Solutions to the Hamilton-Jacobi equation for Bolza problems with discontinuous time dependent data

We consider a class of optimal control problems in which the cost to minimize comprises both a ﬁnal cost and an integral term, and the data can be discontinuous with respect to the time variable in the following sense: they are continuous w.r.t. t on a set of full measure and have everywhere left and right limits. For this class of Bolza problems, employing techniques coming from viability theory, we give characterizations of the value function in the class of lower semicontinuous functions as the unique generalized solution to the corresponding Hamilton-Jacobi equation in the class of lower semicontinuous functions: if the ﬁnal cost term is extended valued, the generalized solution to the Hamilton-Jacobi equation involves the concepts of lower Dini derivative and the proximal normal vectors; if the ﬁnal cost term is a locally bounded lower semicontinuous function, then we can show that this has an equivalent characterization in a viscosity sense.


Introduction
Consider the non autonomous Bolza problem: Minimize T S L(t, x(t),ẋ(t))dt + g(x(T )) over arcs x ∈ W 1,1 ([S, T ], R n ) satisfyinġ x(t) ∈ F (t, x(t)) for almost every t ∈ [S, T ], in which [S, T ] is a given interval, x 0 ∈ R n is a given initial datum, g : R n → R ∪ {+∞} and L : [S, T ] × R n × R n → R are given functions, and F : [S, T ] × R n R n is a given multivalued function. The reference problem (P S,x 0 ) can be embedded in a family of problems (P t,x ) parametrized by pairs of initial data (t, x) ∈ [S, T ] × R n . This leads to the concept of the value function for (P t,x ) V : [S, T ] × R n → R ∪ {+∞}, which, for all (t, x) ∈ [S, T ] × R n , is defined taking the infimum cost for (P t,x ): Here, an F -trajectory on the interval [s, t] ⊂ [S, T ] is an absolutely continuous arc x(·) : [s, t] → R n which satisfies the reference differential inclusionẋ(σ) ∈ F (σ, x(σ)) for a.e. σ ∈ [s, t]. We shall consider characterizations of V (·, ·) as the unique solution -in a suitable generalized sense -to the Hamilton-Jacobi equation: when we may have a discontinuous behaviour of F and L w.r.t. the time variable t. Many techniques have been employed to characterize the value function as solution to (HJE), mainly coming from viscosity solutions theory and viability theory. In both contexts a lot of work has been done including the case of discontinuous time dependence problems (see for instance the monographs [2,9,5,20] and the references therein). In this paper we employ nonsmooth analysis tools and a viability approach to provide value function characterizations involving the notions of lower Dini derivative (also called contingent epiderivative), proximal subdifferential, and Fréchet subdifferential and superdifferential. An important feature is that we allow the final cost function g to be a lower semicontinuous function, possibly extended valued, incorporating implicit terminal constraints. As a consequence the natural class of functions in which we study the value function is the set of lower semicontinuous functions.
In presence of extended terminal costs, the first result, using viability theory, characterizing lower semicontinuous value functions as solutions to (HJE) in a generalized sense which involves the contingent epiderivatives, is obtained in [12]. In the same paper we can find also characterizations using (Fréchet) subdifferentials, and eventually both subdifferentials and superdifferentials leading to a comparison with viscosity solutions for continuous value functions. These results have been achieved for the Mayer problem (i.e. for L = 0) assuming velocity sets F which are continuous in (t, x). A further significant contribution is [9], in which appropriate invariance theorems allow to characterize the value function also considering proximal subdifferentials.
Passing to discontinuous time-dependent optimal control problems, the relevance of the role of lower Dini derivatives to deal with measurable time-dependence was highlighted by [21]. Simple examples illustrate that the value function might not be the unique lower semicontinuous generalized (according to the concepts above-mentioned) solution to (HJE) in an 'almost everywhere' sense (cf. the discussion in [4]). However, uniqueness properties of the solution can be derived for the mere measurable time dependent case imposing additional conditions on the class of functions which are candidate to be solutions, such as the epigraph of the candidate solution is absolutely continuous w.r.t. t, see [14].
A different perspective has been recently suggested in [4] for the intermediate case (between the continuous one and the merely measurable one) when the multifunction t ; F (t, x) has everywhere one-sided limits, for all x, and is continuous on the complement of a zero-measure subset of [S, T ] (without necessarily imposing further a priori regularity conditions such as the absolute continuity of the epigraph of the candidate solutions). In this context, considering optimal control problems with a final cost term (i.e. L = 0), the value function turns out to be the unique lower semicontinuous solution to (HJE) taking into account 'everywhere in t' characterizations which involve the concepts of lower Dini derivative and the proximal subdifferential. Further important features of the results obtained in [4] are: the presence of left and right limits F (t + , x) and F (t − , x) (the role of which cannot be exchanged) in the characterizing conditions and the presence of the horizontal proximal subdifferentials in the concept of the proximal solution.
The main objective of this paper is to explore lower semicontinuous characterizations of the value function in the context of non-autonomous Bolza problems, in which the velocity set F satisfies the same assumptions as in [4]. The Lagrangian L is assumed to have the same behaviour in t (L(·, x, v) is continuous on a set of full measure and has everywhere left and right limits), but is just continuous w.r.t. x. In addition, L satisfies conditions in v (such as convexity and boundedness on bounded sets). We observe that it would be natural to invoke a well-known augmentation technique and rewrite the reference Bolza problem (P S,x 0 ) in a Mayer form: Minimize g(x(T )) + y(T ) over arcs (x, y) ∈ W 1,1 ([S, T ], R n+1 ) satisfying (ẋ(t),ẏ(t)) ∈ G(t, x(t)) for almost every t ∈ [S, T ], x(S) = x 0 , y(S) = 0 where G(t, x) := {(v, w) | v ∈ F (t, x), w ≥ L(t, x, v)}. Even if this method provides a good insight of a correct outcome, previous results on the Mayer problem are not applicable in our case. On the other hand, keeping the Bolza formulation of the reference problem allows us, for instance, to impose weaker assumptions on the Lagrangian L, avoiding additional (and more restrictive) Lipschitz continuity conditions of L w.r.t. the state variable x, that would be otherwise necessary to impose if we passed to the Mayer form, and which is typically required in previous work for the Mayer problem (cf. [4], [12], [14] and [13]). Therefore, the mere state augmentation technique does not simplify the task: we would add a step in the analysis and eventually end up with a (possibly more involved) problem, with exactly the same difficulties as we left the reference optimal control problem in the Bolza form.
Our first main result (see Theorem 1.1 below) provides a characterization of lower semicontinuous extended valued value functions involving both the notions of generalized solution in terms of lower Dini derivatives and in terms of proximal normals to epigraph sets (confirming that a result consistent with [4,Theorem 2.2] can be obtained also for the class of Bolza problems considered here). The second main result of our paper gives a positive answer to an important question (highlighted in [4]): it was not known whether to achieve an extended-sense viscosity solution characterization of lower semicontinuous value functions would require employing horizontal Fréchet subderivatives (and superderivatives). Theorem 1.2 below gives (together with the examples in Section 1.4) an answer to this issue and represents, at the same time, an extension to earlier viscosity solutions characterizations such as in [12], [14] (and [13] for the state constraints free case), to locally bounded lower semicontinuous value functions for Bolza problems with F and L discontinuous in t and a discontinuous final cost term g.
To complete the huge picture of this strand, we recall that the viability approach is applicable also to characterize value functions for state constrained optimal control problems (cf. [15], [20], [13] and [4]). In this case, the analysis requires some compatibility conditions of the velocity sets F with the state constraint (called 'existence of inward/outward pointing conditions'), which conveys more restrictive assumptions on F and is based on some distance estimates results. The discussion on these technical aspects together with the appropriate assumptions which allow to revisit our results in the context of the state constrained Bolza problems goes beyond the main purpose of the present paper and is part of an ongoing project (cf. [3]).
The paper is organized as follows. In Section 1 we display the employed notation, the invoked assumptions (together with an hypotheses reduction technique), our main results (Theorem 1.1 and Theorem 1.2) accompanied by some refinements and a discussion based on three illustrative examples. The second section is dedicated to some preliminary results. Section 3 provides the proof of Theorem 1.1, which is split into three main steps. The proof of Theorem 1.2 is provided in Section 4.
1 Main results

Notation
In the paper we write R + the set of non negative real numbers, i.e. {x ∈ R | r ≥ 0}, and B for the closed unit ball in R n . We denote the Lebesgue subsets of [S, T ] and the Borel subsets of R m by L and B m respectively. The (associated) product σ-algebra of sets in [S, T ] × R m is written L × B m . We denote by L p ([α, β], R n ) the space of L p functions for the Lebesgue measure, that are defined on [α, β], and take values in R n . We write W 1,1 ([α, β], R n ), the space of absolutely continuous function for the Lebesgue measure endowed with the norm: Let D ⊂ R m , we denote by co D, D and co D respectively the convex hull, the closure and the closed convex hull of D. The polar cone D * to a subset D is given by: For arbitrary nonempty closed sets in R n , C and C, we denote by d H (C, C ) the 'Hausdorff distance' between C and C : Take a closed set C ⊂ R m and x ∈ R m . Then min y∈C {|x − y|} is the distance of x from the set C and is written d C (x). If f : C ⊂ R m → R is a locally bounded function, we denote its lower (resp. upper) semicontinous envelope by: The notation y C → x means that we are considering convergent sequences (y i ) i∈N such that y i → x, and each element y i belongs to C. An increasing function ω : R + → R + is a modulus of continuity if lim s→0 ω(s) = 0.
We also recall some basic concepts and tools coming from nonsmooth analysis (detailed dissertations of which can be found in the monographs [1,5,9,7,20]). Consider a set D ⊂ R m , a point x ∈ D and a multifunction G(·) : D ; R m . The limit inferior and the limit superior of G(·) at x along D (in the Kuratowski sense) are the sets lim inf The Bouligand tangent cone (alternatively referred to as contingent cone) T C (x) to a closed set C ⊂ R m at x ∈ C is defined by: The proximal normal cone to C at x ∈ C, denoted N P C (x), is defined by: The strict normal coneN C (x) to C at x is defined as followŝ Consider an extended valued function ϕ : . Take x ∈ dom (ϕ) and d ∈ R m . The lower Dini derivative (also called the contingent epiderivative, cf. [1,5,16]) of ϕ at x in the direction d ∈ R m , denoted D ↑ ϕ(x, d), is defined by: Similarly, one can also define the upper Dini derivative (alternatively referred to as the contingent hypoderivative), of ϕ at x in the direction d ∈ R m , denoted D ↓ ϕ(x, d): We evoke the following useful relations (see [1]): and We recall also that if U is an extended valued function defined on [S, T ]×R n , taking ∈ {1, −1}, then for (t, x) ∈ dom (U ) we can use a simpler expression for D ↑ U ((t, x), ( , d)): x)).
ii) The function g : R n → R ∪ {+∞} is lower semicontinuous, with nonempty domain.
(H2): i) For almost every t ∈ [S, T ] and x ∈ R n ii) ii) For almost every s ∈ [S, T [ and t ∈]S, T ], and every x ∈ R n we have For every t ∈ [S, T ] and x ∈ R n , L(t, x, ·) is convex.
ii) L is locally bounded in the following sense A priori boundedness and hypotheses reduction technique. We observe that condition (H2) guarantees a well-known a priori uniform boundedness property for the F -trajectories. More precisely, if we take initial data (t, x) ∈ [S, T ] × R n and an F -trajectory y ∈ W 1,1 ([t, T ], R n ) such that y(t) = x, then for every s ∈ [t, T ], y(s) T S c F (s)ds , then, owing to (H2) ii), for almost every s ∈ [t, T ],ẏ(s) ∈ c 0 B. As a consequence, once we fix the initial data (t, x), using a standard hypotheses reduction argument (cf. [4] or [20]), when we are interested in studying the behaviour of the value function at (t, x), we can impose much stronger assumptions. More precisely, we introduce the multifunction F : The multifunction F (·, ·) and the function L(·, ·, ·) satisfy hypotheses (H1), (H2) * , (H3) * , (H4), (H5) * and (H6) * , where we denote by (H2) * , (H3) * , (H5) * and (H6) * the global (stronger) version of conditions (H2), (H3), (H5) and (H6), in which we have removed the constant R 0 . The data of the problem (P t,x ) involving either (F, L) or ( F , L) do coincide in a neighbourhood of the reference point (t, x). It follows that in the forthcoming analysis we can invoke the more restrictive version of conditions (H1)-(H6) without loss of generality.

Characterizations of lower semicontinuous value functions
We consider the following family of minimization problems indexed by initial data (t, x) ∈ [S, T ] × R n : Minimize T t L(s, x(s),ẋ(s))ds + g(x(T )) over the arcs x ∈ W 1,1 ([t, T ], R n ) satisfyinġ x(s) ∈ F (s, x(s)), for almost every s ∈ [t, T ], We recall that the value function V : [S, T ] × R n → R ∪ {+∞} is defined by the infimum cost for (P t,x ): The first result provides a characterization of lower semicontinuous extended valued value functions in a generalized sense involving the concepts of Dini derivative and proximal normal (to the epigraph); these are sometimes referred to as 'lower Dini solutions' and 'proximal solutions' (cf. [20,7]). (b) The function U is lower semicontinuous and satisfies: .
(c) The function U is lower semicontinuous and satisfies: i) for every (t, x) ∈ (]S, T [×R n ) ∩ dom(U ) iii) for every x ∈ R n , lim inf We consider now the case when the final cost is lower semicontinuous and locally bounded. In this case it is immediate to see that the value function acquires the same properties. In presence of a locally bounded candidate U to be a solution to an Hamilton-Jacobi equation, a well-known approach in viscosity solutions theory suggests to consider its lower and upper semicontinuous envelopes and check whether the properties of supersolution and subsolution in the viscosity sense are satisfied (cf. [2]). From the perspective developed in our paper, this idea leads to a notion of viscosity solution expressed in terms of strict normals to the epigraph and the hypograph of the candidate solution U .
iii) for every x ∈ R n , lim inf If the condition (g * ) * = g in Theorem 1.2 is removed, then the implication is valid only in one sense: Imposing the lower semicontinuity of L w.r.t. t, we obtain the following result.
Proposition 1.4 Assume (H1)-(H6). If, in addition, we suppose that L( · , x, v) is a lower semicontinuous function for all (x, v) ∈ R 0 B×c 0 B, then the assertions of Theorem 1.1, Theorem 1.2 and Proposition 1.3 remain valid when we replace Remark 1.5 (i) The characterizations (c) and (d) of the value function V (·, ·) are expressed in terms of proximal normals to its epigraph, and strict normals to its epigraph and hypograph. Invoking the well-known relations between subdifferentials (superdifferentials) of a given function and the normal vectors to its epigraph (and hypograph) these properties can be alternatively rewritten considering proximal subdifferentials, and (Fréchet) subdifferentials and superdifferentials of V (·, ·) at points (t, x) belonging to the domain of V (·, ·).
In [4], for instance, where the velocity set F has the same discontinuous behaviour, characterizations of the values function are provided by conditions involving both horizontal and non-horizontal proximal subdifferentials. Here, we prefer to use the formulation with normal vectors because it summarizes in a concise way the characterization of interest, which in our case has to take into consideration also the Lagrangian term L. Moreover, the normal vectors expression highlights the somewhat 'abnormal' feature of the horizontal normal vectors (ξ 0 , ξ 1 , λ = 0), which corresponds to the case in which the Lagrangian disappears in conditions (c) and (d). The contribution of horizontal normals can be easily removed when 'F is continuous' (cf. [12,20]) owing to the well-known (Rockafellar) horizontal approximation theorem (cf. [9]), and it is not clear whether this simplification procedure would be in general applicable in the discontinuous context (cf. the issue raised in [4, Remark 2.2-(d)]).
(ii) Conditions in (b), (c) and (d) are formulated taking into account particular left and right limits w.r.t. t of F and L. For the Mayer problem, in [4] it is shown that the role of the left/right limits is crucial to characterize the value function, and assertions (b) and (c) become in general false if we try to exchange the role of those limits. As one may expect, our results for Bolza problems are consistent with [4]. We underline the fact that also for the viscosity solutions characterization (d) the role of the right limit is crucial as illustrated by Example 1 below. Finally we observe that, in (b) i), (c) i) and (d) i) of Theorems 1.1 and 1.2 we can avoid consideration of the limits of L w.r.t. t, owing to the lower semicontinuity of L (in t).
(iii) The characterization (d) provided by Theorem 1.2 concerns lower semicontinuous value functions for optimal control problems having a terminal cost g which is locally bounded and satisfies the condition (g * ) * = g. A natural question would be: Is that possible to characterize V (·, ·) in the sense of Theorem 1.2 for optimal control problems removing the conditions 'g is locally bounded or (g * ) * = g' ?
If g is a lower semicontinuous extended valued function (taking the value +∞ at some points), the issue of interpreting the viscosity subsolution replacing the condition (d)-ii) immediately arises and it is not clear how we have to interpret U * . Taking the lim sup operator we would lose crucial information on the boundary of dom(V ) and the viability approach would not be applicable or give the desired information. On the other hand, if we consider the smaller (extended valued) upper semicontinuous function bigger than V on the domain of V , under some circumstances (such as V is continuous on its domain and dom(V ) is a closed set) we would be induced to end up with the function V − which coincides with V on dom(V ) and takes the value −∞ on [S, T ] × R n \ dom(V ). The latter technique would not help either, as clarified by Example 2. Condition (g * ) * = g can be removed if we are interested in proving that the value function is a viscosity solution in the sense of (d) of Theorem 1.2 (as established by Proposition 1.3). However, condition (g * ) * = g becomes far from being just a technical hypothesis and emerges as crucial if we want a characterization (comparison result) for the value function. This point is illustrated in Example 3.
(iv) The results above are still valid if we start from a slightly more general context in which the Lagrangian in now extended valued L : [S, T ] × R n × R n −→ R ∪ {+∞} and assumption (H5)-ii) is replaced by a 'local boundedness a.e. in t' in the following sense: there exists a set of full measure E ⊂ [S, T ] such that Indeed, using the lower semicontinuity of L we can reduce attention to the case in which L is locally bounded in the sense of (H5), and, then, the analysis remains the same.
(v) Assertions (c) and (d) of Theorems 1.1 and 1.2 can be easily reformulated in terms of an Hamiltonian function Observe that under our assumptions H λ (·, x, p) turns out to be continuous on the complement of a zero-measure subset of [S, T ] and has everywhere one-sided limits H λ (t + , x, p) and H λ (t − , x, p).

Examples
Example 1. Consider the optimal control problem As a result of a routine analysis, one can see that conditions (b)-(c) of Theorem 1.1 and condition (d) of Theorem 1.2 are satisfied by V . Here, we only display some calculations at the point (t 0 , x 0 ) = ( 1 2 , 1 4 ), which is of particular interest since it carries information about the discontinuous behaviour of the data F , L and g at the same time. Consider, for instance, (d) ii) of Theorem 1.2. Take any (u, v) ∈ R 2 . We have: and hence: By polarity, we deduce that: Consistently with condition (d) ii) in Theorem 1.2, the value function satisfies: On the other hand, switching the roles of F 1 2 + and F 1 2 − in the analysis above, we would not obtain the validity of condition (d) ii) since, taking the vector Similarly, switching the roles of L 1 2 , x 0 , v and L ( 1 2 ) + , x 0 , v for the same normal vector, we would not obtain condition (d) ii) either: Even if we switched limits for both L and F , condition (d) ii) would not be satisfied since we have: This example shows that condition (d) ii) must involve the right limits F (t + , x) and L(t + , x, v), for, if the limits were taken from the other side, the assertion would be false in general. Similar considerations show the fundamental significance of the right limits also in condition (d) i) of Theorem 1.2.
Theorem 1.2 provides a characterization for the class of lower semicontinuous functions which are also locally bounded. One might wonder whether this result can be generalized to the class of lower semicontinuous extended valued functions, like for the characterization provided by Theorem 1.1. The major difficulty comes from interpreting the concept of viscosity subsolution (which would correspond to condition (d) ii)) on the boundary of the domain of the candidate to be value function. The notion of viscosity subsolution used in our paper involves consideration of the upper semicontinuous envelope V * , which has a clear meaning if V is locally bounded.
On the other hand, if V were extended valued (with a closed nonempty domain), one might be tempted to take into account the upper semicontinuous extended valued function V − : The following simple example shows that this would not provide the desired effect, even if F is continuous and the value function V is continuous on dom(V ).
Example 2. Consider the optimal control problem: Let us consider (t 0 , x 0 ) ∈ ]0, 1[×R such that x 0 + t 0 + 1 = 0. We have: Observe that the issue here is not due to the fact that horizontal vectors might be involved in the characterization, indeed the vector (1, 1, 1) ∈N hyp V − ((t 0 , x 0 ) , V (t 0 , x 0 )), considered above, is definitely non-horizontal and corresponds to the superdifferential p = (−1, The value function V : [0, 1] × R → R is: One can easily check that V is a vicosity solution, i.e. satisfies (d) i)-iii). Consider the function U : [0, 1] × R → R: Then U is also a viscosity solution in the sense of condition (d). This shows that, if we do not have the property (g * ) * = g, we do not obtain the uniqueness of the viscosity solution in the sense of (d).

Preliminary results
We observe that, under our reference assumptions (H1)-(H6) (or under their more restrictive form provided by the a priori boundedness argument), for every (t, has a minimizer. This is due to the fact that, with respect to the W 1,1 topology, the set of [7,Theorem 6.39] or [20, Theorem 2.5.3]) and the functional J(·) is lower semicontinuous. Taking into account (H5) * , we can state a local Lipschitz regularity lemma for the function L(s, y, ·) (locally uniformly with respect to (s, y)), the proof of which is based on standard arguments on convex functions, and therefore it is omitted. We just observe that the role of the number 2c 0 (instead of the simpler c 0 ) allows to deduce the Lipschitz regularity of L in v in a ball with the smaller radius c 0 .
In the following lemma we establish a further (uniform) regularity property of the Lagrangian, which we will invoke several times in our analysis.
As a consequence, from this inequality and from (14), we deduce the validity of (8). The proofs of ii) and iii) follow along the same lines. Indeed, in the first step of the proof, we can use respectively (H6) * ii) and the lower semicontinuity of L( · , x, v) instead of (H6) * iii) to obtain (11) on the suitable time interval.

2
We now introduce the auxiliary Lagrangian L − which will be used as a technical tool in the characterization of solutions to the Hamilton-Jacobi equation. Take any (t, x) ∈ ]S, T ] × R n . We consider the following modulus of continuity of F with respect to time (from the left) Write K := exp T S k F (s)ds . If we take also a vector v ∈ F (t − , x), we define the following set: and The map L − arises in a somehow natural way in some crucial steps of our analysis (cf. the proofs of Proposition 2.3 and Theorem 1.1 below). A similar auxiliary Lagrangian function was introduced in [10,11] to investigate characterization of solutions to Hamilton-Jacobi equations in the context of calculus of variations. In our framework the expression of L − is more involved since we have to take account of the velocity constraint given by the differential inclusionż(s) ∈ F (s, z(s)) and the possible different (from the left and from the right) limit behaviour of F w.r.t. t.
ii) If, in addition, L satisfies (H6) * i)-ii), then for every (t, Proof. i) Consider (t, x) ∈ ]S, T ] × R n . We can assume that v ∈ F (t − , x), since otherwise the stated inequality immediately follows from the definition of L − . Using Filippov existence theorem (cf. [20, Theorem 2.4.3]), we have Z(t, x, v) = ∅. As a consequence, we obtain inf t t−h L(s, z(s),ż(s))ds | z ∈ Z(t, x, v) = +∞, for all h ∈ ]0, t − S]. Invoking the a priori uniform boundedness of the F -trajectories, it is straightforward to see that all the arcs in Z(t, x, v) are uniformly bounded and uniformly Lipschitz continuous. Since L is bounded in the sense of condition (H5) * , we deduce that there exists a constant M 0 > 0 such that, for every z(·) ∈ Z(t, x, v), |L(s, z(s),ż(s))| ≤ M 0 , for almost every s ∈ [S, t].

It follows that, for all
We now establish (18). Let ε > 0. For every h ∈ ]0, t − S], small enough, we choose z h ∈ W 1,1 ([S, t], R n ) an εh-minimizer for the following Lagrange problem: Invoking again the a priori boundedness of F -trajectories, the family (ż h ) h∈]0,t−S] is bounded in L ∞ by c 0 . Using (9) of Lemma 2.2 and Jensen's inequality, we obtain for all h ∈ ]0, t − S]: From standard analysis, we also know that lim h↓0 Passing to the limit inferior in the last equation, we have: which confirms (18) since ε is arbitrary.
ii) Consider (t, x) ∈ ]S, T ] × R n . Again we can restrict our attention to the case v ∈ F (t − , x), since otherwise the assertion easily follows from the definition of L − , and claim that Indeed, using Filippov existence theorem, we can find an F -trajectory z ∈ W 1,1 ([S, t], R n ), such that z(t) = x and for every h ∈ [0, t − S]: where K = exp As a consequence, for every h ∈ ]0, t − S], we have: Dividing across by h, passing to the limit inferior as h goes to 0, yields (20). If L satisfies satisfies also (H6) * i)-ii), then: which confirms (19).

2
We conclude this section recalling a well-known result, referred to as the Weak Invariance/Global Viability Theorem (cf. [1] or [20]). i) The graph of Γ is closed and Γ(x) is a nonempty, convex set for each x ∈ R k ; ii) there exists c > 0 such that Then, given any x 0 ∈ D, there exists an absolutely continuous function x(·) satisfying

Proof of Theorem 1.1
The proof has the following structure: we first show that the value function satisfies property (b) of Theorem 1.1. We subsequently prove that condition (b) implies condition (c). Finally, if a lower semicontinuous function U satisfies (c) then we show that it coincides with the value function. Each step is highlighted by a proposition or a theorem statement. Proof. From the definition of V it immediately follows that V (T, ·) = g(·) confirming (b) iii). The lower semicontinuity of V can be deduced by standard arguments (see for instance [17, Theorem 1.1]). We have to prove that V satisfies (b) i) and (b) ii) of Theorem 1.1.
Step 1 The first part of this step is somewhat standard (cf. [20,4]). We briefly reproduce this analysis since, in the second part of this step, it has to be properly combined with suitable properties on the Lagrangian L, mainly described by Lemma 2.2.
T ], R n ) be a minimizing F -trajectory for (P t,x ), whose existence is guaranteed by our assumptions on F and L (see Section 2). Using the principle of optimality, for every δ ∈ ]0, T − t], we have: From the fundamental theorem of calculus we also have for every δ ∈ ]0, T − t]: Let (δ i ) i∈N be a strictly decreasing sequence of positive real numbers that converges to 0. For every integer i ∈ N, let us define v i ∈ R n by: From the a priori boundedness of the F -trajectories guaranteed by the hypotheses reduction of Section 1, |ẏ(s)| ≤ c 0 for almost every s ∈ [t, T ]. From this inequality, we deduce that the sequence (v i ) i∈N is bounded by c 0 . Then, there exists a vectorv ∈ c 0 B such that, up to a subsequence, v i − −−− → i→+∞v . Take any p ∈ R n and i ∈ N. Since F (s + , y(s)) = F (s, y(s)) almost everywhere for each i ∈ N, we have: But since the function s −→ max v∈F (s + ,y(s)) p · v is right continuous at s = t, letting i go to +∞ in equation (22), we have: Employing the characterization of the closed convex hull of a set by the support function [19,Thm 13.1], we deduce thatv ∈ F (t + , y(t)) = F (t + , x).
Thus, for any integer i ≥ N : Applying Jensen's inequality to the convex function L(t, x, ·), we also obtain: We deduce that −δ −1 i t+δ i t L(s, y(s),ẏ(s))ds + L(t + , x,v) ≤ 2ε for every integer i ≥ N , and so, from (23) we obtain: Since ε is arbitrary, this confirms (b)-i). Step Hypotheses on the multifunction F allow us to use the Filippov existence theorem: there exists an F -trajectoryz(·) that satisfiesz(t) = x, such that for every h ∈]0, t − S], where K = exp T S k F (s)ds and θ − t is the modulus of continuity defined in (15). Recalling the hypotheses reduction and definition of Z(t, x,ṽ) given in (16) (see Section 1), it follows that z(·) ∈ Z(t, x,ṽ) = ∅. For any h ∈ ]0, t − S], there exists an h 2 minimizer z h (·) ∈ Z(t, x,ṽ) of the Lagrange problem: inf t t−h L(s, z(s),ż(s))ds z ∈ Z(t, x,ṽ) Using the principle of optimality applied to the F -trajectories z h (·), we also have: It follows that for every h ∈]0, t − S[, L(s, z(s),ż(s))ds z ∈ Z(t, x,ṽ) + h.
Hence, passing to the limit inferior when h goes to 0 and recalling the definition of L − in (17), As a consequence, owing to ii) of Proposition 2.3, we obtain: which establishes the validity of (b)-ii), concluding the proof of Proposition 3.1. We shall make use of two technical lemmas, which provide consequences of properties (b) i) and (b) ii) of Theorem 1.1. . Then, there exists v ∈ F (t + , x), a sequence (v i ) i∈N in R n converging to v, and a strictly decreasing sequence (h i ) i∈N in R + , converging to 0 as i goes to +∞, such that: Assume, in addition, that U satisfies (b) i). Then we have: . Let (ε j ) j∈N in R + be a strictly decreasing sequence that converges to 0. For any j ∈ N, there exists a vector v j ∈ F (t + , x) such that: Since F (t + , x) is compact, there existsṽ in F (t + , x) for which, up to a subsequence, (v j ) j∈N converges toṽ. By definition of the limit inferior, for each j ∈ N, there exists a sequence (v j,i ) i∈N in R n converging to v j and a strictly decreasing sequence (h j,i ) i∈N in R + converging to 0 such that: x)).
there exists a sequence (v i ) i∈N in R n converging to v and a decreasing sequence (h i ) i∈N in R + which converges to 0, such that: Proof. Consider any (t, x) ∈ (]S, T ] × R n ) ∩ dom(U ) and v ∈ F (t − , x). We have: Using the definition of D ↑ U , there exists a sequence (v i ) i∈N in R n converging to v and a decreasing sequence (h i ) i∈N in R + , converging to 0, such that: which concludes the proof.

2
We are now ready to prove Proposition 3.2. The proof is split into three steps.
Step 1 We first claim that (c) i) from Theorem 1.1 holds: for every (t, x) ∈ (]S, T [×R n )∩dom(U ), for every proximal vector (ξ 0 , ξ 1 , −λ) ∈ N P epi U ((t, x), U (t, x)), we have Take any (t, x) ∈ (]S, T [×R n ) ∩ dom(U ) and (ξ 0 , ξ 1 , −λ) ∈ N P epi U ((t, x), U (t, x)). We necessarily have λ ∈ R + . Invoking the definition of the proximal normal cone, there exists M ∈ R + such that for every (t , x , α ) ∈ epi U : From Lemma 3.3, there exists v ∈ F (t + , x), a sequence (v i ) i∈N in R n that converges to v and a strictly decreasing sequence (h i ) i∈N in R + , converging to 0, such that: In particular we have: x))] 2 ) i∈N , which converges and is therefore bounded, with the sequence (h i ) i∈N , that converges to 0. Taking the particular values (t+h i , x+h i v i , U (t+h i , x+h i v i )) for (t , x , α ) in (27), and dividing across by h i , for every i ∈ N, we obtain: Letting the integer i go to +∞, we have: , and thus we obtain: ξ 0 + min v∈F (t + ,x) ξ 1 · v + λL(t + , x, v) ≤ 0, which confirms the claim of step 1.
Step 2 We now prove that (c) ii) is satisfied: for every (t, x) ∈ (]S, T [×R n ) ∩ dom(U ), for every proximal vector (ξ 0 , ξ 1 , −λ) ∈ N P epi U ((t, x), U (t, x)), we have Consider any (t, x) ∈ (]S, T [×R n ) ∩ dom(U ) and (ξ 0 , ξ 1 , −λ) ∈ N P epi U ((t, x), U (t, x)). Take any v ∈ F (t − , x). Owing to Lemma 3.4, we can find two sequences (v i ) i∈N and (h i ) i∈N satisfying (26). Employing the same arguments used in the first step, there exists M ∈ R + such that for every i ∈ N: Bearing in mind (26), letting i go to +∞, we obtain: Thus we have ξ 0 + ξ 1 · v + λL(t − , x, v) ≥ 0 and consequently: Step 3 To conclude the proof we have to consider the boundary conditions (c) iii). Take any x ∈ R n . Using the lower semicontinuity of U , we have: If (S, x) / ∈ dom(U ), then we immediately obtain: If (S, x) ∈ dom(U ), then using Lemma 3.3, we can find v ∈ F (S + , x), a sequence (v i ) i∈N in R n converging to v, and a strictly decreasing sequence (h i ) i∈N in R + converging to 0, such that (24) holds at t = S. As a consequence lim x). Thus we have: which gives the first equality in (c) iii) from Theorem 1.1.
, so we consider the case when (T, x) ∈ dom(U ). Then fix any v ∈ F (T − , x). Using Lemma 3.4, we deduce the existence of a sequence (v i ) i∈N in R n converging to v and a decreasing sequence (h i ) i∈N in R + , that converges to 0, such that (26) holds at t = T .
Using the relation U (T, x) = g(x), given by the fact U satisfies (b) iii), we obtain the last desired boundary condition at t = T .

A proximal solution coincides with the value function: comparison results
We display the last part of the proof which consists in showing that if a lower semicontinuous function U : [S, T ] × R n → R ∪ {+∞} satisfies (c) of Theorem 1.1, then it coincides with the value function for (P t,x ). We observe that for the inequality V (t, x) ≤ U (t, x) conditions (H6) ii) and iii) are not necessary, but they are required for the opposite inequality. More precisely we will prove the following result. i) Suppose that U satisfies (c) i), and that, for all x ∈ R n , .
ii) Assume, in addition, that L satisfies (H6) ii) and iii). Suppose that U satisfies (c) ii), and for all x ∈ R n lim inf . Proof of Theorem 3.5 i). In order to establish the first comparison result, bearing in mind the hypotheses reduction of Section 1, we introduce an auxiliary multivalued function: Q : [S, T ] × R n ; R n × R defined by: A routine analysis allows to verify that the multifunction Q takes as values nonempty convex sets with elements which are (uniformly) bounded by c := c 2 0 + M 2 0 ; moreover the graph of Q is closed. Take any (t 0 , x 0 ) ∈ (]S, T [×R n ) ∩ dom(U ). The crucial point of Theorem 3.5 i) is establishing the applicability of the Weak Invariance Theorem 2.4 for the following differential inclusion: where Γ : [S, T ] × R n+1 ; R n+2 is defined by Clearly the multifunction Γ inherits the following properties from Q: the graph of Γ is closed, for all (τ, x, ) ∈ [S, T ]×R n+1 , Γ(τ, x, ) is nonempty convex set and Γ(τ, x, ) ⊂ (c+1)B. The 'inward pointing condition' iii) of Theorem 2.4 is also satisfied. Indeed, for τ = S, T the construction of Γ immediately yields the required inequality (for instance taking w = 0 that belongs to both Γ(S, x, ) and Γ (T, x, )). On the other hand, since U satisfies (c) i) of Theorem 1.1, for every (τ, x) ∈ dom(U ), and every (ξ 0 , ξ 1 , −λ) ∈ N P epi U ((τ, x), U (τ, x)), there existsv ∈ F (τ + , x) (recall that F (τ + , x) is compact) such that: and min w∈Γ(τ,x, ) (ξ 0 , ξ 1 , −λ) · w ≤ ξ 0 + ξ 1 ·v + λL(τ + , x,v). Then, the Weak Invariance Theorem 2.4 is applicable and there exists (τ (·), x(·), (·)) ∈ W 1,1 ([t 0 , T ], R × R n × R) satisfying τ (t) = t and        (ẋ(t),˙ (t)) ∈ Q(t, x(t)), for a.e. t ∈ [t 0 , T ] Taking into account the definition of the multivalued function Q and the hypotheses on both F and L, we deduce that x(·) is an F -trajectory and that˙ (s) ≤ −L(s, x(s),ẋ(s)) for a.e. s ∈ [t 0 , T ]. Hence we have: L(s, x(s),ẋ(s))ds, which implies: Thus we obtain: If (S, x 0 ) belongs to dom(U ), we pick a decreasing sequence (h i ) i∈N in R + that converges to 0 and a sequence (y i ) i∈N in R n that converges to x 0 such that: From what precedes, for every integer i ∈ N, we have: Passing to the limit inferior in that last equation yields: Note that we also have g( Proof of Theorem 3.5 ii). Pick (t,x) ∈ ([S, T ] × R n ) ∩ dom(U ) and let x ∈ W 1,1 ([t, T ], R n ) be an F -trajectory such that x(t ) =x. We want to prove: U (t,x) ≤ g(x(T )) + T t L(s, x(s),ẋ(s))ds.
We can assume that g(x(T )) < +∞, otherwise we automatically have the desired inequality. Using the fact that lim inf {(t ,x )→(T,x) | t <T } U (t , x ) = U (T, x) = g(x), we can find a sequence of points (T i , y i ) in ]t, T [×R n such that lim i→+∞ (T i , y i ) = (T, x(T )) and lim i→+∞ U (T i , y i ) = U (T, x(T )). Invoking Filippov's Existence Theorem and arguing as in [4], we obtain a subse- The multivalued function F satisfies the assumptions which allow to apply Carathéodory's parametrization theorems [1, Theorems 9.6.2 and 9.7.2]. Hence there exists a mesurable function: such that: For every (x, u) ∈ R n × B, f (·, x, u) is mesurable ; Fix i ≥ 0. Since x i (·) is an F -trajectory, for almost every t ∈ [t, T ]: and, using Filippov's selection theorem (cf. [20,Theorem 2.3.13]), there exists a mesurable selection u i : [t, T ] → B such that: Let ε > 0. Lusin's theorem (cf. [7, Proposition 6.14]) allows us to find a pair of functions ( For The multivalued function Γ ε i is convex, compact valued and has closed graph. We consider the following differential inclusion: We define the arc The arc (τ ε i , y ε i , ε i ) is the unique Γ ε i -trajectory with initial condition (0, x(T i ), U (T i , x i (T i ))). Owing to the 'hypotheses reduction' argument of Section 1, we deduce that there exists a constant c := c 2 0 + M 2 0 such that Γ ε i (t, x, ) ⊂ (c + 1)B, for every (t, x, ).
For every (τ, x) ∈ [0, T i −t ] × R n , we set U (τ, x) := U (T i − τ, x). Therefore the last condition in (31) can be interpreted as the inclusion (τ (t), y(t), (t)) ∈ epi U for all t ∈ [0, T i −t ]. We claim that the Weak Invariance Theorem 2.4 is applicable to the differential inclusion (31). We have already observed that the assumptions i) and ii) of this theorem are satisfied. We show now that Γ ε i also satisfies the last ('inward pointing') condition iii). That is, for every pair (τ, x) ∈ ([0, T i −t [×R n ) ∩ dom( U ) and every ≥ U (τ, x): x)) (we recall that we can always reduce to the case = U (τ, x)), which is equivalent to say: and, bearing in mind U satisfies (5) of condition (c) ii), we obtain: Hence we can confirm (32) by choosing w . As a consequence we can apply the Weak Invariance Theorem obtaining that the arc (τ ε i (·), y ε i (·), ε i (·)) is the solution to (31). For t = T i −t, by a change of variable, we have: Invoking condition (H2) * , for all t ∈ [t, T ], max v∈F (t,x i (t)) |v| ≤ c 0 . So, for almost every s ∈ [t, T ] we have: As a consequence, up to a subsequence,ẋ ε i (·) converges toẋ i (·) almost everywhere in [t, T ]. Using (H6) * and (7), we can apply Lebesgue's dominated convergence theorem, and we obtain: Passing to the limit inferior in (33), bearing in mind that x ε i (t) − −− → ε→0 x i (t), and U is lower semicontinuous, we obtain , for all i.
Since x(·) was an arbitrary F -trajectory satisfying x(t) =x, we deduce that This concludes the proof. Step 3 we prove that if we impose the additional assumption (g * ) * = g, then any lower semicontinuous function U satisfying (d) ii) and (d) iii) satisfies U ≤ V .
Assume that hypotheses (H1)-(H6) are satisfied. We first observe that, from the a priori boundedness of the F -trajectories, and the local boundedness of g and L, it immediately follows that V is locally bounded. Let (t, x) ∈ ]S, T [×R n . We invoke the 'hypotheses reduction' of Section 1, and we can apply the same argument of the proof of Step 1 of Proposition 3.1, obtaining the existence of a vector v ∈ F (t + , x) such that: This and the relation (2) imply that: x)) is the polar cone to the set T epi V ((t, x), V (t, x)) we have: This easily implies that: which confirms (d) i).
Let (t, x) ∈ ]S, T [×R n andṽ ∈ F (t + , x). There exists a sequence (t i , x i ) i∈N in ]S, T [×R n \{(t, x)} that converges to (t, x) such that: We claim that we can extract a subsequence such that t i > t for all i ∈ N. Let us assume that t i ≤ t for every i ∈ N and take a strictly decreasing sequence (τ i ) i∈N in ]t, T ] that converges to t. Fix any i ∈ N, and take an F -trajectory x i (·) ∈ W 1,1 ([t i , T ], R n ) such that x i (t i ) = x i . Using the principle of optimality, we obtain: Using the local boundedness of L given by condition (H5) * , there exists M 0 > 0 such that for every i ∈ N: Passing to the limit superior and using the upper semicontinuity of V * , we obtain: Hence lim sup i→+∞ V (τ i , x i (τ i )) = V * (t, x) and there exists a subsequence (i k ) k∈N for which: Fix anyṽ ∈ F (t + , x). Then for every i ∈ N, there exists v i ∈ F (t + i , x i ) such that lim i→+∞ v i =ṽ. For every i ∈ N, we consider the arc Using the Filippov existence theorem, for every i ∈ N there exists an F -trajectory z i (·) that satisfies z i (t i ) = x i and such that for every h ∈ ]0, T − t i ] where K = exp T S k F (s)ds . From the a priori boundedness of F -trajectories, we can pick R 0 > 0 such that, for every i ∈ N, |y i (s)| ≤ R 0 for every s ∈ [t i , T ]. Observe also that |ẏ i (s)| ≤ c 0 , for any i ∈ N and for almost every s ∈ [t i , T ]. For every i ∈ N, we define δ i = max{|V (t i , x i ) − V * (t, x)|, |x i − x|, |t i − t|}. Take a strictly decreasing sequence (h i ) i∈N that converges to 0 such that h i ≥ √ δ i .
We recall the definition of D ↓ V * ((t, x), (1,ṽ)): Fix any i ∈ N, and set w i = 1 h i t i +h i t iż i (s)ds. Note that we have: There exists τ ∈ [t i , t i + h i ] and z ∈ R n verifying |x i − z| ≤ c 0 h i such that: Hence we obtain: We notice that: This yields: we obtain: For every i ∈ N, we define: and immediately notice that lim i→+∞ (e i ,w i ) = (1,ṽ). This yields: Fix i ∈ N. We have: Using the principle of optimality, we obtain: Hence, dividing across equation (34) by h i , passing to the limit superior in this inequality while recalling δ i h i ≤ √ δ i , we obtain: We recall that from Lemma 2.1, there exists k L > 0 such that for every (t , x ) ∈ [S, T ] × R n , and v, v ∈ c 0 B: As a consequence, for every i ∈ N we have: Dividing across by h i , passing to the limit inferior as i goes to +∞ gives: Combining (35) and (36) we obtain: From the relation (3), this implies that: x)). Necessarily we have λ ∈ R + . Hence using the polarity relation between the contingent cone and the strict normal cone we have: This relation being valid for all v ∈ F (t + , x), we obtain: which confirms (d) ii).
To prove that V satisfies (d) iii), only the assertion V * (T, ·) = g * (·) remains to be proved. Since V (T, ·) = g(·), it is obvious that V * (T, x) ≥ g * (x) for every x ∈ R n . We prove that the converse inequality is also satisfied.
Fix any x ∈ R n . There exists a sequence (t i , For every i ∈ N there exists an F -trajectory x i (·) ∈ W 1,1 ([t i , T ], R n ) such that x i (t i ) = x i . By the principle of optimality: L(s, x i (s),ẋ i (s))ds ≤ V (t, x i (t)), for all t ∈ [t i , T ].
Using again condition (H5) * , we know that there exists a constant M 0 > 0 such that for every i ∈ N: V (t i , x i ) − M 0 |T − t i | ≤ V (T, x i (T )) = g(x i (T )).
Using the fact lim i→+∞ x i (T ) = x, we pass to the limit superior as i tends to +∞ and obtain: V * (T, x) ≤ lim sup i→+∞ g(x i (T )) ≤ lim sup y→x g(y) = g * (x), which achieves to show that V satisfies (d) iii).
This implies that U satisfies (c) i) from Theorem 1.1. Since U satisfies (d) iii), we can use the same arguments employed in the proof of Theorem 3.5 i) we have: V (t,x) ≤ U (t,x), for every (t,x) ∈ [S, T ] × R n .
Step 3 We prove that if U satisfies (d) ii) and (d) iii), then for every (t,x) ∈ [S, T ] × R n we have U (t,x) ≤ V (t,x). Using (d) iii), we can restrict attention to the case when (t,x) ∈ ]S, T [×R n . Let x ∈ W 1,1 ([t, T ], R n ) be an F -trajectory such that x(t ) =x. We want to prove prove that: U (t,x) ≤ g(x(T )) + T t L(s, x(s),ẋ(s))ds.
As a consequence, the Weak Invariance Theorem 2.4 is applicable to the differential inclusion (38), and we can conclude that the arc (τ j , x j , j ) is a solution to the constrained differential inclusion (38). It follows that at t = T : U * (t,x j ) − T t L(s, x j (s),ẋ j (s))ds ≤ U * (T, x j (T )) = g * (x j (T )), for every j. Since U ≤ U * , and U is lower semicontinuous, passing to the limit inferior in (40) yields: U (t,x) ≤ lim inf j→+∞ g * (ξ j ) + T t L(s, x(s),ẋ(s))ds.

2
Proof of Proposition 1.3. The proof immediately follows from the proof of Theorem 1.2, observing that condition (g * ) * = g is used only in Step 3.