , h N (t, x(t)) ≤ 0 forallt ∈ [t 1 , t 2 ], v(t 1 ) = 0, v(t 2 ) ≤ K, (t 1 , x(t 1 ), t 2 , x(t 2 )) ∈ T 0 , where K > 0 is a fixed constant (possibly equal to +∞), A ⊂ R q is a compact subset, C ⊆ R m is a closed convex cone, T 0 ⊆ R 1+n+1+n is a closed subset, and the data are locally Lipschitz continuous in t, x (the precise assumptions will be given in Section 2). Problem (P) is a free end-time minimization problem depending on an ordinary control a and on a control u whose derivatives appear linearly in the dynamics. Furthermore, there are timedependent state constraints in the form of N inequalities, endpoint constraints, and we may have a bound K on the total variation of u -notice that v is nothing but the total variation function of u. Due to a lack of coerciveness, minimizers for problem (P) do not exist in general. Hence, adopting a by now standard extension, we embed the original problem into the space-time problem (P e ) below, where the extended state variable is (y 0 , y, ν) := (t, x, v), and the extended trajectories are (t, x, v)-paths which are (reparameterized) C 0 -limits of graphs of the original trajectories [10,21,23,32,37] To any process (t 1 , t 2 , u, a, x, v) of problem (P), by setting σ (t) := t − t 1 + v(t), t ∈ [t 1 , t 2 ], through the time-change y 0 := σ −1 we can associate a process (S, ω 0 , ω, α, y 0 , y, ν) of the extended problem with ω 0 = dy 0 (s)/ds > 0 a.e.. In particular, problem (P) can be identified with the restriction of problem (P e ) to the set of processes with ω 0 > 0 a.e. (see Section 2). Let us refer to such processes as embedded strict sense processes in the following. The extension consists therefore in considering extended sense processes with ω 0 = 0 on non-degenerate intervals, where the time t = y 0 (s) is constant but the extended state y(s) evolves according to the 'fast' dynamics dy(s)/ds = m j =1 g j (y 0 (s), y(s))ω j (s). This explains why (P e ) is also called the impulsive problem, although it is a conventional optimization problem with bounded controls. In fact, one could give an equivalent t-based description of this extension using bounded variation trajectories and controls [2,7,19,22,24,25,33,41].
The main result of the paper, obtained in Theorem 4.1 below, establishes that the existence of an extended sense minimizer for problem (P e ) which is a normal extremal for a constrained version of the Maximum Principle, is a sufficient condition for the infimum gap avoidance. The occurrence of a gap is strictly related to the presence of endpoints and state constraints. In particular, since the set of trajectories corresponding to embedded strict sense processes is C 0 -dense in the set of trajectories of the extended system, the infimum gap phenomenon can show up only when some extended sense process verifying the constraints is isolated, namely cannot be approximated by trajectories of the original system that satisfy the constraints. From this observation, Theorem 4.1 will be derived from a general result on the properties of isolated processes (see Theorem 3.2). The proof makes use of perturbation and penalization techniques and of the Ekeland's variational principle, in the same spirit of [26,29]. This approach is very different from that of [28,[38][39][40], which is based on the construction of approximating cones to reachable sets and on set separation arguments. We recall that normality is not necessary to exclude the gap phenomenon: for example, it is known that without the drift f in the dynamics, gap never occurs [26,Lemma 4.1] (see also [20]).
The normality criterium for the absence of an infimum gap has some disadvantages. First of all, it requires to know a priori a minimizing extended sense process, information that is not always available. Then, it is necessary to verify that all sets of multipliers associated to the minimizer that meet the conditions of the Maximum Principle have λ > 0. In addition, in the presence of state constraints the normality condition may never be met, making the criterium in fact useless. In particular, it is well known that when the state constraint is active at the initial point of a minimizing process, sets of degenerate multipliers with λ = 0 may always exist. Rather surprisingly, it seems that no attention has been paid to this 'degeneracy question' in previous articles on the relationship between gap and normality in the presence of state constraints. Based on the above considerations, in the second part of the paper we first introduce a nondegenerate version of the Maximum Principle and provide simple geometrical conditions on endpoint and state constraints, under which abnormal -namely, not normalextremals for the original Maximum Principle turn out to be abnormal extremals also for the nondegenerate Maximum Principle. In this case, Theorem 4.1 can be rephrased as follows: 'normality among nondegenerate multipliers implies no infimum gap'. This 'nondegenerate normality test' is useful especially because in certain special cases it allows to deduce the absence of the infimum gap from easily verifiable conditions, some examples of which we will provide. In particular, these are constraint and endpoint qualification conditions.
Although this article is mainly focused on the infimum gap phenomenon, it also establishes some new sufficient conditions for normality which extend previous conditions in [25]. In the literature on conventional, non-impulsive problems with state constraints, a variety of constraint qualifications to avoid degeneracy as well as to ensure normality are known (see e.g. [4, 5, 8, 9, 11-14, 18, 30, 31] and the references therein). In impulsive control, instead, some nondegenerate Maximum Principles have been obtained in [6,7,17,25], while a Maximum Principle in normal form has only recently been introduced in [25].
The paper is organized as follows: in Section 2 we introduce precisely problems (P), (P e ) and a constrained version of the Maximum Principle for the extended problem. Section 3 is devoted to prove that an isolated extended sense extremal cannot be normal and, as a corollary, we deduce that presence of an infimum gap implies abnormality of any extended sense minimizer. In Section 4 we provide sufficient conditions for normality, which guarantee a priori, without any knowledge of the multipliers associated with the given extended sense minimizer, the non occurrence of gap-phenomena. In Section 5, we present some examples to illustrate the results.

Notations and Preliminaries
Given an interval I ⊆ R and a set X ⊆ R k , we write C 0 (I ; X), W 1,1 (I ; X), C 0,1 (I ; X), C 0,1 loc (I ; X) for the space of continuous functions, absolutely continuous functions, Lipschitz continuous functions, locally Lipschitz continuous functions defined on I and with values in X, respectively. For all the classes of functions introduced so far, we will not specify domain and codomain when the meaning is clear. Furthermore, we denote by (X), co(X), Int(X), ∂X the Lebesgue measure, the convex hull, the interior and the boundary of X, respectively. As customary, χ X is the characteristic function of X, namely χ X (x) = 1 if x ∈ X and χ X (x) = 0 if x ∈ R k \ X; I · X denotes the set {r x | r ∈ I, x ∈ X}. Given two nonempty subsets X 1 , X 2 of R k , we denote by X 1 +X 2 the set {x 1 +x 2 | x 1 ∈ X 1 , x 2 ∈ X 2 }. Let X ⊆ R k 1 +k 2 for some natural numbers k 1 , k 2 , and write x = (x 1 , x 2 ) ∈ R k 1 × R k 2 for any x ∈ X. Then proj x i X will denote the projection of X on R k i , for i = 1, 2. We denote the closed unit ball in R k by B k , omitting the dimension when it is clear from the context. Given a closed set O ⊆ R k and a point z ∈ R k , we define the distance of z from O as d O (z) := min y∈O |z − y|. We set R ≥0 := [0, +∞[. For any a, b ∈ R, we write a ∨ b := max{a, b}. For all τ 1 , τ 2 ,τ 1 ,τ 2 ∈ R, τ 1 < τ 2 ,τ 1 <τ 2 , and for any pair (z 1 , where for any z ∈ C 0 ([a, b], R k ),z denotes its continuous constant extension to R and · L ∞ (I ) is the ess-sup norm on I ⊆ R interval. When the domain is clear, we will sometimes simply write · L ∞ . We denote by NBV + ([0, S]; R) the space of increasing, real valued functions μ on [0, S] of bounded variation, vanishing at the point 0 and right continuous on ]0, S[. Each μ ∈ NBV + ([0, S]; R) defines a Borel measure on [0, S], still denoted by μ, its total variation function is indicated by μ T V or equivalently by μ([0, S]), and its support by spt{μ}.
Some standard constructs from nonsmooth analysis are employed in this paper. For background material we refer the reader for instance to [35]. A set K ⊆ R k is a cone if αk ∈ K for any α > 0, whenever k ∈ K. Take a closed set D ⊆ R k and a pointx ∈ D, the proximal normal cone N P D (x) of D atx is defined as The limiting normal cone N D (x) of D atx is given by in which the notation x i D −→x is used to indicate that all points in the converging sequence (x i ) i∈N lay in D. In general, N P D (x) ⊆ N D (x). Take a lower semicontinuous function G : R k → R and a pointx ∈ R k , the limiting subdifferential of G atx is while the reachable gradient of G atx is where diff(G) denotes the set of differentiability points of G and ∇ is the usual gradient operator. We define the hybrid subdifferential as ∂ > G(x) :=co ∂ * > G(x). The set ∂ * G(x) is nonempty, closed, in general non convex, and its convex hull coincides with the Clarke subdifferential

Optimal Control Problems and a Maximum Principle
In this section we introduce rigorously the constrained optimization problem over W 1,1controls u and its embedding in an extended, or impulsive, problem. Furthermore, we state a Maximum Principle for the extended problem. For simplicity, we will establish all the results for a single state constraint, explaining from time to time with remarks how to adapt these results to the case with N constraints. Throughout the paper we shall consider the following hypotheses.

(H0)
The control set C ⊆ R m is a convex cone, the set of ordinary controls A ⊂ R q is compact, and the endpoint constraint set T 0 ⊆ R 1+n+1+n is closed.

The Original Optimal Control Problem
We set T := T 0 ×] − ∞, K] and define the set U of strict sense controls as .
The original optimal control problem is defined as over the set of feasible strict sense processes (t 1 , t 2 , u, a, x, v).

(P)
We consider the following concept of local minimizer.

Definition 2.2
We call a feasible strict sense process (t 1 ,t 2 ,ū,ā,x,v) a local strict sense minimizer of (P) if there exists δ > 0 such that (2) is satisfied for all feasible strict sense processes, we say that (t 1 , Remark 2.1 Arguing similarly to [25], we could consider a more general cost of the form where 0 , 1 are nonnegative and the extended Lagrangian L, defined by The results of this article can also be applied to the case where dynamics, cost, and constraints depend on the variable u. In fact, it is sufficient to add to the control system in (P) the equations dx n+1 (t)/dt = du 1 (t)/dt, . . . , dx n+m (t)/dt = du m (t)/dt.

Remark 2.2 As is not difficult to see, given a closed, Hausdorff-Lipschitz continuous multifunction
. Therefore, we could allow implicit time-dependent state constraints of the form

The Extended Optimal Control Problem
We set and introduce the set of extended sense controls, defined as follows:
The set of strict sense processes, say Σ, can be embedded into the set of extended sense processes, Σ e , through the following map I : Σ → Σ e , defined as where, setting σ (t) , we associate to any strict sense process (t 1 , t 2 , u, a, x, v) the extended sense process where clearly ω 0 > 0 a.e.. Conversely, if (S, ω 0 , ω, α, y 0 , y, ν) is an extended sense process with ω 0 > 0 a.e., the absolutely continuous, increasing and surjective inverse σ : [t 1 , t 2 ] → [0, S] of y 0 , allows us to define the strict sense process Therefore, I is injective, 3 I(Σ) = {(S, ω 0 , ω, α, y 0 , y, ν) ∈ Σ e : ω 0 > 0 a.e.}, and the extension consists in considering also extended sense processes with w 0 possibly zero on some non-degenerate intervals. As anticipated in the Introduction, we will sometimes refer to the processes in I(Σ) as embedded strict sense processes.

A Maximum Principle for the Extended Problem
Consider the unmaximized Hamiltonian H , defined by Theorem 2.1 (PMP) Assume (H0)-(H2). Let (S,ω 0 ,ω,ᾱ,ȳ 0 ,ȳ,ν) be an extended sense local minimizer for (P e ) . Then there exist a path Borel measurable and μintegrable functions, verifying the following conditions: Furthermore: Proof The extended problem (P e ) is a conventional optimization problem in the state-space where H s , Ψ s denote the partial derivatives w.r.t. s of H and Ψ , respectively. However, since the vector fields f and (g j ) j =1,...,m , the cost function Ψ , and the constraints do not depend explicitly on the pseudo-time s, this yields the constancy of the Hamiltonian with constant equal to 0 in (iv). Finally, the strengthened non-triviality condition (9), which does not involve the multiplier π associated to ν, and the refinements (vii), (viii), can be proved as in [26,  given an extended sense local minimizer (S,ω 0 ,ω,ᾱ,ȳ 0 ,ȳ,ν), there exist (p 0 , p) ∈ W 1,1 , , and conditions (i)-(iv), (vii) and (viii) of Theorem 2.1 are met with (q 0 , q) and μ verifying Definition 2.5 (Normal and abnormal extremal) We say that a feasible extended sense process (S,ω 0 ,ω,ᾱ,ȳ 0 ,ȳ,ν) is an (extended sense) extremal if there exists a set of multipliers (p 0 , p, π, λ, μ) and functions m 0 and m which meet the conditions of Theorem 2.1. We will call an extremal normal if all possible choices of multipliers as above have λ > 0, and abnormal when there exists at least one set of such multipliers with λ = 0.

Infimum Gap and Abnormality
Write ) for the cost of a strict sense process (t 1 , t 2 , u, a, x, v) in problem (P), and J e (S, ω 0 , ω, α, y 0 , y, ν) := Ψ (y 0 (0), y(0), y 0 (S), y(S), ν(S)) for the cost of an extended sense process (S, ω 0 , ω, α, y 0 , y, ν) in problem (P e ). We also write Σ f ⊆ Σ and Σ f e ⊆ Σ e for the subset of feasible strict sense processes and for the subset of feasible extended sense processes, respectively. Furthermore, if (S,ω 0 ,ω,ᾱ,ȳ 0 ,ȳ,ν) is an extended sense local minimizer, we shall say that there is local infimum gap at (S,ω 0 ,ω,ᾱ,ȳ 0 ,ȳ,ν) if, for some δ > 0, where we have set To prove that, in the presence of a gap, extended sense local minimizers for problem (P e ) are abnormal extremals, it is convenient to rephrase Definition 3.1 only in terms of extended sense processes. Precisely, using the above notation, by the properties of the map I (see Eq. 6) it follows that Even if the set of embedded strict sense processes is dense into the set of extended sense processes with respect to the distance d ∞ , the infimum gap can actually occur, since all embedded strict sense processes close to a given feasible extended sense process might violate either the endpoint constraints or the state constraint. This leads us to the following definition: The following result relates isolated feasible extended sense processes and infimum gap. Proof Suppose by contradiction that (S,ω 0 ,ω,ᾱ,ȳ 0 ,ȳ,ν) is not isolated. Then we can take a sequence δ j ↓ 0 and, for each j ∈ N, there exists S j , ω 0 j , ω j , α j , y 0 j , y j , ν j ∈ B δ j e (S,ω 0 ,ω,ᾱ,ȳ 0 ,ȳ,ν) ∩ I(Σ f ). By the definition of d ∞ and the continuity of the cost function Ψ , this implies that no infimum gap may occur.
Thanks to Proposition 3.1, Theorem 3.1 is a straightforward consequence of the following result, which extends [26,Th. 4.4] to the case with state constraints, an additional ordinary control in the drift, and nonsmooth data. (H0)-(H1). If (S,ω 0 ,ω,ᾱ,ȳ 0 ,ȳ,ν) is an isolated feasible extended sense process, then it is an abnormal extremal.

Theorem 3.2 Assume
Proof Since the proof involves only space-time trajectories (y 0 , y) which are close to the reference space-time trajectory (ȳ 0 ,ȳ) and the controls assume values in a compact set, using standard truncation and mollification arguments we can assume that there exists some L > 0 such that the functions f , g 1 , . . . , g m , and h are L-Lipschitz continuous and bounded by L. The proof is divided into several steps in which successive sequences of optimization problems are introduced that have as eligible controls only embedded strict controls, and costs that measure how much a process violates the constraints. Using the Ekeland Principle, minimizers are then built for these problems, which converge to the initial isolated process. Furthermore, applying the PMP to these approximate problems with reference to the above mentioned minimizers, we obtain in the limit a set of multipliers with λ = 0 for problem (P e ), with reference to the isolated process (S,ω 0 ,ω,ᾱ,ȳ 0 ,ȳ,ν).
Let us now show that, through suitable reparameterization techniques, the sequence of minimizing processes (ζ i , ω i , α i , y 0 i , y i , ν i ) can be associated to a sequence of embedded strict processes converging to the original isolated process (S,ω 0 ,ω,ᾱ,ȳ 0 ,ȳ,ν).
Precisely, for each i ∈ N, let us consider the surjective, bi-Lipschitz continuous, and strictly increasing function is an embedded strict sense process for problem (P e ). In particular, we havẽ
Hence, we deduce from Eq. 17 that, for i sufficiently large, where δ > 0 is the constant appearing in Definition 3.2, with reference to the isolated feasible extended sense process (S,ω 0 ,ω,ᾱ,ȳ 0 ,ȳ,ν). As a consequence, for all i large enough, (S i ,ω 0 i ,ω i ,α i ,ỹ 0 i ,ỹ i ,ν i ) cannot be a feasible embedded strict sense process, namely, it must violate either the endpoint constraints or the state constraint. By Eqs. 19 and 20 this implies that J (y 0 i , y i , ν i ) > 0, namely, at least one of the following three inequalities holds true: ] h(y 0 i (s), y i (s)) > 0. (22) In the following, as is clearly not restrictive, we will always assume that the properties valid from a certain index onwards, apply to each index i ∈ N.
The process (ζ i , ω i , α i , y 0 i , y i , ν i , c i ) turns out to be a minimizer for Our aim is now to apply the Pontryagin Maximum Principle to problem (Q i ) with reference to the minimizer (ζ i , ω i , α i , y 0 i , y i , ν i , c i ). Preliminarily, let us observe that, passing eventually to a subsequence, we may assume that either c i > 0 for each i ∈ N or c i ≤ 0 for each i ∈ N.
Assume first that c i > 0 for each i ∈ N.
and σ k i = 0 when the maximum in Φ y 0 i (0), y i (0), y 0 i (S), y i (S), ν i (S) ∨ c i (S) is strictly greater than the k-th term in the maximization. Thus, the Maximum Principle in [35,Th. 9.3.1] yields the existence of some multipliers Observe that, for each i ∈ N, by (ii) and (iii) we have furthermore, (m 0 i , m i ) L ∞ ≤ L by (v) and the L-Lipschitz continuity of h. Then by (iii) and Eq. 23, we get By this estimate, Eq. 23, the non-triviality condition (i) , and using the facts that ρ i ≤ 1 2 for i sufficiently large and λ i ∈ [0, 1], for such i we get Hence, scaling the multipliers, we obtain Suppose now c i ≤ 0 for each i ∈ N. In this case, by Eq. 22, either ν i (S) > K or d T 0 (y 0 i (0), y i (0), y 0 i (S), y i (S)) > 0. Thus, for ε > 0 suitably small, the process (ζ i , ω i , α i , y 0 i , y i , ν i ,ĉ i ) withĉ i := c i + ε is still a minimizer for problem (Q i ) and, in addition, it verifies h(y 0 i (s), y i (s)) −ĉ i < 0 for all s ∈ [0,S] (namely, the state constraint is inactive on [0,S]). Hence, by applying the Maximum Principle for problem (Q i ) with reference to this minimizer we deduce the existence of multipliers (p 0 i , p i , π i , r i ) ∈ W 1,1 ([0,S]; R 1+n+1+1 ), which satisfy conditions (i) -(vi) with μ i = 0, σ 3 i = 0, π i ≤ 0, and λ i > 0. In this case, by considering again i sufficiently large to have ρ i ≤ 1 2 , from (iii) we get and, scaling the multipliers appropriately after summing (i) , we finally obtain Step 4. From the previous step, we arrive at the following properties (for either the case where c i > 0 for each i ∈ N or the case where c i ≤ 0 for each i ∈ N): for any i ∈ N, there exist (p 0 i , p i ) ∈ W 1,1 ([0,S]; R 1+n ), π i ≤ 0, μ i ∈ NBV + ([0,S]; R) and Borelmeasurable, μ i -integrable functions (m 0 i , m i ) : [0,S] → R 1+n , such that: 9]) such that, eventually for a further subsequence, π i → π, (p 0 i , p i ) → (p 0 , p) in L ∞ , and dp 0 i ds , dp i ds dp 0 ds , dp ds weakly in L 1 , as i → +∞. By this analysis it also follows that the functions (q 0 i , q i ) are uniformly integrably bounded and verify for a.e. s ∈ [0,S], Hence, by the dominated convergence theorem, one has Passing to the limit as i → +∞ and using Eq. 17, by (i),(v), and (vi) we get Furthermore, using that ȳ 0 (0),ȳ(0),ȳ 0 (S),ȳ(S),ν(S) ∈ T , the properties of distance function, and the 'max-rule' for subdifferentials, by (iii) we have Incidentally, from this relation we immediately deduce that π = 0 ifν(S) < K. Passing to the limit in (iv), with the help of a measurable selection theorem, using Eqs. 17 and 18 and the dominated convergence Theorem, we deduce that, for a.e. s ∈ [0,S], In view of relations Eqs. 28, 29, 30, 31 and 32, to conclude the proof that the isolated feasible process (S,ω 0 ,ω,ᾱ,ȳ 0 ,ȳ,ν) is an abnormal extremal, it remains only to show that Suppose by contradiction that Eq. 33 is not true. Then q 0 ≡ 0, q ≡ 0 a.e., and by Eq. 27 we deduce that π = 0, which in turn impliesν(S) = K > 0. Thanks to these information and integrating (31) in [0,S] we find that 0 = S 0 π |ω| ds = πν(S) = π K, which is not possible.

Nondegeneracy, Normality and No Infimum Gap
As a consequence of Theorem 3.1, 'normality implies no infimum gap'. Precisely, as a corollary of the results in Section 3, we have: (i) Suppose that there exists an extended sense minimizer for (P e ) which is a normal extremal. Then there is no infimum gap. (ii) Let (S,ω 0 ,ω,ᾱ,ȳ 0 ,ȳ,ν) be an extended sense local minimizer for the extended problem (P e ) which is a normal extremal. Then there is no local infimum gap at (S,ω 0 ,ω,ᾱ,ȳ 0 ,ȳ,ν).
As observed in the Introduction, the above 'normality test' is of more theoretical than practical interest (specially in the presence of state constraints). In this section we identify some verifiable conditions guaranteeing that every set of multipliers is normal. To begin with, let us introduce the notion of nondegenerate estremal.

Definition 4.1 (Nondegenerate Maximum Principle)
Given an extended sense local minimizer for problem (P e ), (S,ω 0 ,ω,ᾱ,ȳ 0 ,ȳ,ν), we say that the Maximum Principle is nondegenerate when there is a choice of the multipliers (p 0 , p, π, λ, μ) and of the functions m 0 , m that meets the conditions (i)-(vi) of Theorem 2.1, and such that where q 0 , q are defined as in Theorem 2.1. As it is easy to see, a nondegenerate abnormal extremal is always an abnormal extremal, and, on the contrary, any normal extremal is also nondegenerate normal. To obtain the converse implications, we introduce condition (CNa) below. In the following, we will often use the notation Ω := {(t, x) : h(t, x) ≤ 0}.

Constraint Qualifications for Normality (CQn) b , (CQn) f Letz be a feasible extended sense
process for the extended optimization problem (P e ).
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