Maria-Vittoria Vittoria Salvetti

Hilde Ouvrard

Bruno Koobus

Simone Camarri

Thème Num

Alain Dervieux

A Hybrid Model Based on Continuous Fluctuation Corrections

Keywords: Turbulence, Hybrid RANS/LES approach, blu-body ows, variational multi-scale, unstructured meshes Turbulence, Hybride RANS/LES, écoulements autour d'obstacles, Variational Multiscale, maillages non-structurés

published or not. The documents may come

Introduction

The most widely used approach for the simulation of high-Reynolds number turbulent ows is the one based on the Reynolds-Averaged Navier-Stokes equations (RANS). However, RANS models usually have diculties in providing accurate predictions for ows with massive separations, as for instance the ow around blu bodies. An alternative approach is the Large-Eddy simulation (LES), which, for massively separated ows, is generally more accurate, but also computationally more expensive, than RANS. Moreover, the cost of LES simulations increases as the ow Reynolds number is increased. Indeed, the grid has to be ne enough to resolve a signicant part of the turbulent scales, and this becomes particularly critical in the near-wall regions. A new class of models has recently been proposed in the literature in which RANS and LES approaches are combined together in order to obtain simulations as accurate as in the LES case but at reasonable computational costs. These hybrid methods can be divided in zonal approaches (e.g. [START_REF] Davidson | Hybrid LES-RANS using synthesized turbulent uctuations for forcing in the interface region[END_REF], [START_REF] Keating | Interface conditions for hybrid RANS/LES calculations[END_REF], [START_REF] Terracol | A zonal RANS/LES approach for noise sources prediction[END_REF] and [START_REF] Tessicini | Approximate near-wall treatments based on zonal and hybrid RANS-LES methods for LES at high Reynolds numbers[END_REF]), in which the regions where RANS and LES are used are a-priori dened, and the so called universal models, which should be able to automatically switch from RANS to LES throughout the computational domain. In the perspective of the simulation of massively separated unsteady ows in complex geometry, as occur in many cases of engineering or industrial interest, we are primarily interested in universal hybrid models. Among the universal hybrid models described in the literature, the Detached Eddy Simulation (DES) has received the largest attention. This approach [START_REF] Spalart | Advances in DNS/LES, chapter Comments on the feasibility of LES for wings and on a hybrid RANS/LES approach[END_REF] is generally based on the Spalart-Allmaras RANS model, modied in such a way that, in the near-wall region and with RANS-like grids the Spalart-Allmaras RANS model is used, while far from solid walls and with rened grids, the simulation switches to the LES mode with a one-equation SGS closure. Another universal hybrid approach is the Limited Numerical Scales (LNS, [START_REF] Batten | Interfacing statistical turbulence closures with large-eddy simulation[END_REF]), in which the blending parameter depends on the values of the eddy-viscosity given by a RANS model and of the SGS viscosity given by a LES closure. In practice, the minimum of the two eddy-viscosities is used (see also [START_REF] Camarri | Hybrid RANS/LES simulations of a blu-body ow[END_REF] for an example of application of LNS to the simulation of blu-body ows).

A major diculty in combining a standard RANS model with a LES one is due to the fact that the RANS equations govern the averaged ow eld, while the LES equations describe the evolution of spatially ltered, uctuating variables. RANS solution can be steady or only contains the unsteadiness of the largest scales, if large-scale unsteadiness is present in the ow (e.g. blu-body ows). On the other hand, LES needs a signicant level of uctuations in order to model the ow with a sucient accuracy. This leads to the need of special treatment at the RANS/LES interface regions in zonal approaches (see, e.g., [START_REF] Davidson | Hybrid LES-RANS using synthesized turbulent uctuations for forcing in the interface region[END_REF], [START_REF] Keating | Interface conditions for hybrid RANS/LES calculations[END_REF], [START_REF] Terracol | A zonal RANS/LES approach for noise sources prediction[END_REF] and [START_REF] Tessicini | Approximate near-wall treatments based on zonal and hybrid RANS-LES methods for LES at high Reynolds numbers[END_REF]) and to the so-called grey-zones or modeled stress depletion ([28] or [START_REF] Batten | Interfacing statistical turbulence closures with large-eddy simulation[END_REF]) for universal models. More generally, nding strategies building universal hybridization models, i.e. able to combine a large class of statistical models with a large class of LES is still an open issue.

In the present work, such a strategy for blending RANS and LES approaches in a hybrid model is proposed. To this purpose, the NLDE (Non-Linear Disturbance Equations) technique is used [START_REF] Labourasse | Reconstruction of turbulent uctuations using a hybrid RANS/LES approach[END_REF], in which the ow variables are decomposed in a RANS part (i.e. the averaged ow eld), a correction part that takes into account the turbulent large-scale uc-RR n°6564 tuations, and a third part made of the unresolved or SGS uctuations. The basic idea is to solve the RANS equations in the whole computational domain and to correct the obtained averaged ow eld by adding, where the grid is adequately rened, the remaining resolved uctuations. The NLDE technique was formulated and used as a zonal approach ( [START_REF] Labourasse | Reconstruction of turbulent uctuations using a hybrid RANS/LES approach[END_REF], [START_REF] Terracol | Hybrid methods for airframe noise numerical predictions[END_REF], [START_REF] Terracol | A zonal RANS/LES approach for noise sources prediction[END_REF]). In contrast, we search here for a hybridization strategy in which the RANS and LES models are blended in the computational domain following a given criterion. To this aim, a blending function is introduced, θ, which smoothly varies between 0 and 1. The correction term which is added to the averaged ow eld is thus damped by a factor (1-θ), obtaining a model which coincides with the RANS approach when θ = 1 and recovers the LES approach in the limit of θ → 0. For θ going from 0 to 1, i.e. when the model switches from the RANS to the LES mode, a progressive addition of uctuations is obtained and is aimed at avoiding, although in an empirical manner, the previously mentioned problems occuring for abrupt switches from RANS to LES (and vice versa). Following strictly these guidelines would imply that the two elds, RANS and LES correction, need to be computed separately. In this rst paper, we explore a single eld version and investigate several other ingredients of the proposed hybrid family. In particular, three dierent denitions of the blending function θ are proposed and will be examined in this paper. They are based on the ratios between (i) two eddy viscosities, (ii) two characteristic length scales and (iii) two characteristic time scales given by the RANS and the LES models, respectively. A low-Reynolds version of the k -ε model [START_REF] Goldberg | A wall-distance-free k -ε model with enhanced near-wall treatment[END_REF] is used here as RANS closure, while for the LES part the Variational Multi-Scale approach (VMS) is adopted [START_REF] Hughes | Large eddy simulation and the variational multiscale method[END_REF]. Note that the proposed hybridization strategy does not require any particular RANS or LES closure as soon as characteristic length scales or time scales can be dened. It permits a natural integration of the VMS concept, while this is not the case for other existing approaches, as LNS or DES. The VMS approach can be compared in terms of accuracy to the dynamic Smagorinsky model, but its computational cost is denitely lower and comparable to that of the simple Smagorinsky model (see [START_REF] Koobus | A variational multiscale method for the large eddy simulation of compressible turbulent ows on unstructured meshes-application to vortex shedding[END_REF]). The extension of VMS to unstructured meshes [START_REF] Koobus | A variational multiscale method for the large eddy simulation of compressible turbulent ows on unstructured meshes-application to vortex shedding[END_REF] [START_REF] Sirnivas | A study of LES models for the simulation of a turbulent ow around a truss spar geometry[END_REF] opens the way for industrial applications.

The capabilities of the proposed hybrid approach, refered in the sequel as the Fluctuation Correction Model (FCM), are appraised in the simulation of the ow around a circular cylinder at Re = 140000 (based on the far-eld velocity and the cylinder diameter). Comparisons with experimental data and DES simulations in the literature are provided. The sensitivity to the blending function denition, to the closure model for the LES part and to grid renement are investigated.

Hybrid RANS/LES coupling

The Navier-Stokes equations for compressible ows of (calorically and thermally) perfect Newtonian gases are considered here, in conservative form and using the following variables: W =(ρ, ρu i (i = 1, 2, 3), E), where ρ is the density, ρu i is momentum and E is the total energy per unit volume (E = ρe + 1/2ρu i u i , e being the internal energy).

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As in [START_REF] Labourasse | Reconstruction of turbulent uctuations using a hybrid RANS/LES approach[END_REF], the following decomposition of the ow variables is adopted:

W = < W > RAN S + W c correction +W SGS
where < W > are the RANS ow variables, obtained by applying an averaging operator to the Navier-Stokes equations, W c are the remaining resolved uctuations (i.e. < W > +W c are the ow variables in LES) and W SGS are the unresolved or SGS uctuations.

If we write the Navier-Stokes equations in the following compact conservative form:

∂W ∂t + ∇ • F (W ) = 0
in which F represents both the viscous and the convective uxes, for the averaged ow W we get:

∂ W ∂t + ∇ • F ( W ) = -τ RAN S ( W ) (1)
where τ RAN S ( W ) is the closure term given by a RANS turbulence model.

As well known, by applying a ltering operator to the Navier-Stokes equations, the LES equations are obtained, which can be written as follows:

∂ W + W c ∂t + ∇ • F ( W + W c ) = -τ LES ( W + W c ) (2)
where τ LES is the SGS term.

An equation for the resolved uctuations W c can thus be derived (see also [START_REF] Labourasse | Reconstruction of turbulent uctuations using a hybrid RANS/LES approach[END_REF]):

∂W c ∂t + ∇ • F ( W + W c ) -∇ • F ( W ) = τ RAN S ( W ) -τ LES ( W + W c ) (3)
The basic idea of the proposed hybrid model is to solve Eq. (1) in the whole domain and to correct the obtained averaged ow by adding the remaining resolved uctuations (computed through Eq. ( 3)), wherever the grid resolution is adequate for a LES. To identify the regions where the additional uctuations must be computed, we introduce a blending function, θ, smoothly varying between 0 and 1. When θ = 1, no correction to W is computed and, thus, the RANS approach is recovered. Conversely, wherever θ < 1, additional resolved uctuations are computed; in the limit of θ → 0 we want to recover a full LES approach.

The denition of the blending function is discussed in details in Sec. 6. Thus, the following equation is used here for the correction term:

∂W c ∂t + ∇ • F ( W + W c ) -∇ • F ( W ) = (1 -θ) τ RAN S ( W ) -τ LES ( W + W c ) (4)
Note that for θ → 1 the RANS limit is actually recovered; indeed, for θ = 1 the right-hand side of Eq. ( 4) vanishes and, hence, a trivial solution is W c = 0. As required, for θ = 0

Eq. ( 4) becomes identical to Eq. ( 3) and the remaining resolved uctuations are added RR n°6564 to the averaged ow; the model, thus, works in LES mode. For θ going from 1 to 0, i.e.

when, following the denition of the blending function (see Sec. 6), the grid resolution is intermediate between one adequate for RANS and one adequate for LES, the right-hand side term in Eq. ( 4) is damped through multiplication by (1 -θ). Although it could seem rather arbitrary from a physical point of view, this is aimed to obtain a smooth transition between RANS and LES. More specically, we wish to obtain a progressive addition of uctuations when the grid resolution increases and the model switches from the RANS to the LES mode.

Summarizing, the ingredients of the proposed approach are: a RANS closure model, a SGS model for LES and the denition of the blending function. These will be described in Secs. 4, 5 and 6.

Basic numerical ingredients

The governing equations are discretized in space using a mixed nite-volume/nite-element method applied to unstructured tetrahedrizations. The adopted scheme is vertex centered, i.e. all degrees of freedom are located at the vertices. P1 Galerkin nite elements are used to discretize the diusive terms. A dual nite-volume grid is obtained by building a cell C i around each vertex i. The convective uxes are discretized in terms of uxes through the common boundaries shared by neighboring cells. The Roe scheme [START_REF] Roe | Approximate Riemann solvers, parameters, vectors and dierence schemes[END_REF] represents the basic upwind component for the numerical evaluation of the convective uxes. A Turkel-type preconditioning term is introduced to avoid accuracy problems at low Mach numbers [START_REF] Guillard | On the behaviour of upwind schemes in the low Mach number limit[END_REF].

A parameter γ s multiplies the upwind part of the scheme and permits a direct control of the numerical viscosity, leading to a full upwind scheme for γ s = 1 and to a centered scheme when γ s = 0. The resulting numerical approximation of the convective ux between the i-th and the j-th cells is the following:

Φ R (W i , W j , n) = F (W i , n) + F (W j , n) 2 -γ s P -1 |P R| W j -W i 2 (5)
in which W i is the solution vector at the i-th node, n is the outward normal to the cell boundary and R (W i , W j , n) is the Roe Matrix. The matrix P (W i , W j ) is the preconditioning term. Note that, since it only appears in the upwind part of the numerical uxes, the scheme remains consistent in time and can thus be used for unsteady ow simulations.

The MUSCL linear reconstruction method (Monotone Upwind Schemes for Conservation Laws [START_REF] Van Leer | Towards the ultimate conservative scheme. IV: A new approach to numerical convection[END_REF]) is employed to increase the order of accuracy of the Roe scheme. This is obtained by expressing the Roe ux between two cells, centered on two generic nodes i and j, as a function of the reconstructed values of W at their interface. A reconstruction using a combination of dierent families of approximate gradients is adopted [START_REF] Camarri | A low diusion MUSCL scheme for LES on unstructured grids[END_REF], which allows a numerical dissipation made of sixth-order space derivatives, and, thus, concentrated on a narrow-band of the highest resolved frequencies, to be obtained. This is important in LES simulations to limit as far as possible the interactions between numerical and SGS dissipation, which could deteriorate the accuracy of the results.

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An implicit time marching algorithm is used, based on a second-order time-accurate backward dierence scheme.

More details on the numerical ingredients used in the present work can be found in [START_REF] Camarri | A low diusion MUSCL scheme for LES on unstructured grids[END_REF] and [START_REF] Farhat | Simulation of vortex shedding dominated ows past rigid and exible structures[END_REF].

RANS closure model

As far the closure of the RANS equations is concerned, the Low Reynolds k -ε model proposed in [START_REF] Goldberg | A wall-distance-free k -ε model with enhanced near-wall treatment[END_REF] is considered. The Reynolds stress tensor has the same form as in the standard k -ε model but here the turbulent eddy-viscosity µ t is dened as follows:

µ t = C µ f µ k 2 (6)
where C µ has the same value as in the standard k -ε multiplied by a damping function f µ , chosen as follows:

f µ = 1 -e -AµRt 1 -e -R 1/2 t max(1, ψ -1 ) (7)
where

ψ = R 1/2 t /C τ , R t = k 2 /(νε)
is the turbulent Reynolds number (ν being the kinematic viscosity) and A µ = 0.01; and k and ε are determined by ad-hoc modeled transport equations (see [START_REF] Goldberg | A wall-distance-free k -ε model with enhanced near-wall treatment[END_REF]).

Variational Multiscale LES formulation and SGS models

For the LES mode, we wish to recover the Variational Multi-Scale approach [START_REF] Hughes | Large eddy simulation and the variational multiscale method[END_REF], in which the ow variables are decomposed as follows:

W = W LRS + W SRS +W SGS (8)
where W are the large resolved scales (LRS) and W are the small resolved scales (SRS).

This decomposition is obtained by variational projection in the LRS and SRS spaces respectively. In the present study, we follow the VMS approach proposed in [START_REF] Koobus | A variational multiscale method for the large eddy simulation of compressible turbulent ows on unstructured meshes-application to vortex shedding[END_REF] for the simulation of compressible turbulent ows through a nite volume/nite element discretization on unstructured tetrahedral grids. Let χ l and φ l be the N nite-volume and nite-element basis functions associated to the used grid. In order to obtain the VMS ow decomposition, the nite dimensional spaces V F V and V F E , respectively spanned by χ l and φ l , can be in turn decomposed in V F V and V F V and in V F E and V F E [START_REF] Koobus | A variational multiscale method for the large eddy simulation of compressible turbulent ows on unstructured meshes-application to vortex shedding[END_REF], V F V and V F V being the nite volume spaces associated to the largest and smallest resolved scales, spanned by the basis RR n°6564 functions χ l and χ l ; V F E and V F E are the nite element analogous. In [START_REF] Koobus | A variational multiscale method for the large eddy simulation of compressible turbulent ows on unstructured meshes-application to vortex shedding[END_REF], the basis functions of the LRS space are dened through a projector operator in the LRS space, based on spatial average on macro cells:

ψ k = V ol(C k ) j I k V ol(C j ) j I k ψ j (9)
for nite volumes, and:

φ k = V ol(C k ) j I k V ol(C j ) j I k φ j (10)
for the diusive terms, discretized by nite elements. In both Eqs. ( 9) and [START_REF] Pagano | Strategies for RANS/VMS-LES coupling[END_REF],

I k = {j/C j ∈ C m(k) }, C m(k)
being the macro-cell containing the cell C k . The macro-cells are obtained by a process known as agglomeration [START_REF] Lallemand | Unstructured multigridding by volume agglomeration: current status[END_REF]. The basis functions for the smallest resolved scales (SRS) space are clearly obtained as follows: ψ l = ψ l -ψ l and φ l = φ l -φ l .

Finally, a key feature of the VMS approach is that the SGS model is added only to the smallest resolved scales. As in [START_REF] Koobus | A variational multiscale method for the large eddy simulation of compressible turbulent ows on unstructured meshes-application to vortex shedding[END_REF], eddy-viscosity models are used here, and, hence, the SGS terms are discretized analogously to the viscous uxes. Thus, the Galerkin projection of Eq. ( 2) becomes:

∂ W + W c ∂t , χ l + (∇ • F c ( W + W c ), χ l ) + (∇ • F v ( W + W c ), φ l ) = -τ LES (W ), φ l l = 1, N (11)
in which (•, •) denotes the L 2 scalar product. In this context, the Galerkin projection of Eqs. ( 1) and ( 4) for the computation of W and of the additional uctuations in the proposed hybrid model become respectively:

∂ W ∂t , χ l + (∇ • F c ( W ), χ l ) + (∇ • F v ( W ), φ l ) = -τ RAN S ( W ), φ l l = 1, N (12)
∂W c ∂t , χ l + (∇ • F c ( W + W c ), χ l ) -(∇ • F c ( W ), χ l ) + (∇ • F v (W c ), φ l ) = (1 -θ) τ RAN S ( W ), φ l -τ LES (W ), φ l l = 1, N (13)
As for the closure model for LES, giving the expression of τ LES in Eqs. ( 12) and ( 13), three dierent eddy-viscosity models have been considered. The rst one is the Smagorinsky model [START_REF] Smagorinsky | General circulation experiments with the primitive equations[END_REF] giving the following expression for the eddy viscosity:

ν (S) s = (C s ∆) 2 S ij S ij (14)

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where C s is the model input parameter, S ij is the strain-rate tensor (computed in the VMS approach as a function of W ) and ∆ is a length which should be representative of the size of the resolved turbulent scales. Here, ∆ has been selected as V ol(i) 1/3 (V ol(i) being the volume of the i-th tetrahedron) and C s has been set equal to 0.1.

The model proposed by Vreman [35] is also considered, which gives the following expression of the eddy-viscosity:

ν (V ) s = C V ( B β α ij α ij ) 1 2 (15)
with

α ij = ∂u j /∂x i β ij = ∆ 2 α mi α mj B β = β 11 β 22 -β 2 12 + β 11 β 33 -β 2 13 + β 22 β 33 -β 2 23
The constant C V ≈ 2.5C 2 s where C s denotes the Smagorinsky constant (see [START_REF] Vreman | An eddy-viscosity subgrid-scale model for turbulent shear ow: algebraic theory and application[END_REF]); ∆ is the same as for the Smagorinsky model.

The last considered SGS closure is the Wall-Adapting Local Eddy-Viscosity (WALE) SGS model proposed by Nicoud and Ducros [START_REF] Nicoud | Subgrid-scale stress modelling based on the square of the velocity gradient tensor[END_REF]. The eddy-viscosity term µ w in this model is dened by:

ν (W ) s = (C w ∆) 2 (S ij d S ij d ) 3 2 (S ij S ij ) 5 2 + (S ij d S ij d ) 5 4 (16)
with

S ij d = 1 2 (g ij 2 + g ji 2 ) - 1 3 δ ij g kk 2
being the symmetric part of the tensor g ij 2 = g ik g kj , where g ij = ∂u i /∂x j . The constant C w is set to 0.1.

Denition of the blending function and simplied model

As a possible choice for θ, the following function is used in the present study:

θ = F (ξ) = tanh(ξ 2 ) ( 17
)
where ξ is the blending parameter, which should indicate whether the grid resolution is ne enough to resolve a signicant part of the turbulence uctuations, i.e. to obtain a LES-like simulation. The choice of the blending parameter is clearly a key point for the denition of the present hybrid model. In the present study, dierent options are proposed and investigated, namely: the ratio between the eddy viscosities given by the LES and the RANS closures, ξ V R = µ s /µ t , which is also used as a blending parameter in LNS Ref. [START_REF] Batten | Interfacing statistical turbulence closures with large-eddy simulation[END_REF], ξ LR = ∆/l RAN S , l RAN S being a typical length in the RANS approach, i.e. l RAN S = k 3/2 -1 and, nally, ξ T R = t LES /t RAN S , t LES and t RAN S being characteristic times of the LES and RANS approaches respectively, t LES = (S ij S ij ) -1/2 and t RAN S = k -1 . 12) and ( 13) can be recast together as:

RR n°6564

∂W ∂t , χ l + (∇ • F c (W ), χ l ) + (∇ • F v (W ), φ l ) = -θ τ RAN S ( W ), φ l -(1 -θ) τ LES (W ), φ l l = 1, N (18)
Clearly, if only Eq. ( 18) is solved, W is not available at each time step. Two dierent options are possible: either to use an approximation of W obtained by averaging and smoothing of W , in the spirit of VMS, or to simply use in Eq. ( 18) τ RAN S (W ). Relaying on the well known and previously discussed tendency of RANS model of naturally damping the uctuations, the second option is adopted here as a rst approximation.

Hybrid simulations of the ow around a circular cylinder

The proposed approach has been applied to the simulation of the ow around a circular cylinder at Re = 140000 (based on the far-eld velocity and the cylinder diameter, D).

With reference to Fig. 1, the domain dimensions are L i /D = 5, L o /D = 15, L y /D = 7 and L z /D = 2, L z being the domain size in the spanwise direction. On the side surfaces free-slip is imposed and the ow is assumed to be periodic in the spanwise direction. Approximate boundary conditions based on the Reichardt wall-law are used on the cylinder surface (see [START_REF] Camarri | Large-eddy simulation of a blu-body ow on unstructured grids[END_REF] and [START_REF] Camarri | On the approximate treatment of wall boundary conditions in large-eddy simulation[END_REF]). Boundary conditions based on Steger-Warming decomposition [START_REF] Steger | Flux vector splitting for the inviscid gas dynamic equations with applications to the nite dierence methods[END_REF] are used at

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the inow and at the outow surfaces. The inow conditions are the same as in the DES simulations of [START_REF] Travin | Detached-eddy simulations past a circular cylinder[END_REF]. In particular, the ow is assumed to be highly turbulent by setting the inow value of eddy-viscosity to about 5 times the molecular viscosity as in the DES simulation of [START_REF] Travin | Detached-eddy simulations past a circular cylinder[END_REF]. This setting corresponds to a free-stream turbulence level T u = u 2 /U 0 (where u is the inlet uctuation velocity and U 0 is the free-stream mean velocity) of the order of 4%. As discussed also in [START_REF] Travin | Detached-eddy simulations past a circular cylinder[END_REF], the eect of such a high level of free-stream turbulence is to make the boundary layer almost entirely turbulent also at the relatively moderate considered Reynolds number and this allows the proposed hybridization strategy to be tested in a case in which it should actually work in RANS mode in the whole boundary layer. directions, with the rened grid GR2 having a number of nodes in each direction twice the one in GR1. On the other hand, the radial distribution of nodes inside the structured zone is logarithmic and the resolution is the same for both grids, with a distance of the nearest point to the cylinder boundary of 0.004D which corresponds to y + ≈ 28. Note that, to limit the total number of nodes, for GR2 the radial thickness of the structured crown is half of that for GR1.

The main parameters characterizing the simulations carried out with the FCM are summarized in Tab. 2. The numerical method is the one briey described in Sec. Reynolds k -ε model on the GR1 grid are also reported. We did not carry out a LES simulation, due to the diculty in imposing in LES the highly turbulent inow conditions. However, based on previous studies in the literature (see e.g. [START_REF] Breuer | A challenging test case for large eddy simulation: high reynolds number circular cylinder ow[END_REF]), it can be inferred that the used grids are plausibly too coarse to obtain satisfactory results in LES at this Reynolds number.

As a general consideration, the agreement with the DES results is fairly good for all the simulations carried out with the FCM approach. As for the comparison with the experiments, as also stated in Travin et al. [START_REF] Travin | Detached-eddy simulations past a circular cylinder[END_REF], since our simulations are characterized by a high level of turbulence intensity at the inow, it makes sense to compare the results signicant discrepancies with respect to the DES and the experimental data and, namely, a noticeably higher absolute value of the C p in the wake, which results in an overestimated mean drag coecient. Moreover, the mean recirculation bubble is signicantly shorter than in the reference experiments and in the hybrid simulations, and this is connected with an overestimation of the amplitude of lift oscillations (see the values of C l in Tab. 3). An example of the time behavior of the lift coecient for a hybrid simulation (FCM2) is given in Fig. 4a and can be compared with the one obtained in the unsteady RANS simulation shown in Fig. 4b. Besides the larger amplitude of the C l oscillations, it is evident that in the RANS simulation the modulations of the amplitude of these oscillations are signicantly less important than in the hybrid simulations. Since such modulations are due to 3D eects in the wake, this is related to the known tendency of the RANS approach to damp 3D phenomena due to the too large introduced turbulent viscosity. This is conrmed also by the instantaneous eld values of the spanwise velocity and of the vorticity components ω x and ω y , which are much lower in the RANS simulation than in the hybrid ones (not shown here for sake of brevity).

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RR n°6564 As for the behavior of the hybridization strategy in the eld, for all the denitions of the blending parameter, the model actually works in RANS mode in the boundary layer and in the shear-layers detaching from the cylinder, while in the wake a full LES correction is recovered. This is shown, for instance, in Fig. 5, in which the instantaneous isocontours of spanwise vorticity obtained in the simulation FCM2 are reported, to which the isolines of the blending function θ = 0.1 and θ = 0.9 are superimposed. This a-posteriori conrms that the introduced blending function is actually able to give, at least qualitatively, the desired behavior in hybrid simulations of blu-body ows, without any need of imposing it a priori.

Note that there is a thin layer around the cylinder wake in which the model works in a hybrid way, i.e. damped uctuations are added. From Fig. 6a it appears that a smooth passage from a RANS modeling in the boundary layer and in the detaching shear-layers to a LES one in the wake is obtained, consistently with one of the aims of the proposed hybridization strategy. This corresponds to a progressive addition of uctuations, as shown in Fig. 7a, in which the isocontours of the mean resolved uctuating kinetic energy obtained in the FCM2 simulation are reported.

Figs. 8 show instantaneous elds of the SGS viscosity (µ s ) and RANS viscosity (µ t )

for the FCM2 simulation. These plots conrm the behaviour of the hybrid model in the wake. Indeed, µ s is lower than µ t and, since the model works in LES mode, the turbulence uctuations and the ow unsteadiness are less damped than with the RANS model.

Let us analyze now the sensitivity of the results obtained in the FCM simulations to dierent parameters and, rst, the sensitivity to the denition of the blending parameter, by comparing the results of the simulations FCM1 and FCM2. From Tab. 3, it appears that the results are practically insensitive to the denition of the blending parameter. This is also conrmed, for instance, by the distribution over the cylinder of the mean pressure coecient, C p , reported in Fig. 3a. As for the behavior of hybridization strategy, it is also similar for the two denitions of the blending parameter. Analogous results were also found RR n°6564

RR n°6564

with the denition based on the ratio of characteristic times and they are not shown here for sake of brevity. A very low sensitivity of the results to the denition of the blending parameter was also found in the FCM simulations of the ow around a square cylinder described in [START_REF] Pagano | Strategies for RANS/VMS-LES coupling[END_REF] and [START_REF] Salvetti | A hybrid model combining RANS and variational multiscale LES[END_REF].

As for the eect of grid renement, from Tab. 3 it can be seen that it leads to a decrease of Cd (compare FCM2 and FCM3). This is due to a decrease of the absolute value of the pressure coecient in the separated wake (see Tab. 3 and Fig. 3b). However, note that, as it is quite a usual situation for unstructured grids, the renement carried out to obtain grid GR2 has signicantly changed the local quality of the grid (in terms of homogeneity and regularity of the elements) and this may enhance the sensitivity of the results. As for the behavior of the hybridization strategy with grid renement, it is, at least qualitatively, correct. Indeed, the extension of the zone in the detaching shear-layers in which the model works in RANS mode decreases with grid renement, as shown for instance in Figs. 6, reporting a zoom near the cylinder of the instantaneous isocontours of the blending function θ, obtained in the simulations FCM2 and FCM3. Furthermore, Fig. 7b shows that, as expected with grid renement and with the reduced extent of the RANS zone, the regions where the amount of the mean resolved uctuating kinetic energy starts to become signicantly larger are dierent. With the ner grid, this event happens closer to the cylinder than in the simulation on the coarse grid (FCM2 in Fig. 7a).

Finally, the sensitivity to the SGS model is also very low (compare FCM2, FCM4 and FCM5 in Tab. 3 and Fig. 3c). This very low sensitivity has been observed also in VMS-LES simulations of the same ow carried out at low Reynolds number (see [START_REF] Ouvrard | Computation of complex unsteady ows around blu bodies through vms-les modeling[END_REF]) and, thus, it seems more peculiar to the VMS-LES approach rather than to the hybrid model.

Concluding remarks

A strategy for blending RANS and LES has been proposed, which is based on a decomposition of the ow variables in a RANS part and a correction part, which takes into account the resolved uctuations. The zones in which the correction must be computed and added to the RANS part are not a priori specied, but a blending function is introduced, in order to automatically switch from RANS to LES and vice versa. This function is such that the model works in RANS mode where, based on some criterion, the grid is considered too coarse for LES and tends with continuity to LES as the grid renement becomes adequate. For complex geometries and complex grid topologies, as e.g. for unstructured grids, this automatic switch may be advantageous with respect to zonal approaches or to methods which requires the denition of the distance from the wall, as DES. Another positive feature is that the proposed strategy does not require any specic RANS or LES closure. In particular, in the present paper, the VMS approach has been integrated in the proposed hybridization strategy for the closure of the LES part, with dierent eddy-viscosity subgrid scale models.

As a rst choice, we use here a simplied version of the model, in which only one set of unknowns is computed.

INRIA

The proposed method has been applied to the hybrid simulations of the ow around a circular cylinder at Re=140000. As in the DES simulations in [START_REF] Travin | Detached-eddy simulations past a circular cylinder[END_REF], high turbulence level was assumed at the inow, in order to have a turbulent boundary layer before separation.

The sensitivity to dierent parameters, viz. the grid resolution, the SGS model and the denition of the blending parameter, has also been investigated. Unstructured meshes allow for renement which signicantly change the local quality of the grid in terms of homogeneity and regularity of the elements. In that case, results are fairly sensitive to grid resolution and attention need be paid to this eect. The sensitivity to the SGS model in the LES closure is very low. This was also observed (see [START_REF] Ouvrard | Computation of complex unsteady ows around blu bodies through vms-les modeling[END_REF]) for non-hybrid VMS-LES simulations and may be a peculiar feature of the VMS-LES approach rather than of the proposed hybrid approach. The denition of the blending parameter is the most empirical step of the proposed hybridization strategy. Then it is an interesting and positive output of this study that the results are practically insensitive to the denition of the blending parameter, as far as it is chosen among the three options proposed. The behavior of the new method is sound. The blending function, fullls its role in detecting subregions of the eld.

The model works in RANS mode in the boundary layer and in the shear layers detaching from the cylinder while complete LES uctuations are added in the wake, passing through a layer in which the model works in hybrid way, i.e. damped uctuations are introduced.

The quality of prediction is promising. It is in in good agreement with experimental data and compares well to DES simulations in the literature.

Figure 1 :

 1 Figure 1: Computational domain

Figure 2 :

 2 Figure 2: Zoom of the computational grids near the cylinder in the plane z = 0; (a) GR1, (b) GR2.

  C d is the mean drag coecient, C l is the r.m.s. of the lift coecient, St the vortex shedding frequency made nondimensional by the cylinder diameter and the freestream velocity, l r is the length of the mean recirculation bubble and C pb the value of the mean pressure coecient in the separated wake. with experiments at higher Reynolds number, in which, although the level of turbulence intensity of the incoming ow is very low, the transition to turbulence of the boundary layer occurs upstream separation. The agreement with these high Re experiments is indeed fairly good, as shown in Tab. 3 and in Figs. 3, showing the distribution of the mean pressure coecient over the cylinder surface. Conversely, the results of the RANS simulation show

Figure 3 :Figure 4 :

 34 Figure 3: Distribution over the cylinder surface of the mean pressure coecient obtained in the dierent simulations compared to experimental data and to DES results. Dierent sets of simulations are shown in dierent panels.

Figure 5 :

 5 Figure 5: Instantaneous isocontours of spanwise vorticity (simulation FCM2). The isocontours go from the adimensionalized value of -1 (black) to 1 (white). The black and white lines are the isolines of the blending function θ = 0.1 and θ = 0.9.

Figure 6 :Figure 7 :Figure 8 :

 678 Figure 6: Instantaneous isocontours of the blending function θ. The isocontours go from 0 (white), corresponding to a full LES correction, to 1 (black), corresponding to pure RANS. Simulations FCM2 (a) and FCM3 (b).

Table 1 :

 1 Main characteristics of the computational grids. N θ is the number of nodes over the cylinder in the azimuthal direction, N z is the number of nodes in the spanwise direction, N r is the number of nodes in the radial direction inside the crown of thickness t around the cylinder.

	Two dierent grids are considered; the rst one (GR1) has 4.6×10 5 nodes, while the
	second one has (GR2) 1.4×10 6 nodes. The main characteristics of the grids are summarized
	in Tab. 1. Both grids are composed of a structured part in a circular crown around the
	Grid	Elements	Nodes	N θ N z N r t/D
	GR1	2.6 10 6	4.6 10 5 180 40 25	0.1
	GR2	8.1 10 6	1.4 10 6 360 80 13 0.05

cylinder boundary (see Figs. 2) and a unstructured part in the rest of the domain. Inside the structured zone, the nodes are uniformely distributed in the azimuthal and spanwise

Table 2 :

 2 Main characteristics of the hybrid RANS/VMS-LES simulations

	3. The RANS

Table 3 :

 3 Main bulk ow quantities for the circular cylinder test case.

	Simulations FCM1	Re 1.4 10 5	C d 0.62	C l 0.083	St 0.30	l r 1.20	-C pb 0.72
	FCM2	1.4 10 5	0.62	0.083	0.30	1.19	0.65
	FCM3	1.4 10 5	0.54	0.065	0.33	1.13	0.63
	FCM4	1.4 10 5	0.65	0.077	0.28	1.14	0.72
	FCM5	1.4 10 5	0.66	0.094	0.28	1.24	0.72
	RANS	1.4 10 5	0.756	0.433	0.275	0.67	0.85
	DES [33]	1.4 10	5	0.57-0.65	0.06-0.1	0.28-0.31	1.1 -1.4	0.65-0.7
	DES [19]	1.4 10	5	0.6-0.81		0.29-0.3	0.6-0.81	0.85-0.91
	Exp. [14]	3.8 10 6	0.58		0.25		0.58
	Exp. [1]	5 10 6	0.7
	Exp. [23]	8.4 10 6	0.7		0.27		0.8
	Exp. [25]	8 10 6	0.52	0.06	0.28

http://www.inria.fr ISSN 0249-6399


Helene Frankowska

Nobusumi Sagara

Value Function of an Infinite Dimensional Infinite Horizon Problem

We investigate the value function of an infinite horizon problem in the setting of an infinite-dimensional differential inclusion. In particular, we provide an upper estimate of its Gateaux subdifferential in terms of the Clarke subdifferential of the integrand and the Clarke normal cone to the graph of the set-valued dynamics. We also derive a necessary optimality condition in the form of an Euler-Lagrange inclusion, the maximum principle and a sensitivity relation.

I. INTRODUCTION

Optimal control and dynamic programming are instrumental cornerstones of modern economic growth theory originated in [START_REF] Ramsey | A mathematical theory of saving[END_REF]. In the general reduced model of capital accumulation, necessary (and sufficient) conditions for optimality are employed under the convexity assumptions on utility functions and technologies to investigate the existence of competitive equilibria and support prices [START_REF] Benveniste | Duality theory for dynamic optimization models of economics: The continuous time case[END_REF], [START_REF] Magill | Pricing infinite horizon programs[END_REF], [START_REF] Takekuma | Support price theorem for the continuous time model of capital accumulation[END_REF], transversality conditions at infinity [START_REF] Araujo | Maximum principle and transversality condition for concave infinite horizon economic models[END_REF], [START_REF] Benveniste | Duality theory for dynamic optimization models of economics: The continuous time case[END_REF], and the uniqueness and global stability of stationary states [START_REF] Brock | Global asymptotic stability of optimal control systems with applications to the theory of economic growth[END_REF], [START_REF] Cass | The structure and stability of competitive dynamical systems[END_REF], [START_REF] Rockafellar | Saddle points of Hamiltonian systems in convex problems of Lagrange[END_REF], [START_REF] Rockafellar | Saddle points of Hamiltonian systems in convex Lagrange problems having a nonzero discount rate[END_REF]. One of the advantages of convex economic models lies in the crucial observation that the differentiability of the value function is guaranteed under the smoothness assumptions on data, see [START_REF] Benveniste | On the differentiability of the value function in dynamic models of economics[END_REF], [START_REF] Benveniste | Duality theory for dynamic optimization models of economics: The continuous time case[END_REF], [START_REF] Bonnisseau | On the subdifferential of the value function in economic optimization problems[END_REF], [START_REF] Rincón-Zapatero | Differentiability of the value function in continuous-time economic models[END_REF].

On the contrary, the absence of convexity and smoothness are two major sources of complex economic dynamics in continuous time as illustrated in [START_REF] Askenazy | A model of optimal growth strategy[END_REF], [START_REF] Beyn | Dynamical optimization and Skiba sets in economic examples[END_REF], [START_REF] Davidson | Non-convexities in continuous-time investment theory[END_REF], [START_REF] Hartl | History dependence without unstable steady state: A non-differentiable framework[END_REF], [START_REF] Skiba | Optimal growth with a convex-concave production function[END_REF], [START_REF] Wagener | Skiba points and heteroclinic bifurcations, with applications to the shallow lake system[END_REF]. The lack of convexity results in the failure of the differentiability of the value function even if the underlying data are smooth. One can expect at best the local Lipschitz continuity of the value function. This causes problems with expressing optimal synthesis in many nonconvex economic growth models when one attempts to apply the Hamilton-Jacobi-Bellman (HJB) theory.

The failure of differentiability of the value function has stimulated two alternative approaches. One is the application of a "generalized" subdifferential calculus along the lines of [START_REF] Clarke | Optimization and Nonsmooth Analysis[END_REF], leading to the formulation of a relation between the maximum principle and the value function [START_REF] Clarke | Local optimality conditions and Lipschitzian solutions to the Hamilton-Jacobi equation[END_REF], [START_REF] Clarke | The relationship between the maximum principle and dynamic programming[END_REF]. The other one is the concept of "viscosity solutions" to the associated HJB equation (initiated in [START_REF] Crandall | Some properties of viscosity solutions of Hamilton-Jacobi equations[END_REF], [START_REF] Crandall | Viscosity solutions of Hamilton-Jacobi equations[END_REF]), whose unique (viscosity) solution is the value function. For the connections between the maximum principle and the superdifferentials of the value function, see also [START_REF] Cannarsa | Value functions, relaxation, and transversality conditions in infinite horizon optimal control[END_REF], [START_REF] Frankowska | Optimal trajectories associated with a solution of the contingent Hamilton-Jacobi equation[END_REF], [START_REF] Frankowska | On relations of the adjoint state to the value function for optimal control problems with state constraints[END_REF].

*Research of the first author benefited from the support of the FJMH Program PGMO and from the support to this program from EDF-THALES-ORANGE-CRITEO. Research of the second author benefited from the support of the JSPS KAKENHI grant 26380246 from the Ministry of Education, Culture, Sports, Science and Technology, Japan. 1 Frankowska H. is with CNRS, IMJ-PRG, Sorbonne Université, Paris, France helene.frankowska@imj-prg.fr 2 Sagara N. is with Department of Economics, Hosei University, Tokyo, Japan nsagara@hosei.ac.jp

The purpose of this paper is twofold. Firstly, we provide an upper estimate of the Gateaux subdifferential of the value function in terms of the Clarke subdifferential and the Clarke normal cone to the set-valued mapping describing dynamics.

Secondly, we derive a necessary condition for optimality in the form of an adjoint inclusion, the maximum principle and a relation of the adjoint variable to the subdifferentials of the value function along the optimal trajectory. To deal with the adjoint variable in dual spaces, we use the Gelfand integrals of the Gateaux and Clarke subdifferential mappings, which is a new feature that does not arise in the context of finitedimensional control systems. We also consider the relaxed variational problem, which is a suitable convexification of the original variational problem, and derive the necessary condition that narrows the class of candidates for optimality. Our dynamical system in the form of a differential inclusion being very general, these results are derived under quite restrictive "interiority assumption" (H 8 ). In the particular case of dynamics described by a control system we may remove assumption (H 8 ).

For the finite-dimensional infinite horizon problem, necessary conditions without convexity assumptions using Gateaux, Clarke, and limiting subdifferentials were derived in [START_REF] Aubin | Shadow prices and duality for a class of optimal control problems[END_REF], [START_REF] Cannarsa | Value functions, relaxation, and transversality conditions in infinite horizon optimal control[END_REF], [START_REF] Sagara | Value functions and transversality conditions for infinitehorizon optimal control problems[END_REF], [START_REF] Ye | Nonsmooth maximum principle for infinite-horizon problems[END_REF], while for the control systems in Hilbert spaces, a necessary condition under the convexity assumption was obtained in [START_REF] Barbu | Convex control problems and Hamiltonian systems on infinite intervals[END_REF]. For the semilinear control systems in Banach spaces, the necessary and sufficient conditions were derived in [START_REF] Cannarsa | Value function and optimality conditions for semilinear control problems[END_REF] in the finite horizon setting when the sets of admissible velocities are convex.

The outline of the paper is as follows. Section 2 collects some definitions. In Section 3 we formulate the nonconvex variational problem and investigate the Lipschitz continuity and subdifferentiability of the value function. We derive in Sections 4, 5, 6 some necessary optimality conditions for the original and relaxed variational problems. Because of the lack of space some of proofs are omitted. They will be published elsewhere.

II. PRELIMINARIES

Let (E, • ) be a real separable Banach space with the dual system E * , E , where E * is the norm dual of E. For a nonempty subset C ⊂ E denote by d C : E → IR + the distance function from the set C, by K C (x) and T C (x) respectively the contingent and Clarke tangent cones to C at x ∈ C and let N C (x) stand for the Clarke normal cone to C at x ∈ C, see for instance [START_REF] Aubin | Set-Valued Analysis[END_REF]Chapter 4] or [START_REF] Clarke | Optimization and Nonsmooth Analysis[END_REF] for the precise definitions. The lower (resp. upper) directional derivative of ϕ :

E → IR ∪ {+∞} at x ∈ dom ϕ in the direction v ∈ E is defined by ϕ -(x; v) := lim inf u→v θ↓0 ϕ(x + θu) -ϕ(x) θ ∈ IR ∪ {±∞}, ϕ + (x; v) := lim sup u→v θ↓0 ϕ(x + θu) -ϕ(x) θ ∈ IR ∪ {±∞}.
respectively. The Dini-Hadamard subdifferential of ϕ at x is

∂ -ϕ(x) := {x * ∈ E * | x * , v ≤ ϕ -(x; v) ∀ v ∈ E}
and the Dini-Hadamard superdifferential of ϕ at x is defined by

∂ + ϕ(x) := {x * ∈ E * | x * , v ≥ ϕ + (x; v) ∀ v ∈ E}.
If ϕ is locally Lipschitz at x, denote by ϕ • (x; v) the Clarke directional derivative of ϕ at x in the direction v, see [START_REF] Clarke | Optimization and Nonsmooth Analysis[END_REF] for the definition. We denote by IR + the set of all nonnegative reals. Below, by measurable, we shall always mean Lebesgue measurable. Let W

L : IR + × E × E → IR ∪ {+∞} be an integrand. Given an arc x(•) ∈ W 1,1 loc ([t, ∞), E), define ∞ t L(s, x(s), ẋ(s))ds = lim T →∞ T t L(s, x(s), ẋ(s))ds
for every t ∈ IR + provided the above limit does exist. Let Γ : IR + ×E E be a multifunction. Define the set of admissible trajectories starting at time t ∈ IR + from a given initial condition ξ ∈ E by

A (t,ξ) = x(•) ∈ W 1,1 loc ([t, ∞), E) | ẋ(s) ∈ Γ(s, x(s)) a.e., x(t) = ξ .
Then the value function

V : IR + × E → IR ∪ {±∞} is V (t, ξ) = inf x(•)∈A (t,ξ) ∞ t L(s, x(s), ẋ(s))ds. (P t )
In the above we set

V (t, ξ) = +∞ if A (t,ξ) is empty or if for every x(•) ∈ A (t,ξ) the integral ∞ t L(s, x(s), ẋ(s))ds is not well-defined. The effective domain of V is dom V = {(t, x) ∈ IR + × E | V (t, x) < +∞}. For every (t, ξ) ∈ dom V , an admissible trajectory x(•) ∈ A (t,ξ) is said to be optimal for (P t ) if ∞ t L(s, x(s), ẋ(s))ds = V (t, ξ) > -∞.
The standing hypothesis are as follows.

(H 1 )

A (t,ξ) is nonempty for every (t, ξ) ∈ IR + × E. (H 2 ) L(•, x, y) is measurable for every (x, y) ∈ E × E.
(H 3 ) There exist integrable functions l 0 , l 1 : IR + → IR + , and a locally bounded, integrable function

l 2 : IR + → IR + such that |L(t, 0, 0)| ≤ l 0 (t), |L(t, x, y)-L(t, x , y )| ≤ l 1 (t) x-x +l 2 (t)
+ → IR + such that Γ(t, 0) ⊂ γ 0 (t)B, Γ(t, x) ⊂ Γ(t, x ) + γ 1 (t) x -x B for a.e. t ∈ IR + and all x, x ∈ E. (H 7 ) ∞ 0 exp s 0 γ 1 (τ )dτ 1 + s 0 γ 0 (τ )dτ × (l 1 (s) + l 2 (s)γ 1 (s))ds < ∞.
The above assumptions and the results from [START_REF] Frankowska | A priori estimates for operational differential inclusions[END_REF] imply Theorem 1: andL • y (t, x, ȳ; v), and the Clarke partial subdifferentials ∂ •

If (H 1 )-(H 7 ) hold, then V is bounded, lower semicontinuous on IR + ×E, and V (t, • ) is Lipschitz of rank k(t) on E for every t ∈ IR + with a continuous decreasing function k : IR + → IR + satisfying k(t) → 0 as t → ∞.

A. Subdifferentiability of the Value Function

Let us denote by L

- x (t, x, ȳ; v) (resp. L + x (t, x, ȳ; v)) the lower (resp. upper) partial directional derivative of L(t, •, ȳ) at x ∈ E in the direction v ∈ E; L - y (t, x, ȳ; v) and L + y (t, x, ȳ; v) have obvious meanings. Then ∂ - x L(t, x, ȳ) (resp. ∂ + x L(t, x, ȳ)) denotes the Dini-Hadamard partial subdifferential (resp. superdifferential) of L(t, •, ȳ) at x; ∂ - y L(t, x, ȳ) and ∂ + y L(t, x, ȳ) have similar meanings. The Clarke partial directional derivatives L • x (t, x, ȳ; v)
x L(t, x, ȳ) and ∂ • y L(t, x, ȳ) are defined in a similar way. We need another continuity assumption on Γ:

(H 5 ) Γ(•, x) is lower semicontinuous for every x ∈ E.
Using the Filippov type theorem from [START_REF] Frankowska | A priori estimates for operational differential inclusions[END_REF] we prove Lemma 1: Suppose (H 4 ), (H 5 ), (H 6 ). Let x 0 (•) ∈ A (0,ξ0) and t ∈ IR + be such that ẋ0 (t) ∈ Γ(t, x 0 (t)). Fix any v ∈ Γ(t, x 0 (t)). Then for every h > 0 there exists

x h (•) ∈ W 1,1 ([t, t + h]; E) such that: (i) ẋh (s) ∈ Γ(s, x h (s)) a.e., x h (t + h) = x 0 (t + h); (ii) x h (t) -x 0 (t) -h( ẋ0 (t) -v) = o(h); (iii) ẋh (•) -v L 1 ([t,t+h]) = o(h).
The results below substantially improve those from [START_REF] Benveniste | On the differentiability of the value function in dynamic models of economics[END_REF], [START_REF] Benveniste | Duality theory for dynamic optimization models of economics: The continuous time case[END_REF], [START_REF] Bonnisseau | On the subdifferential of the value function in economic optimization problems[END_REF], [START_REF] Rincón-Zapatero | Differentiability of the value function in continuous-time economic models[END_REF].

Let x 0 (•) ∈ A (0,ξ0) be optimal and [t] := (t, x 0 (t), ẋ0 (t)). Theorem 2: Assume (H 1 )-(H 4 ), (H 5 ), (H 6 ) and (H 7 ). Then for a.e. t ≥ 0 and any v ∈ Γ(t, x 0 (t)),

(i) V - x (t, x 0 (t); ẋ0 (t) -v) ≤ L(t, x 0 (t), v) -L[t] ; (ii) -∂ - x V (t, x 0 (t)) ⊂ ∂ • y L[t] + N Γ(t,x0(t)) ( ẋ0 (t)). Moreover, if for a.e. t ≥ 0, L(t, x 0 (t), •) is Gateaux differ- entiable at ẋ0 (t), then for a.e. t ≥ 0, -∂ - x V (t, x 0 (t)) ⊂ {∇ y L[t]} + N Γ(t,x0(t)) ( ẋ0 (t)). Proof: (i): Let t > 0 be such that the two inclusions in (H6) hold, v ∈ Γ(t, x 0 (t)) and x h (•) ∈ W 1,1 ([t, t + h]; E) be as in the claim of Lemma 1. Then, for every s ∈ [t, t + h] we have x h (s) -x 0 (t) ≤ x h (s) -x h (t) + x h (t) -x 0 (t) ≤ h v + h ẋ0 (t) -v + o(h) ≤ h(γ 0 (t) + γ 1 (t) x h (t) + ẋ0 (t) -v ) + o(h) ≤ h(γ 0 (t) + γ 1 (t)( x 0 (t) + h ẋ0 (t) -v ) +h ẋ0 (t) -v + o(h) = h(γ 0 (t) + γ 1 (t) x 0 (t) + ẋ0 (t) -v ) + o(h)
which yields the following estimates:

t+h t L(s, x h (s), ẋh (s))ds - t+h t L(s, x 0 (t), v)ds ≤ t+h t (l 1 (s) x h (s) -x 0 (t) + l 2 (s) ẋh (s) -v ) ds ≤ (h(γ 0 (t) + γ 1 (t) x 0 (t) + ẋ0 (t) -v ) + o(h))× t+h t l 1 (s)ds + sup s∈[t,t+h] l 2 (s) ẋh (•) -v L 1 ([t,t+h]) .
So the above is equal to o(h). By the separability of E and [24, Theorem 2.5], there exists a subset I of IR + such that the Lebesgue measure of its complement

IR + \ I is zero with lim h↓0 h -1 t+h t L(s, x, v)ds = L(t, x, v) for every (t, x, v) ∈ I × E × E.
We thus obtain: for every t ∈ I,

lim h↓0 1 h t+h t L(s, x h (s), ẋh (s))ds = L(t, x 0 (t), v). Let t ∈ IR + be a Lebesgue point of L(•, x 0 (•), ẋ0 (•)). By the dynamic programming, V (t, x h (t)) ≤ t+h t L(s, x h (s), ẋh (s))ds+V (t+h, x 0 (t+h)). Subtracting V (t, x 0 (t)) = t+h t L[s]ds + V (t + h, x 0 (t + h)) and using that V (t, •) is locally Lipschitz, we get t+h t L(s, x h (s), ẋh (s))ds - t+h t L[s]ds ≥ V (t, x h (t)) -V (t, x 0 (t)) ≥ V (t, x 0 (t) + h( ẋ0 (t) -v)) -V (t, x 0 (t)) -o(h).
Dividing by h > 0 and taking the limit as h → 0 yields

V - x (t, x 0 (t); ẋ0 (t) -v) ≤ L(t, x 0 (t), v) -L[t]. Since v ∈ Γ(t, x 0 (t)) is arbitrary, we proved (i). (ii): Let x * ∈ ∂ - x V (t, x 0 (t)), u ∈ K Γ(t,x0(t)) ( ẋ0 (t)). Then for some θ n → 0+ and u n ∈ E with u n → u we have ẋ0 (t) + θ n u n ∈ Γ(t, x 0 (t)) for each n ∈ N. Since V - x (t, x 0 (t); -θ n u n ) ≤ L(t, x 0 (t), ẋ0 (t) + θ n u n ) -L[t], x * , -u n ≤ L(t, x 0 (t), ẋ0 (t) + θ n u n ) -L[t] θ n .
Letting n → ∞ yields

-x * , u ≤ L + y (t, x 0 (t), ẋ0 (t); u) ≤ L • y (t, x 0 (t), ẋ0 (t); u) (1) for every u ∈ K Γ(t,x0(t)) ( ẋ0 (t)). Suppose for a moment that -x * ∈ ∂ • y L[t] + N Γ(t,x0(t)) ( ẋ0 (t)). Since ∂ • y L[t] is weakly * compact, and N Γ(t,x0(t)) ( ẋ0 (t)) is weakly * closed, ∂ • y L[t] + N Γ(t,x0(t)) ( ẋ0 (t)
) is weakly * closed. Then, by the separation theorem, there exists v ∈ E such that

-x * , v > sup y * ∈∂ • y L[t] y * , v + sup z * ∈N Γ(t,x 0 (t)) ( ẋ0(t)) z * , v . Thus z * , v ≤ 0 for every z * ∈ N Γ(t,x0(t)) ( ẋ0 (t)). This means that v ∈ T Γ(t,x0(t)) ( ẋ0 (t)) ⊂ K Γ(t,x0(t)) ( ẋ0 (t)).
The above inequality also implies that -x * , v > L • y (t, x 0 (t), ẋ0 (t); v), in contradiction with (1). This proves (ii). If t is so that L(t, x 0 (t), •) is Gateaux differentiable and locally Lipschitz at ẋ0 (t), then (1) can be replaced by -x * , u ≤ ∇ y L(t, x 0 (t), ẋ0 (t); u) for every u ∈ K Γ(t,x0(t)) ( ẋ0 (t)). The above arguments are also valid when we replace ∂

lim h→0 p(t + h) -p(t) h , x = x * , x .
for every x ∈ E. Then x * is called the weak * derivative of p at t and is denoted by ṗ(t) with d dt p(t), x = ṗ(t), x . The weak * differentiability of locally absolutely continuous functions is fundamental in the sequel and can be deduced from the proof of [START_REF] Kōmura | Nonlinear semi-groups in Hilbert space[END_REF]Lemma].

(H 8 ) For every t ∈ IR + there exists η > 0 such that

(x 0 (s) + ηB, ẋ0 (s)) ⊂ gph Γ(s, •) a.e. s ∈ [0, t],
where gph Γ(s, •) stands for the graph of Γ(s, •).

Define the Hamiltonian H : IR

+ × E × E * → IR ∪ {+∞} H(t, x, x * ) := sup y∈Γ(t,x)
{ x * , y -L(t, x, y)} .

(2)

Theorem 3: Assume (H 1 )-(H 4 ), (H 5 ), (H 6 )-(H 8 ). If ∂ - x V (0, x 0 (0)) = ∅ and ∂ + x L[t]
is nonempty a.e., then for every x * ∈ ∂ - x V (0, x 0 (0)) there exists a locally absolutely continuous p :

IR + → E * with p(0) = -x * such that (i) -p(t) ∈ ∂ - x V (t, x 0 (t)) for every t ∈ IR + ; (ii) p(t) ∈ ∂ • y L[t] + N Γ(t,x0(t)) ( ẋ0 (t)) a.e.; (iii) ṗ(t) ∈ ∂ + x L[t] a.e.; (iv) H(t, x 0 (t), p(t)) = p(t), ẋ0 (t) -L(t, x 0 (t)), ẋ0 (t)) a.e.; (v) lim t→∞ p(t) = 0,
where ṗ(t) denotes the weak * derivative of p(•). In particular, if ∂ - x V (0, x 0 (0)) is nonempty, then ∂ - x V (t, x 0 (t)) is nonempty for every t ∈ IR + .

Proof: Let t ∈ IR + and η > 0 be as in (H 8 ). Fix any x * ∈ ∂ - x V (0, x 0 (0)) and let f : [0, t] → E * be a Gelfand integrable selector of the mapping s → ∂ +

x L[s] over the interval [0, t]. Define p(t) = t 0 f (s)ds -x * as a Gelfand integral. We claim that -p(t) ∈ ∂ - x V (t, x 0 (t)). To this end, fix any v ∈ E and consider the local perturbation of x 0 (•) over [0, t] given by x θ (s) := x 0 (s) + θv for s ∈ [0, t]. By construction, ẋθ (s) = ẋ0 (s) a.e. s ∈ [0, t] and x θ (s) ∈ x 0 (s) + ηB whenever 0 < θ ≤ (1 + v ) -1 η, and hence, (x θ (s), ẋθ (s)) ∈ gph Γ(s, •) a.e. s ∈ [0, t]. Thus

V (0, x θ (0)) ≤ t 0 L(s, x θ (s), ẋθ (s))ds + V (t, x θ (t)). Subtracting V (0, x 0 (0)) = t 0 L[s]ds + V (t, x 0 (t)) yields V (0, x θ (0)) -V (0, x 0 (0)) ≤ V (t, x θ (t)) -V (t, x 0 (t))+ t 0 (L(s, x θ (s), ẋθ (s)) -L[s]) ds. Let θ n → 0+ be such that V - x (t, x 0 (t); v) = lim n→∞ V (t, x 0 (t) + θ n v) -V (t, x 0 (t)) θ n .
Using the Lebesgue dominated convergence theorem and Fatou's lemma and taking the limit as n → ∞ yields

V - x (0, x 0 (0); v) ≤ t 0 L + x (s, x 0 (s), ẋ0 (s); v)ds+ V - x (t, x 0 (t); v) ≤ t 0 f (s), v ds + V - x (t, x 0 (t); v) = p(t), v + x * , v + V - x (t, x 0 (t); v) ∀ v ∈ E. Since x * , v ≤ V - x (0, x 0 (0); v), we obtain -p(t), v ≤ V - x (t, x 0 (t); v) for every v ∈ E and (i) follows. Since p(t), y = t 0 f (s), y ds -x * , y for all t ∈ IR + , y ∈ E and | f (s), y | ≤ f (s) y ≤ l 1 (s) y||, we get | p(t + h) -p(t), y | ≤ y t+h t
l 1 (s)ds, and therefore, p(t + h) -p(t) ≤ t+h t l 1 (s)ds for every h > 0. Hence p is locally absolutely continuous and its weak * derivative ṗ(t) = f (t) exists a.e. This proves (iii). By Theorem 2 and (i), p(t), ẋ0 (t) -L[t] ≥ p(t), y -L(t, x 0 (t), y) for every y ∈ Γ(t, x 0 (t)) implying the maximum principle (iv). Thus, for a.e. t ∈ IR + and every v ∈ T Γ(t,x0(t)) ( ẋ0 (t)), we have p(t), v ≤ L - y (t, x 0 (t), ẋ0 (t); v) ≤ L • y (t, x 0 (t), ẋ0 (t); v) and condition (ii) follows from the separation argument as in the proof of Theorem 2. To get (v) recall that by Theorem 1, V (t, •) is Lipschitz of rank k(t) with k(t) → 0 as t → ∞. Therefore, p(t) ≤ k(t) → 0.

V. RELAXED VARIATIONAL PROBLEMS

The significance of Theorems 2 and 3 is severely limited whenever, ẋ0 (t) happens to be an isolated point in Γ(t, x 0 (t)) on a set of positive measure, because on this set N Γ(t,x0(t)) ( ẋ0 (t)) is the whole space. To overcome this limitation, the relaxation technique provides some additional information on the subdifferentiability of the value function.

Define the multifunction Γ :

IR + × E IR × E by Γ(t, x) = {(r, y) ∈ IR × E | r = L(t, x, y), y ∈ Γ(t, x)} .
Then Γ(t, x) is the augmented velocity set. The relaxed variational problem, which is a convexified problem corresponding to (P t ), is as follows.

Ṽ (t, ξ) = inf ψ(•)∈L 1 (IR+) x(•)∈W 1,1 loc ([t,∞);E) ∞ t ψ(s)ds (ψ, ẋ)(s) ∈ co Γ(s, x(s)) a.e., x(t) = ξ.
(RP t )

Ṽ : IR×E → IR∪{±∞} is the value function of the relaxed problem. Define the set of admissible relaxed trajectories starting at time t ≥ 0 from the initial condition ξ ∈ E by

Ã(t,ξ) := • 0 ψ(τ )dτ, x(•) ∈ W 1,1 loc ([t, ∞); IR × E) | (ψ(s), ẋ(s)) ∈ co Γ(s, x(s)) a.e. s ∈ [t, ∞), x(t) = ξ .
For (t, ξ) ∈ dom Ṽ , a trajectory (

• 0 ψ(τ )dτ, x(•)) ∈ Ã(t,ξ) is said to be optimal for (RP t ) if it satisfies ∞ t ψ(s)ds = Ṽ (t, ξ) > -∞. Then Ṽ (t, ξ) ≤ V (t, ξ), ∀ (t, ξ) ∈ dom Ṽ .
Denote by B the open unit ball in IR × E, where the norm in IR × E is given by (r, x) := |r| + x . It is easy to see that if (H 2 )-(H 6 ) hold, then Γ has nonempty closed values, is measurable with respect to the first variable and satisfies assumptions similar to (H 6 ).

The following relaxation result for differential inclusions is a special case of [START_REF] Frankowska | A priori estimates for operational differential inclusions[END_REF]Theorem 2.5].

Lemma 2: Let 0 ≤ t 0 < t 1 . If (H 4 ), (H 5 ) and (H 6 ) hold, then for any y(•) ∈ W 1,1 ([t 0 , t 1 ]; E) with ẏ(t) ∈ co Γ(t, y(t)) a.e. and every ε > 0 there exists x(•) ∈ W 1,1 ([t 0 , t 1 ]; E) with ẋ(t) ∈ Γ(t, x(t)) a.e. such that x(t 0 ) = y(t 0 ) and sup t∈[t0,t1] x(t) -y(t) < ε.

It implies the following theorem. Theorem 4: If (H 1 )-(H 7 ) hold, then V coincides with Ṽ . In particular, If x(•) ∈ A (t,ξ) is optimal for (P t ), then

( • t L(τ, x(τ ), ẋ(τ ))dτ, x(•)) ∈ Ã(t,ξ) is optimal for (RP t ). Define the "relaxed" Lagrangian L : IR + × E × E → IR ∪ {+∞} of L by: for any y ∈ co Γ(t, x) L(t, x, y) = inf r ∈ IR | (r, y) ∈ co Γ(t, x)
and set L(t, x, y) = +∞ if y / ∈ co Γ(t, x). Then L ≤ L on gph co Γ, and L(t, x, •) is lower semicontinuous and convex on E for every (t, x) ∈ IR + .

(H 2 ) L(•, x, y) is continuous for every (x, y) ∈ E × E. Theorem 5: If (H 1 ), (H 2 ), (H 3 ), (H 4 ), (H 5 ), (H 6 ) and (H 7 ) hold, then:

(i) V - x (t, x 0 (t); ẋ0 (t) -v) ≤ L(t, x 0 (t), v) -L[t] for a.e. t ∈ IR + and every v ∈ co Γ(t, x 0 (t)); (ii) -∂ - x V (t, x 0 (t)) ⊂ ∂ - y L[t] a.e. t ∈ IR + ; (iii) ∂ - y L[t] ⊂ ∂ • y L[t] + N co Γ(t,x0(t)) ( ẋ0 (t)) a.e. t ∈ IR + . Proof: (i): (H 3 ) implies that for a.e. t ≥ 0, L(t, •, •) has finite values on gph co Γ(t, •). Since l 2 (•) is locally bounded, (H 2 ) guarantees that L(•, x, •) is continuous on IR + × E for every x ∈ E. Then Γ(•, x
) is lower semicontinuous, and hence, co Γ(•, x) is also lower semicontinuous. Let t ∈ IR + be a Lebesgue point of s → s 0 L(τ, x 0 (τ ), ẋ0 (τ ))dτ such that ẋ0 (t) ∈ Γ(t, x 0 (t)). Take any (r, v) ∈ co Γ(t, x 0 (t)). Applying Lemma 1 to co Γ, for every h > 0 there exists

(a h (•), x h (•)) ∈ W 1,1 ([t, t + h]; IR × E) such that: (a) ( ȧh , ẋh )(s) ∈ co Γ(s, x h (s)) a.e., x h (t + h) = x 0 (t + h); (b) x h (t) -x 0 (t) -h( ẋ0 (t) -v) = o(h); (c) ȧh (•) -r L 1 ([t,t+h]) + ẋh (•) -v L 1 ([t,t+h]) = o(h). By Theorem 4, V (t, x h (t)) = Ṽ (t, x h (t)) ≤ t+h t ȧh (s)ds + V (t + h, x 0 (t + h)). Subtracting V (t, x 0 (t)) = t+h t L[s]ds + V (t + h, x 0 (t + h))
from the both sides of the above inequality yields

t+h t ȧh (s)ds - t+h t L[s]ds ≥ V (t, x h (t)) -V (t, x 0 (t)) ≥ V (t, x 0 (t) + h( ẋ0 (t) -v)) -V (t, x 0 (t)) -o(h). Thus V - x (t, x 0 (t); ẋ0 (t) -v) ≤ r -L[t].
We proved that for every v ∈ co Γ(t, x 0 (t))

V - x (t, x 0 (t); ẋ0 (t) -v) ≤ L(t, x 0 (t), v) -L[t]. (ii): Take any x * ∈ ∂ - x V (t, x 0 (t)).
In the same way as in the proof of Theorem 2(ii), we obtain x V (0, x 0 (0)) there exists a locally absolutely continuous function p : IR + → E * with p(0) = -x * such that all conclusions of Theorem 3 hold with Γ replaced by co Γ and moreover p(t) ∈ ∂ - y L[t] a.e. In particular, if ∂ - x V (0, x 0 (0)) is nonempty, then ∂ - x V (t, x 0 (t)) is nonempty for every t ∈ IR + .

-x * , v ≤ L- y (t, x 0 (t), ẋ0 (t); v) (3)
∀ v ∈ K co Γ(t,x0(t)) ( ẋ0 (t)). If v / ∈ K co Γ(t,

VI. NECESSARY CONDITIONS WITHOUT THE

INTERIORITY ASSUMPTION Hypothesis (H 8 ) in Theorem 3 is stringent, mostly because the velocity multifunction Γ is too general. Hence, a "structural assumption" on x 0 (•) compensates this generality. If instead some "structural" assumptions are imposed on Γ, then hypothesis (H 8 ) can be omitted. To illustrate this observation, we consider a standard optimal control problem.

Let X be a complete separable metric space, f : IR + × E × X → E be a velocity function, and U : IR + X be a control multifunction. Denote by M(IR + , X) the space of measurable functions on IR + with values in X. Define the integrand L : IR + × E × X → IR by L(t, x, u) := L(t, x, f (t, x, u)). The optimal control problem under consideration is as follows:

inf x(•)∈W 1,1 loc (IR+,E) u(•)∈M(IR+,X) ∞ 0 L(t, x(t), u(t))dt s.t. u(t) ∈ U (t) a.e. t ∈ IR + , ẋ(t) = f (t, x(t), u(t)) a.e. t ∈ IR + , x(0) = ξ. ( P0
)
The Hamiltonian for problem ( P0 ) is given by:

H(t, x, x * ) = sup u∈U (t)
x * , f (t, x, u) -L(t, x, u) .

The velocity multifunction is defined by Γ(t, x) := f (t, x, U (t)). Now impose "usual" assumptions on f and U in order that Γ satisfies (H Let (x 0 (•), u 0 (•)) ∈ W 1,1 loc (IR + , E) × M(IR + , X) be an optimal trajectory-control pair for optimal control problem ( P0 ). Denote by ∇ x f (s, x 0 (s), u 0 (s)) * ∈ L(E * ) the adjoint operator of ∇ x f (s, x 0 (s), u 0 (s)) ∈ L(E).

Theorem 6: Suppose that (H 1 )-(H 4 ), (H 5 ), (H 6 ), (H 7 ), and where ṗ(t) denotes the strong derivative of p(•) at t ∈ IR + . In particular, if ∂ - x V (0, x 0 (0)) is nonempty, then ∂ - x V (t, x 0 (t)) is nonempty for every t ∈ IR + . The proof of the above theorem is quite technical and will be published elsewhere.

(H 9 ) hold with Γ(t, x) = f (t, x, U (t)). If ∂ - x V (0, x 0 ( 0
Remark 1: The existence of locally Bochner integrable selectors from the superdifferential mapping t ∂ +

x L(t, x 0 (t), u 0 (t)) follows from (H 3 ) and (H 9 ) whenever E * is separable in the dual norm. For the case with nonseparable E * , the Fréchet differentiability of the integrand L(t, •, •) on E × E and the continuity of (t, x, y) → (∇ x L(t, x, y), ∇ y L(t, x, y)) in the dual norm of E * × E * guarantee the local Bochner integrability of t → ∇ x L(t, x 0 (t), u 0 (t)) in E * under (H 3 ) and (H 9 ). If ∂ +

x,y L(t, x 0 (t), ẋ0 (t)) is nonempty, then take any (p, q) ∈ ∂ + x,y L(t, x 0 (t), ẋ0 (t)) and observe that for every v ∈ E and u ∈ U (t), we have:

L+ x (t, x, u; v) ≤ L +
x,y (t, x, f (t, x, u); v, ∇ x f (t, x, u)v) ≤ p, v + q, ∇ x f (t, x, u)v = p + ∇ x f (t, x, u) * q, v .

VII. CONCLUSIONS

It is well known that in optimal control of finite horizon problems the adjoint state can be related to generalized gradients of the value function. In the present paper we considered an infinite horizon problem with dynamics described by a differential inclusion and derived relations of subdifferential of the value function to generalized gradients of the integrand and normals to the set of admissible velocities along the optimal trajectories. Under an additional assumption we have also shown that for any optimal trajectory there exist solutions of the Euler-Lagrange inclusions satisfying in addition the Weierstrass type conditions and a sensitivity relation.

x(t) = t 0 f

 0 (s, x(s), u(s))ds + ξ for every t ∈ IR + for any control u(•) ∈ M(IR + , X). Moreover the locally absolutely continuous function x(•) : IR + → E is a unique mild solution to the ordinary differential equation (ODE) in ( P0 )), which has the strong derivative ẋ(t) a.e. t ∈ IR + (in view of the separability of E and the Lebesgue theorem).

  H 5 ) Γ(•, x) is measurable for every x ∈ E. (H 6 ) There exist locally integrable γ 0 : IR + → IR + and γ 1 : IR

y-y for a.e. t ∈ IR + and all (x, y), (x , y ) ∈ E × E. (H 4 ) Γ has nonempty closed values.

(

  • y L[t] and L • y (t, x 0 (t), ẋ0 (t); v) respectively by ∇ y L[t] and ∇ y L[t], v . IV. MAXIMUM PRINCIPLE A function p : IR + → E * is locally absolutely continuous if its restriction to any bounded closed interval is absolutely continuous. A function p(•) is said to be weakly * differentiable at t > 0 if there exists x * ∈ E * such that

  is empty, then the inclusion is trivial.Assume next that ∂ - y L[t] is nonempty. Observe that L[t] = L[t] a.e. t ∈ IR + , for otherwise we must have V (0, ξ 0 ) = Ṽ (0, ξ 0 ), which contradicts to Theorem 4. Since L ≤ L, Take any x * ∈ ∂ - y L[t] and assume for a moment that x * ∈ ∂ • y L[t] + N co Γ(t,x0(t)) ( ẋ0 (t)).In the same way as before we derive a contradiction.Corollary 1:Assume (H 1 ), (H 2 ), (H 3 ), (H 4 ), (H 5 ), (H 6 )-(H 8 ). If ∂ - x V (0, x 0 (0)) is nonempty and ∂ + x L[t]is nonempty for a.e. t ∈ IR + , then for every x * ∈ ∂ -

	x0(t)) ( ẋ0 (t)), then
	L-y (t, x 0 (t), ẋ0 (t); v) = +∞. Thus inequality (3) is true for every v ∈ E. Therefore, -x * ∈ ∂ -y L[t]. (iii): If ∂ -y L[t] L-y (t, x 0 (t), ẋ0 (t); v) ≤ L • y (t, x 0 (t), ẋ0 (t); v) (4)
	for every v ∈ E.

  [START_REF] Araujo | Maximum principle and transversality condition for concave infinite horizon economic models[END_REF] )-(H 4 ), (H 5 ), (H 6 ), and (H 7 ). Note that ∂ +x L(t, x, u) is nonempty at (t, x, u) ∈ IR + ×E×X whenever so is ∂ +x,y L(t, x, f (t, x, u)) and f (t, •, u) is locally Lipschitz and Gateaux differentiable at x. Denote by L(E) the space of bounded linear operators on E.The following reasonable hypothesis is a "structural assumption" on f that dispenses with (H 8 ).(H 9 )(i) f is a Carathéodory function, i.e., f (•, x, u) is measurable for every (x, u) ∈ E×X and f (t, •, •) is continuous for every t ∈ IR + . (ii) For every R > 0 and T > 0 there exists an integrable function k : [0, T ] → IR such that: Conditions (H 9 )-(i), (ii) guarantee the existence of solutions of the integral equation

a) f (t, x, u) ≤ k(t) for every t ∈ [0, T ], x ∈ RB, and u ∈ U (t); b) f (t, •, u) is Lipschitz of rank k(t) on RB for every t ∈ [0, T ] and u ∈ U (t). (iii) f (t, •, u) is Fréchet differentiable on E for every (t, u) ∈ IR + × X and the mapping (t, x, u) → ∇ x f (t, x, u

) is continuous in the uniform operator topology of L(E).

  )) is nonempty and the mapping ∂ +x L(•, x 0 (•), u 0 (•)) : IR + E * admits a locally Bochner integrable selector, then there exists a locally absolutely continuous function p : IR + → E * such that:-p(t) ∈ ∂ - x V (t, x 0 (t)) for every t ∈ IR + ; p(t) ∈ ∂ • y L(t, x 0 (t), f (t, x 0 (t), u 0 (t)))+ N Γ(t,x0(t)) (f (t, x 0 (t), u 0 (t))) a.e. t ∈ IR

+ ; -ṗ(t) ∈ ∇ x f (t, x 0 (t), u 0 (t)) * p(t)-∂ + x L(t,

x 0 (t), u 0 (t)) a.e. t ∈ IR + ; H(t, x 0 (t), p(t)) = p(t), f (t, x 0 (t), u 0 (t)) -L(t, x 0 (t), u 0 (t)) a.e. t ∈ IR + ; lim t→∞ p(t) = 0,