Fabiola A Camargo
email: fabiola.camargo@institutoptique.fr

Paul Dumont

Iryna Ghozyk

Gaëlle Lucas-Leclin

Patrick Georges

Jean-Marie Danet

David Holleville

Stéphane Guerandel

Isabelle Sagnes

Ghaya Baili

Fabiola Almeida Camargo

Loic Morvan

Gregoire Pillet

Daniel Dolfi

Two-cross-polarized-frequency VECSEL at 852nm

Keywords: cavity surface emitting lasers; (270.2500) Fluctuations, relaxations, and noise

Introduction

Among the different kind of compact atomic clocks, those using the technique of coherent population trapping (CPT) to interrogate the atoms could achieve significantly better performance than the conventional doubleresonance clocks [START_REF] Vanier | Atomic clocks based on coherent population trapping: a review[END_REF]. In an atomic clock based on CPT, the microwave interrogation between two atomic levels is caused by two coherent, in-phase, laser fields resonant with a common excited state, the frequency difference of the two laser fields corresponding exactly to the frequency microwave clock. The atoms are trapped in a superposition of states decoupled from the laser fields, called a "dark" state, and stop interacting with the laser light. With 133 Cs atoms, the pair of laser fields has to be tuned either at the D1 (894 nm) or D2 (852 nm) optical transition line, and the hyperfine frequency splitting in the microwave range is equal to 9.192631770 Hz. It corresponds to a narrow transmission line in the Cs absorption spectrum, which is used as a frequency reference. The common solutions for the realization of the coherent pair of laser beams are sideband generation by microwave modulation of a single-frequency diode laser [START_REF] Merimaa | All-optical atomic clock based on coherent population trapping in 85 Rb[END_REF] and optical phase-locking of two independent lasers [START_REF] Zanon | High Contrast Ramsey Fringes with Coherent -Population-Trapping Pulses in a Double Lambda Atomic System[END_REF]. The latter configuration is ideal as regards the atomic signal and it is commonly used nowadays in atomic clock experiments; however it requires a complex and costly wideband phase-locked electronics loop. Dualfrequency operation of an optically-pumped vertical external cavity semiconductor laser (VECSEL) has been proposed as a simple and compact solution for CPT clocks [START_REF] Baili | Experimental demonstration of a tunable dualfrequency semiconductor laser free of relaxation oscillations[END_REF] and it presents many advantages. First, this configuration takes benefit of the intrinsically strong correlation between the two laser lines which share the same cavity; second, the frequency difference, which is proportional to the intracavity phase anisotropy, may be tunable from few tens of MHz to a few THz. Finally, a high-cavity-finesse VECSEL exhibits a class-A dynamical behavior, which results in a flat, shot-noise limited, intensity noise spectrum on a wide frequency range, and so for the phase noise; this should provide a high-purity RF signal.

Description of the dual-frequency laser

The semiconductor active chip has been grown by metal-organic chemical-vapor deposition on a 350 μm thick GaAs substrate and is designed for emission at λ = 852 nm under barrier-pumping at λ ≤ 700 nm. The 30λ/4thick active region is composed of seven 8-nm thick GaAs quantum wells, embedded between Al 0.22 Ga 0.78 As barriers. The active structure is resonant at the design wavelength, which increases its effective optical gain [START_REF] Garnache | Diode-pumped broadband vertical-external-cavity surface-emitting semiconductor laser applied to high-sensitivity intracavity absorption spectroscopy[END_REF]. The bottom Distributed Bragg Reflector (DBR) consists of 32.5 pairs of AlAs/Al 0.225 Ga 0.775 As λ/4-thick layers, and its reflectivity is evaluated to R > 99.95% at 852 nm. As no anti-reflection coating is deposited on the top surface of the structure, ~ 30% of the incident pump beam is reflected on the surface.

The laser cavity is composed by the semiconductor chip and a 15 mm concave output mirror with a transmission of 0.5% at 852 nm. The cavity length is 10 mm, resulting in a laser beam diameter of 70 µm in the structure. The pump source is a 1.6 W-broad-area laser diode emitting at 670 nm coupled into a 100 μm diameter, NA = 0.22, multimode fiber. This source delivers almost 1 W at the fiber end and is focused on the semiconductor chip with two doublets of f 1 = 25 mm and f 2 = 19 mm under a 50° angle, yielding a 70 μm  110μm-elliptical spot on the structure.

The dual frequency emission is obtained by introducing a birefringent element in the laser cavity, which controls the phase anisotropy inside the cavity and results in the emission of two cross-polarized lines. We use a 500 μm-thick antireflection coated YVO 4 plate cut at 45° to its optic axis, which induces a spatial separation of (Figure 1 left). This spatial separation of the two polarized beams reduces the nonlinear coupling between them and is thus necessary to ensure a stable dual frequency emission [START_REF] Pal | Measurement of the coupling constant in a two-frequency VECSEL[END_REF]; nevertheless it is small enough to maintain a good overlap between the pump beam and the two laser spots. The birefringence introduced in the laser cavity results in slightly different optical paths for each polarization, which produce two combs of longitudinal modes distanced by . This frequency difference between the two polarizations is adjusted by a 1 mm-long MgO:SLT electro-optic modulator (Figure 1 right), whose birefringence changes with both the temperature and the applied voltage [START_REF] Camargo | Tunable high-purity microwave signal generation from a dual-frequency VECSEL at 852 nm[END_REF]. A course tunability of the frequency difference ∆ν is possible by changing the temperature of the crystal thanks to the high rate of 1.25 GHz/K. And the fine tunability of ∆ν is obtained with the voltage applied to the crystal, with a rate of 1.59 MHz/V; it allows to achieve the exact microwave transition of Cs atoms, and to compensate for the residual ∆ν fluctuations on a wide frequency range. The laser cavity design has focused on compactness, mechanical and thermal stabilities. The pump optics, the semiconductor chip and the laser cavity elements are integrated in a compact 90 mm × 90 mm × 40 mm casing [START_REF] Camargo | Tunable high-purity microwave signal generation from a dual-frequency VECSEL at 852 nm[END_REF]. This limits mechanical and acoustic vibrations as well as air temperature fluctuations inside the external cavity. The temperature of the whole setup is stabilized to 24° C. The temperature of the semiconductor chip is controlled at 16° C using a Peltier element. The output coupler is glued on a piezo-electric transducer (PZT) for the fine adjustment of the cavity length. In these conditions the laser delivers around 10 mW per polarization at 852 nm.

Laser stabilization and noise

A small portion (less than 0.5 mW) of the ordinary-polarized laser line is used to lock the laser frequency at the top of a Doppler-free transition (F = 4 -F' = 4'/5') of the Cs D2 line at 852 nm using a pump-probe saturated absorption technique, as shown in Figure 2. An acousto-optic crystal is used to modulate the so-called pump beam at 100 kHz to generate the error signal. This method avoids the direct modulation of the pump diode current, which would simultaneously introduce a modulation of the laser intensity. The error signal is used to compensate for the low-frequency fluctuations of the laser cavity length through a 2-integration stage servo loop on the piezoelectric transducer glued to the output coupler. The power spectral density of the frequency noise is measured from the error signal with a Fast Fourier Transform analyzer (Figure 3 left). The servo loop gain reaches 60 dB at low frequencies and its bandwidth is around 600 Hz, limited by the piezoelectric bandwidth. The frequency difference between the two laser lines has been locked to a local oscillator (LO) which produces a stable RF reference (Figure 2). The polarization of the extraordinary line is rotated with a  plate and both lines are superimposed on a fast photodiode (bandwidth > 12.5 GHz). The subsequent beatnote signal is amplified and mixed with the RF reference to generate the error signal, which is amplified. The error signal is applied to the electro-optic crystal to compensate for the frequency-difference fluctuations.

Figure 3 compares the beat-note spectra of the free-running laser with the one of the laser when the frequency difference between the two laser lines is locked to the RF reference at 9.2 GHz. The servo loop bandwidth is about 1 MHz, which depends on the proportional gain as well as on the electro-optic crystal response. The signal-to-noise ratio is 60 dB and the RF beat note linewidth is less than 30 Hz, limited by the analyzer resolution. The two servo-loopsof the absolute frequency difference on a Cs atomic transition and of the frequency difference on a microwave local oscillatoroperate together without perturbing each other. Eventually the beatnote phase noise has been measured with both servo-loops operating. In the frequency range from 100 Hz to 10 kHz, the phase noise is lower than -100 dBrad²/Hz. For frequencies higher than 10 kHz, the phase noise increases to -90 dBrad²/Hz. Using the formalism of the Dick effect [START_REF] Dick | Local oscillator induced instabilities in trapped ion frequency standards[END_REF], the frequency stability of a pulsed CPT atomic clock based on this laser source is estimated as low as 3.10 -13  -1/2 . We have described the design and operation of a new laser source based on the dual-frequency operation of an OP-VECSEL. The performance of the laser source has been assessed with regards to the requirements for the interrogation of Cs atoms in a compact atomic clock experiment. The phase noise level of the 2-frequency 2polarization laser source demonstrated here is already suitable for the realization of a Cs atomic clock with a stability of 3.10 -13  -1/2 .

Figure 1 :

 1 Figure 1: Experimental setup for the OP-VECSEL (right) and the laser and pump beam overlap on the chip surface (left).

Figure 2 :

 2 Figure 2: Experimental set-up of two the servo-loops use to lock the laser frequency at the Cs transition and the frequency difference at a RF reference (left). Doppler-free Cs absorption (right).

Figure 3 :

 3 Figure 3: Spectral density of the frequency noise of the free running laser and of the laser frequency locked at the Cs transition (left). Beat note of the free-running laser and with the frequency difference locked at the RF reference at 9.2GHz.

Acknowledgments :

This work was supported by the French Agence Nationale de la Recherche (ANR-07-BLAN-0320-03) and the french RENATECH network. F. Camargo was funded by RTRA/Triangle de la Physique (2010-089T).


R E Silva

F Harel

S Jemei

R Gouriveau

D Hissel

L Boulon

K Agbossou

PROTON EXCHANGE MEMBRANE FUEL CELL OPERATION AND DEGRADATION IN SHORT-CIRCUIT

Keywords: proton exchange membrane fuel cells, short circuit, degradation mechanism, electrochemical impedance spectroscopy

This paper presents an experimental study dealing with operation and degradation during an electrical short circuit of a proton exchange membrane fuel cell stack. The physical quantities in the fuel cell (electrical voltage and current, gas stoichiometry, pressures, temperatures and gas humidity) are studied before, during and after the failure. After a short circuit occurs, a high peak of current appears but decreases to stabilize in a much lower value. The voltage drops in all the cells and even some cells presents reversal potentials. The degradation is quantified by using electrochemical impedance spectroscopy.

INTRODUCTION

The performance of a Proton Exchange Membrane Fuel Cell (PEMFC) is affected by many direct and indirect factors such as fuel cell design and assembly, material properties, operating conditions, and impurities or contaminants. In order to improve cell lifetime, it is important to identify and understand the potential failures and then analyze its mechanisms to allow mitigating their damage. The research presented in this paper is realized in the framework of a research project aiming to determine the influence on different failures in PEMFC operation. Among the operation modes, electrical short circuit represents a common and particularly undesirable failure in PEMFC [START_REF] Wang | PEM Fuel Cell Failure Mode Analysis, PEM Fuel Cell Durability Handbook[END_REF], [START_REF] Fox | Investigation of Shorting by Penetration in PEM Fuel Cell Membranes[END_REF].

On the one hand, the risks of a long short circuit are a non-reversible degradation e.g. melting of the electrodes or at least an accelerated aging; on the other hand, some researches have demonstrated that under particular conditions, high current peaks for very short periods not only rise the temperature but also generate an excess of liquid water decreasing the membrane resistance and reducing the ohmic losses [START_REF] Fox | Investigation of Shorting by Penetration in PEM Fuel Cell Membranes[END_REF], [START_REF] Mench | Polymer electrolyte fuel cell degradation[END_REF]. Moreover, the performance of the fuel cell increases with the increase of the operating temperature [START_REF] Tohidi | Effect of Primary Parameters on the Performance of PEM Fuel Cell[END_REF].

Short circuits are generally originated by the penetration of conductive materials through the thickness of the membrane. The shorting objects may include the carbon fibers and binders of gas diffusion layers, aggregates in the catalyst layers [START_REF] Mench | Polymer electrolyte fuel cell degradation[END_REF]. Hard shorts can occur suddenly in an operating fuel cell stack where one cell develops a significantly higher ohmic resistance compared to the cells in the rest of the stack. As the stack continues to draw current from all cells, the cell with the abnormally high resistance can develop an excessive voltage drop and even reverse its cell potential [START_REF] Mench | Polymer electrolyte fuel cell degradation[END_REF].

The aim of this paper is to present an experimental study of the operation and degradation in PEMFC stack under electrical short circuit. A controlled short circuit is imposed between the electrodes of the fuel cell stack. Then the electrical and fluidic parameters are studied during and after this operation mode. An analysis of the degradation is realized by performing Electrochemical Impedance Spectrometry (EIS). This paper is organized as follows: Section 2 describes the fuel cell stack as well as explains the experimental setup. Section 3 presents experimental results. Section 4 presents the conclusions of this experimental research.

EXPERIMENTAL SETUP

Fuel cell stack and test bench

The fuel cell stack is fed with air and hydrogen. It is composed of 40 cells connected in series, with an active surface of 220cm 2 each. The nominal current of the PEMFC is 110A, the nominal (electrical) power is 3.8kW and the nominal operating temperature is 80°C. Hydrogen and air stoichiometry rates are fixed to 1.5 and 2 respectively. The test bench illustrated in Figure 1 is used to evaluate the fuel cell in both dynamic and static states [START_REF] Candusso | Characterisation and modelling of a 5 kW PEMFC for transportation applications[END_REF].

Short-circuit procedure

The PEMFC electrodes are directly connected by using a controllable circuit breaker. The current is measured by using a Hall Effect sensor. Fuses are used as redundancy protection.

Measures

Electrical, thermal and fluidic parameters are measured before, during and after the controlled short-circuit. It allows understanding the physical phenomena occurring in the stack. The EIS is used to determine the dynamic behavior of the fuel cell [START_REF] Springer | Characterization of polymer electrolyte fuel cells using AC impedance spectroscopy[END_REF], thus it is used to define reference conditions to quantify the degradation. EIS are performed before and after the short circuit by using the spectrometer designed by S. Wasterlain [START_REF] Wasterlain | Development of test instruments and protocols for the diagnostic of fuel cell stacks[END_REF].

EXPERIMENTAL RESULTS

A short circuit is imposed between the two current outputs of the fuel cell stack. However, the objective is not to perform a destructive test, and then the duration of the short circuit is limited to avoid strong thermal stress which can degrade permanently the fuel cell. This section presents the experimental results obtained for a short circuit with duration of 10s.

Electrical behavior

Figures 3a, and 3b illustrate the current and voltage during the short circuit. Figure 3c presents the voltage in each of the cells of the stack.

Figure 3a shows that the short circuit current (from 0.5 to 10.5 s) reaches a value of 1100A (5A/cm²) this represents near 10 times its nominal value. However, this current then decreases to stabilize at a value of near 2 times the nominal value. This is a relatively low value compared with electrical machines; here the current is limited by the mass transfer losses and by the hydrogen flow. This confirms the results presented by Hinaje et al. which use this phenomenon to use the fuel cell as a current source controlled on the hydrogen flow [START_REF] Hinaje | An innovating application of PEM fuel cell: Current source controlled by hydrogen supply[END_REF]. Figure 3b shows that the stack voltage drops near to 0V, even some cells presents reverse potential as illustrated in figure 3c and explained by [START_REF] Mench | Polymer electrolyte fuel cell degradation[END_REF] .This occurs mainly in the single cells that are far from the air input. The figure also shows that in some cells the voltage drops only about 50% of its nominal value, this is mostly the case in the single cells located near the air input.

Thermal and fluidic behavior

Figure 5 presents the experimental results for the gas stoichiometry, humidity and pressure, air temperature, H 2 temperature and water (cooling) temperature. It is important to highlight that the time scale is not the same than in Figure 3, this is because the fluidic time constants are much higher than the electrical ones; in the figure the short circuit is occurring from 9s to 19s.

During the short circuit the gas are consumed at a very high rate. The stoichiometric factors air/H 2 decrease from 2/1.5 to 1/0.75 respectively. As the supply in gas is proportional to the current supplied to the load, water is produced at a very high rate. The excess of liquid water contributes to the reduction of the membrane resistance while reducing the ohmic losses in the membrane [START_REF] Fox | Investigation of Shorting by Penetration in PEM Fuel Cell Membranes[END_REF][START_REF] Hinaje | An innovating application of PEM fuel cell: Current source controlled by hydrogen supply[END_REF][START_REF] Wasterlain | of Temperature, Air Dew Point Temperature and Reactant Flow Effects on Proton Exchange Membrane Fuel Cell Performances Using Electrochemical Spectroscopy and Voltammetry Techniques[END_REF].

The high current, even during a short time, causes a rise in the stack temperature; the heat must thus be evacuated. This originates a high thermal gradient, the temperature (measured based on the cooling water temperature) rises to 92 °C; 15% higher than the nominal value.

Electrochemical Impedance Spectroscopy

Figure 4 presents the EIS (110A) results before and after the short circuit. Tables 1 and2 summarize the polarization and ohmic resistance of the individual cells (in groups of 4). Experimental results exhibit that the change in the impedance spectra before and after a short circuit is not so pronounced, the value of the internal resistance of the stack increases by about 1.23% (passing from 17.24 to 17.03 mΩ). Similarly, for low frequencies, the polarization resistance of the stack evolves from 93.77 to 94.42 mΩ an increase of 0.69%. The value of the internal resistance of the stack has not really been an indicator of degradation after short-circuit. Nevertheless, the polarization resistance increases during short circuit on the low frequency capacitive arc associated with diffusion phenomena.

CONCLUSIONS

This paper presents an experimental study about operation of a PEM fuel cell stack under electrical short circuit. After a short circuit occurs, a high peak of current appears (near 10 times nominal current) but decreases to stabilize in a much lower value (two times the nominal current), this is due to mass transfer losses limitation and the hydrogen flow as also concluded in [START_REF] Hinaje | An innovating application of PEM fuel cell: Current source controlled by hydrogen supply[END_REF]. The voltage in the cells varies and depends on the relative position regarding the gas input. The voltage drops in all the cells and even some cells present reversal potentials as presented in [START_REF] Mench | Polymer electrolyte fuel cell degradation[END_REF]. The temperature rises but the dynamic is much slower than the electric ones.

It is difficult to conclude the degradation by using EIS because the results before and after the short circuit are very similar. This could be because the short circuits were maintained for a relatively short duration. For further research, it could be interesting to study the effect of long duration short circuits.

Figure 1 .

 1 Figure 1. 5kW Fuel Cell test bench at FCLAB

Figure 2 .

 2 Figure 2. Short-Circuit Schema

Figure 3 .

 3 Figure 3. PEMFC electrical behavior during short circuit (a) current, (b) stack voltage, (c) individual cell voltage

Figure 4 .

 4 Figure 4. Electrochemical impedance spectroscopy before (nominal conditions) and after the short circuit

Figure 5 .

 5 Figure 5. PEMFC fluidic behavior during short circuit (a) Stoichiometry, (b) Gas humidity, (c,d) Air/H 2 pressure, (e,f) Air/H 2 temperature, (g) Water temperature

Table 1 .

 1 Ohmic and polarization resistances. Before Short-Circuit

	Cell n°	[5-8] [9-12] [13-16] [17-20] [21-24] [25-28] [29-32] [33-36] [37-40] Stack
	Rm [mΩ] 1.83	1.76	1.74	1.71	1.70	1.66	1.61	1.57	1.63	17.03
	Rp [mΩ] 9.99	9.63	8.96	9.41	9.21	8.65	8.31	8.41	8.02	93.77

Table 2 .

 2 Ohmic and polarization resistances. After Short-Circuit (10s)

	Cell n°	[5-8] [9-12] [13-16] [17-20] [21-24] [25-28] [29-32] [33-36] [37-40] Stack
	Rm [mΩ] 1.87	1.80	1.72	1.71	1.72	1.68	1.62	1.58	1.63	17.24
	Rp [mΩ] 10.22 9.72	9.02	9.46	9.30	8.69	8.26	8.59	7.87	94.42


Hélène Frankowska

Daniel Hoehener

Daniela Tonon

A Second-Order Maximum Principle in Optimal Control under State Constraints *

Keywords: Optimal control, Second-order necessary optimality conditions, Second-order tangents AMS subject classification: 49K15, 49K21, 34A60, 34K35

A second-order variational inclusion for control systems under state constraints is derived and applied to investigate necessary optimality conditions for the Mayer optimal control problem. A new pointwise condition verified by the adjoint state of the maximum principle is obtained as well as a second-order necessary optimality condition in the integral form. Finally, a new sufficient condition for normality of the maximum principle is proposed. Some extensions to the Mayer optimization problem involving a differential inclusion under state constraints are also provided.

Introduction

This paper is devoted to second-order necessary optimality conditions for control problems in the presence of pure state constraints together with endpoint constraints. Let us consider first an abstract optimization problem min

c∈C φ(c), (1)
where C is a subset of a Banach space X and φ : X → R is a twice Fréchet differentiable function.

Assume that c ∈ C is an optimal solution, i.e.

φ(c) = min c∈C φ(c),
and denote by φ and φ the first and second-order derivatives of φ. If c lies in the interior of C, then the classical results guarantee the first-order necessary optimality condition φ (c) = 0 (Fermat rule) and the second-order condition φ (c) ≥ 0. Let us assume now that c is a boundary point of C and denote by T C (c) the adjacent cone to C at c. Recall that u ∈ T C (c) whenever there exist δ > 0 and a "path"

x : [0, δ] → C satisfying x(0) = c and such that the difference quotients 1 h (x(h) -x(0)) converge to u when h → 0+. The Fermat rule becomes then:

φ (c)u ≥ 0 ∀ u ∈ T C (c).
(

This is a first-order necessary optimality condition for problem [START_REF] Aubin | Set-Valued Analysis, Systems Control Found[END_REF]. Furthermore, if the set C is convex and for some c ∈ C, we have inf

u∈T C (c), u =1 φ (c)u > 0,
then it is not difficult to justify that c is a local minimum for the problem [START_REF] Aubin | Set-Valued Analysis, Systems Control Found[END_REF]. Observe that the second-order derivative of φ has no influence on this conclusion. In particular it may happen that φ (c)(u, u) < 0 for some u ∈ T C (c).

The second derivative starts to play a role in the expression of optimality conditions when u ∈ T C (c) is such that φ (c)u = 0. In this case it is natural to consider not only the second-order derivative of the function φ but also a second-order "linearization" of the constraint. To this end, one uses second-order tangents, whose definition we recall next. With every u ∈ T C (c) we associate the set of second-order tangents to C at (c, u). Namely v ∈ T

C (c, u) if we can find δ > 0 and a "path" x : [0, δ] → C satisfying x(0) = c such that the difference quotients 1 h 2 (x(h) -x(0) -hu) converge to v when h → 0+. The Taylor formula implies then that the following second-order optimality condition holds true:

φ (c)v + 1 2 φ (c)(u, u) ≥ 0 ∀ u ∈ T C (c), v ∈ T (2)
C (c, u) such that φ (c)u = 0.

The optimality conditions (2) and ( 3) are classical in mathematical programming, see for instance [START_REF] Bonnans | Perturbation Analysis of Optimization Problems[END_REF] for an overview of this subject, and were largely explored in the optimal control theory, where they have led to various second-order necessary optimality conditions, mostly in the form of an integral inequality. See for instance [START_REF] Bonnans | Second-order analysis for optimal control problems with pure state constraints and mixed control-state constraints[END_REF][START_REF] Gilbert | Second-order necessary conditions in optimal control: Accessoryproblem results without normality conditions[END_REF][START_REF] Páles | Optimal control problems with set-valued control and state constraints[END_REF][START_REF] Warga | A second-order condition that strengthens Pontryagin's maximum principle[END_REF] and the bibliographies contained therein. The space X is then usually the space of essentially bounded controls and it is assumed that every control determines a unique state trajectory, see for instance [START_REF] Bonnans | Second-order analysis for optimal control problems with pure state constraints and mixed control-state constraints[END_REF][START_REF] Gilbert | Second-order necessary conditions in optimal control: Accessoryproblem results without normality conditions[END_REF]. The map associating to each control the corresponding state trajectory is then, under suitable assumptions on the system dynamics, Fréchet differentiable. The set C incorporates restrictions imposed on trajectories and controls.

To derive necessary optimality conditions, the optimal control problem at hand is usually reformulated as an infinite dimensional abstract optimization problem involving inequality and equality constraints. Then first and second-order optimality conditions that are dual to (2) and (3) are obtained, see for instance [START_REF] Bonnans | Perturbation Analysis of Optimization Problems[END_REF][START_REF] Cominetti | Metric regularity, tangent sets and second order optimality conditions[END_REF][START_REF] Kawasaki | Second order necessary optimality conditions for minimizing a sup-type function[END_REF]. This approach requires Robinson's like constraint qualifications. The obtained necessary conditions have to be translated then in terms of the original optimal control problem. Verification of Robinson's constraint qualifications and translation of the abstract necessary conditions in terms of control problems is not an easy task and it often requires strong assumptions on control systems and on optimal controls. This is the reason why in many published papers it is assumed that optimal controls are piecewise continuous, that the control system depends on time in a continuous way and the equality and inequality constraints satisfy qualification hypotheses like linear independence of gradients of active constraints. Moreover, in some works on second-order necessary optimality conditions in optimal control, see for instance [START_REF] Gilbert | Second-order necessary conditions in optimal control: Accessoryproblem results without normality conditions[END_REF][START_REF] Páles | Optimal control problems with set-valued control and state constraints[END_REF], the proposed first and second-order conditions are strictly linked, that is each u appearing in [START_REF] Bettiol | L ∞ estimates on trajectories confined to a closed subset[END_REF] gives rise to a pair of first/second-order conditions. This is usually inconvenient for the analysis of possible candidates for optimality. Let us also underline that, especially in the context of state constrained problems, it is more natural to expect optimal controls to be merely measurable and very general first-order necessary optimality conditions are already known in this context, see for instance [START_REF] Vinter | Optimal Control[END_REF]. This creates an important gap between generality of the available results on first and second-order necessary optimality conditions. Avoiding the reformulation of optimal control problems under state constraints in an abstract way by using a direct variational approach allows to work under less restrictive hypotheses. In particular, in [START_REF] Vinter | Optimal Control[END_REF], several very general first-order necessary optimality conditions were derived from the Ekeland variational principle stated on metric spaces. The main tools used there are penalization, limiting gradients, generalized normals and their calculus. An alternative approach was proposed in [START_REF] Cernea | A connection between the maximum principle and dynamic programming for constrained optimal control problems[END_REF][START_REF] Tamzali-Lafond | Variational inclusions under state constraints[END_REF] by considering the space X = C([0, 1]; R n ) of continuous mappings defined on the time interval [0, 1] and taking values in R n and the subset C equal to the set of trajectories of the control system satisfying state and endpoint constraints.

It was shown that solutions of a linearized control system under linearized constraints (along a given trajectory) are tangent to the set of trajectories of the original control system under state constraints. This, inequality (2) and the duality theory of convex analysis allowed then to obtain a direct proof of the maximum principle in the presence of state constraints. The aim of our paper is to go beyond these first-order results and to investigate second-order necessary optimality conditions using second-order tangents.

In the context of optimal control theory this approach was applied in [START_REF] Hoehener | Second-order optimality conditions for a Bolza problem with mixed constraints[END_REF][START_REF] Hoehener | Variational approach to second-order optimality conditions for control problems with pure state constraints[END_REF] by considering second-order tangents to the sets of admissible controls. An important difference with the existing literature is the fact that the analysis takes place in the framework of measurable controls and therefore larger sets of second-order tangents are considered, since weaker convergence properties are imposed (L 1 versus L ∞ ).

In particular, an optimal control may be merely measurable. The derived integral type second-order necessary conditions for weak optimality were obtained in primal form for general control and state constraints.

In the present work we address second-order necessary optimality conditions for strong local minima (that is we work with trajectories instead of controls). To investigate the second-order tangents to the set of trajectories of a control system under constraints, we derive a second-order variational differential inclusion. Applying it to the Mayer optimal control problem under state and end point constraints, we obtain a pointwise second-order maximum principle and a second-order necessary optimality condition in the form of an integral inequality which extends some earlier known results to the case of strong local minimizers. The second-order maximum principles derived in Theorems 5.10, 6.9 and Corollaries 5.11, 6.10 seem to be new even in the case without state constraints. Furthermore, for the Mayer optimization problem involving a differential inclusion under state constraints, we also obtain a second-order maximum principle. Since the presence of end point constraints requires some additional assumptions, to simplify the discussion, we first state results not involving the end point constraints, postponing to the last section their analogues in a more general setting. We would like to underline that, in the difference with the traditional approach to the first-order necessary optimality conditions based on normal cones, see for instance [START_REF] Vinter | Optimal Control[END_REF], in the present work we explore first-and second-order tangents and the associated graphical derivatives/variations of set-valued maps. In particular, we prove here several new properties of second order graphical variations, see section 2. The first-order graphical derivatives having a wide range of applications, see for instance [START_REF] Aubin | Set-Valued Analysis, Systems Control Found[END_REF], [START_REF] Dontchev | Implicit Functions and Solution Mappings[END_REF] and the bibliographies contained therein, we believe that further analysis of second-order graphical variations may bring new results also to other areas of variational analysis. The outline of the paper is as follows. In Section 2 we introduce some notations and provide a few preliminary results. In Section 3 a second-order variational differential inclusion is derived. Section 4 is devoted to a second-order approximation of differential inclusions under state constraints and to second-order necessary optimality conditions for a Mayer problem. Section 5 deals with the Mayer optimal control problem under state constraints. Finally Section 6 extends results of Sections 4 and 5 to the case when endpoint constraints are present.

Preliminaries

The spaces of continuous and essentially bounded maps from [0, 1] into R n are denoted respectively by C([0, 1]; R n ) and L ∞ ([0, 1]; R n ) and their norms by • ∞ , while W 1,1 ([0, 1]; R n ) and L 1 ([0, 1]; R n ) stand respectively for the spaces of absolutely continuous and of integrable maps from [0, 1] to R n with the usual norms • W 1,1 and • 1 . The space of mappings from [0, 1] to R n having bounded total variation that are right continuous on ]0, 1[ and vanish at zero is denoted by

N BV ([0, 1]; R n ). The norm f T V of f ∈ N BV ([0, 1]; R n ) is the total variation of f on [0, 1].
Partial derivatives are denoted with subscripts, for instance f u := ∂f ∂u . Similarly, partial second-order derivatives are denoted by a double subscript, i.e. f xu := ∂ 2 f ∂x∂u . Moreover, for y 1 , y 2 ∈ R n we will abbreviate f xx (t 0 , x 0 , u 0 )(y 1 , y 2 ) by f xx (t 0 , x 0 , u 0 )y 1 y 2 . If f : R n → R is differentiable at x, then we denote by ∇f (x) its gradient. We denote the norm in R n by |•| and by •, • the inner product. For a set K ⊂ R n , let K be its closure, ∂K its boundary, int K its interior, K c its complement and co K its convex hull. K -stands for the (negative) polar cone to K, i.e.

K -= {v ∈ R n | v, k ≤ 0, ∀ k ∈ K}. The open unit ball in R n is denoted by B := {x ∈ R n | |x| < 1}. Write B(x, r
) for an open ball with radius r > 0 and center x ∈ R n and S n-1 for the unit sphere. Let X be a Banach space and K ⊂ X. The distance between a point x ∈ X and K is defined by dist

K (x) := inf k∈K x -k X . When K is a nonempty proper subset of R n , the oriented distance b K (•) from K is a real valued function defined by b K (x) := dist K (x) -dist K c (x) for all x ∈ R n . We set b K (•) ≡ 0 if K = R n .
Note that any closed set K ⊂ R n can be represented via an inequality constraint involving the oriented distance function:

K = {x ∈ R n | b K (x) ≤ 0} . Let B 0 := {v = (v 1 , . . . , v n ) ∈ B | v n = 0} & B -:= {v = (v 1 , . . . , v n ) ∈ B | v n < 0} .
A proper closed subset K of R n is said to be of class C 2 at x 0 ∈ ∂K if there exist an open neighborhood N of x 0 and a bijective map ϑ : N → B such that ϑ(•) ∈ C 2 (N ; B), its inverse ϑ -1 (•) ∈ C 2 (B; N ) and

ϑ -1 (B -) = (int K) ∩ N , ϑ -1 (B 0 ) = ∂K ∩ N =: Γ, ϑ(Γ) = B 0 .
We say that K is of class C 2 if it is so for every x 0 ∈ ∂K.

Remark 2.1. If K is of class C 2 at x 0 ∈ ∂K, then there exists a neighborhood W of x 0 such that the oriented distance b K (•) ∈ C 2 (W; R), see for instance [START_REF] Delfour | Shapes and Geometries: Analysis, Differential Calculus and Optimization[END_REF]Theorem 4.3].

We recall next some definitions concerning tangent sets to a nonempty subset K of a Banach space X.

Let T be a metric space and {K τ } τ ∈T be a family of subsets of X. The lower limit of K τ at τ 0 ∈ T is defined by: Liminf

τ →τ 0 K τ := v ∈ X lim τ →τ 0 dist Kτ (v) = 0 .
Moreover, we will write x i K → x, when a sequence x i converges to x and x i ∈ K for all i. First and second-order adjacent subsets are defined next.

Definition. Let K be a closed subset of X and x ∈ K. The adjacent cone to K at x is the set,

T K (x) := Liminf h→0+ K -x h .
The second-order adjacent subset to K at (x, u) ∈ K × X is the set defined by, T

K (x, u) := Liminf h→0+ K -x -hu h 2 . (2)
Note that T

(2)

K (x, u) = ∅ implies u ∈ T K (x). If K is convex, then T K (x)
coincides with the tangent cone of convex analysis to K at x and is denoted by T K (x). Below C K (x) stands for the Clarke tangent cone to K at x and

N K (x) = C K (x) -. If X = R n then it is convenient to abbreviate, N 1 K (x) := N K (x) ∩ S n-1 .
A set K is called sleek if for all x ∈ K the contingent cone and the Clarke tangent cone to K at x do coincide (see [START_REF] Aubin | Set-Valued Analysis, Systems Control Found[END_REF] for the definition of contingent cone). In this case also

T K (x) = C K (x) for every x ∈ K. Remark 2.2. If K ⊂ R n is of class C 2 at some x 0 ∈ ∂K, then for all x ∈ ∂K sufficiently close to x 0 , T K (x) = {u ∈ R n | ∇b K (x), u ≤ 0} and ∇b K (x) is the unit outward normal to K at x. Furthermore v ∈ T (2) K (x, u) if and only if either ∇b K (x), u < 0 or ∇b K (x), u = 0 and ∇b K (x), v + 1 2 b K (x)uu ≤ 0. Consequently, in this case, T (2)
K (x, u) is a closed affine halfspace. Example 2.3. Let K = ∩ k i=1 K i , where K i ⊂ R n are of class C 2 for every i = 1, ..., k.
Denote by b i the oriented distance associated to K i and by I(x 0 ) the set of active indices at x 0 , that is, i ∈ I(x 0 ) if and only if

x 0 ∈ ∂K i . If 0 / ∈ co {∇b i (x 0 ) | i ∈ I(x 0 )} for every x 0 ∈ ∂K, then it is well known that K is sleek and T K (y 0 ) = i∈I(y 0 ) T K i (y 0 ) ∀ y 0 ∈ ∂K.
It is not difficult to check that if x 0 ∈ ∂K, then for every u ∈ T K (x 0 ), a vector v ∈ T

K (x 0 , u) if and only if

∇b i (x 0 ), v + 1 2 b i (x 0 )uu ≤ 0 ∀ i ∈ I (1) (x 0 , u),
where

I (1) (x 0 , u) = {i ∈ I(x 0 ) | ∇b i (x 0 ), u = 0}. That is if I (1) (x 0 , u) = ∅, then T (2)
K (x 0 , u) is a closed convex polytope in R n (an intersection of a finite family of affine halfspaces in R n ).

For any K ⊂ X we adopt the following convention: K + ∅ = ∅ + K = ∅. A useful property of the second-order adjacent set is given in the following lemma.

Lemma 2.4. Let K ⊂ X, x ∈ K and u ∈ T K (x). Then, T (2) K (x, u) = T (2) K (x, u) + C K (x). Proof. Since 0 ∈ C K (x), the inclusion ⊂ is obvious. To prove the opposite, let v ∈ T (2) K (x, u) and ũ ∈ C K (x).
We have to show that for all h i → 0+, there exists a sequence w i → v + ũ such that,

x + h i u + h 2 i w i ∈ K, ∀ i. Fix a sequence h i → 0+. Since v ∈ T (2) K (x, u), there exist v i → v such that, x + h i u + h 2 i v i ∈ K, ∀ i.
Further, by the very definition of C K (x), for every sequence x i K → x, there exists ũ i → ũ such that x i + h 2 i ũ i ∈ K for all i. Thus, in particular, there exists ũi → ũ such that,

x + h i u + h 2 i (v i + ũi ) ∈ K, ∀ i. Setting w i = v i + ũi , we end the proof. Let K ⊂ R n be closed and define K := {x ∈ C([0, 1]; R n ) | x(t) ∈ K ∀ t ∈ [0, 1]}. Lemma 2.5. Let x, y, w, w ∈ C([0, 1]; R n ) be such that w ∈ T (2) K (x, y) and for all t ∈ [0, 1], w(t) ∈ C K (x(t)) and int C K (x(t)) = ∅. Then w + w ∈ T (2) K (x, y).
Proof. Let x, y, w, w be as above. Since w ∈ T

(2) K (x, y), for every h > 0 there exists wh such that wh → w uniformly when h → 0+ and for all h > 0,

x + hy + h 2 wh ∈ K.

(4)

It follows from [6, Lemma 4.1] that there exists ŵ ∈ C([0, 1]; R n ) satisfying ŵ(t) ∈ int C K (x(t)) for all t ∈ [0, 1]
. Thus, by convexity of Clarke's tangent cone, for every i ∈ N, w i (t) :

= 1 i ŵ(t) + 1 -1 i w(t) ∈ int C K (x(t)) for all t ∈ [0, 1]. We claim that for every i ∈ N, there exists ε i > 0 such that for all t ∈ [0, 1], z + [0, ε i ]B(w i (t); ε i ) ⊂ K, ∀ z ∈ B(x(t), ε i ) ∩ K. (5)
Indeed, from [24, Thm. 2] it follows that for all i ∈ N and all t ∈ [0, 1] there exists ε t i > 0 for which (5) is satisfied with ε i replaced by ε t i . Then one can use the compactness of x([0, 1]) to deduce (5) for some ε i and all t. Applying ( 4) and ( 5) and using that x + hy + h 2 wh → x uniformly when h → 0+, we find that for all h small enough,

x(t) + hy(t) + h 2 ( wh (t) + w i (t)) ∈ K ∀ t ∈ [0, 1]. It follows that w + w i ∈ T (2)
K (x, y) for all i ∈ N. Finally, since w + w i → w + w uniformly and the second-order adjacent set T

(2)

K (x, y) is closed, we deduce that w + w ∈ T (2) K (x, y). Lemma 2.6. Let x, w ∈ C([0, 1]; R n ) be such that for all t ∈ [0, 1], w(t) ∈ C K (x(t)) and int C K (x(t)) = ∅. Then w ∈ T K (x).
Proof. Consider a sequence h j > 0 converging to 0+ and let ŵ, w i be as in the proof of Lemma 2.5. Then by [START_REF] Bonnans | Perturbation Analysis of Optimization Problems[END_REF] for every i we have x + h j w i ∈ K for all j large enough and therefore w i ∈ T K (x). Since T K (x) is closed in C([0, 1]; R n ) and w i converge uniformly to w the proof follows.

We shall also need the following tangent sets.

Definition. Let K be a closed subset of R n and x ∈ K. The Dubovitskii-Milyutin cone to K at x is the set, D K (x) := {v ∈ R n | ∃ ε > 0, ∀ h ∈ [0, ε], x + hB(v, ε) ⊂ K}. For any (x, u) ∈ K × R n , define D 2 K (x, u) := {v ∈ R n | ∃ ε > 0, ∀ h ∈ [0, ε], x + hu + h 2 B(v, ε) ⊂ K}.
Remark 2.7. The set D 2 K (x, u) was introduced in [START_REF] Páles | Nonsmooth optimum problems with constraints[END_REF], see also [START_REF] Páles | Optimum problems with certain lower semicontinuous set-valued constraints[END_REF]. Observe that D K (x) and

D 2 K (x, u) are open and int C K (x) ⊂ D K (x) ⊂ T K (x), D 2 K (x, u) ⊂ T (2) K (x, u). Lemma 2.8. Let K be a closed subset of R n , x ∈ K and u ∈ R n . Then, D 2 K (x, u) + int C K (x) ⊂ D 2 K (x, u). Proof. Let v ∈ D 2 K (x, u) and ũ ∈ int C K (x).
It is enough to show that there exists ε > 0 such that for all small h > 0

x

+ hu + h 2 B(v + ũ, ε) ⊂ K. (6) Since v ∈ D 2 K (x, u), for some ε v > 0 and for all h ∈ [0, ε v ], x + hu + h 2 B(v, ε v ) ⊂ K. Moreover, since ũ ∈ int C K (x), there exists ε ũ > 0 such that for all x ∈ K ∩ B(x, ε ũ) we have x + [0, ε ũ]B(ũ, ε ũ) ⊂ K.
Thus, in particular, for ε = min{ε v , ε ũ} and all h ≥ 0 sufficiently small, (6) holds true.

We define next the first and second-order directional derivatives/variations of set-valued maps.

Definition. Let F : R n ; R m be a set-valued map, locally Lipschitz around some x ∈ R n and let y ∈ F (x). The adjacent derivative dF (x, y) is the set-valued map defined by,

dF (x, y)(u) := Liminf h→0+ F (x + hu) -y h ∀ u ∈ R n .
For v 1 ∈ dF (x, y)(u 1 ), the second-order adjacent variation d 2 F (x, y, u 1 , v 1 ) is the set-valued map defined by,

d 2 F (x, y, u 1 , v 1 )(u 2 ) := Liminf h→0+ F (x + hu 1 + h 2 u 2 ) -y -hv 1 h 2 ∀ u 2 ∈ R n .
Remark 2.9. It is well-known that if F has convex images and is Lipschitz around x, then for any y ∈ F (x), the images of dF (x, y) are convex and,

dF (x, y)(0) = T F (x) (y) and dF (x, y)(u) + T F (x) (y) = dF (x, y)(u), (7)
see for instance [1, Prop. 5.2.6].

The adjacent variations can also be expressed using the distance function:

Proposition 2.10. Let F : R n ; R m be Lipschitz around x, y ∈ F (x) and v 1 ∈ dF (x, y)(u 1 ). Then for all u 2 ∈ R n , the following holds true:

v 2 ∈ d 2 F (x, y, u 1 , v 1 )(u 2 ) if and only if, lim h→0+ dist F (x+hu 1 +h 2 u 2 ) (y + hv 1 + h 2 v 2 ) h 2 = 0.
The next result provides a useful property of the second-order adjacent variation.

Proposition 2.11. Let F : R n ; R m be a set-valued map with convex images which is Lipschitz around x, y ∈ F (x) and v 1 ∈ dF (x, y)(u 1 ). Then for all u 2 ∈ R n ,

T F (x) (y) + T dF (x,y)(u 1 ) (v 1 ) + d 2 F (x, y, u 1 , v 1 )(u 2 ) = d 2 F (x, y, u 1 , v 1 )(u 2 ). ( 8
)
In particular,

dF (x, y)(0) + dF (x, y)(u 1 ) + d 2 F (x, y, u 1 , v 1 )(u 2 ) = v 1 + d 2 F (x, y, u 1 , v 1 )(u 2 ). ( 9
)
Proof.

Step 1 : We start by showing that,

T F (x) (y) + d 2 F (x, y, u 1 , v 1 )(u 2 ) = d 2 F (x, y, u 1 , v 1 )(u 2 ). ( 10
)
Since 0 ∈ T F (x) (y), the inclusion "⊃" is obvious. To show the opposite, let

w ∈ T F (x) (y), v 2 ∈ d 2 F (x, y, u 1 , v 1 )(u 2
) and h i → 0+ be an arbitrary sequence. By the very definition of the second-order adjacent derivative, there exists a sequence v i 2 → v 2 such that,

y + h i v 1 + h 2 i v i 2 ∈ F (x + h i u 1 + h 2 i u 2 ) ∀ i ∈ N. (11)
Moreover, since v 1 ∈ dF (x, y)(u 1 ), by [START_REF] Cominetti | Metric regularity, tangent sets and second order optimality conditions[END_REF], w + v 1 ∈ dF (x, y)(u 1 ). Hence for some β i → w + v 1 we have

y + h i β i ∈ F (x + h i u 1 ) ⊂ F (x + h i u 1 + h 2 i u 2 ) + o(h i )B ∀ i ∈ N, (12)
where o(α)/α → 0, when α → 0+. Then, multiplying (11) by 1 -h i and ( 12) by h i , and taking their sum, we obtain, thanks to the convexity of the images of F ,

y + h i v 1 + h 2 i v i 2 + β i -v 1 ∈ F (x + h i u 1 + h 2 i u 2 ) + o(h 2 i )B.
Using that

β i → v 1 + w when i → ∞, we get w + v 2 ∈ d 2 F (x, y, u 1 , v 1 )(u 2 ).
Step 2 : By [START_REF] Fontes | Normal forms of necessary conditions for dynamic optimization problems with pathwise inequality constraints[END_REF], to prove (8) it suffices to show that,

T dF (x,y)(u 1 ) (v 1 ) + d 2 F (x, y, u 1 , v 1 )(u 2 ) = d 2 F (x, y, u 1 , v 1 )(u 2 ). ( 13
)
First of all, since dF (x, y)(u 1 ) is a convex set, we know that T dF (x,y)(u 1 ) (v 1 ) is equal to the closure of the set R + (dF (x, y)(u 1 ) -v 1 ). Hence, it is enough to show that for arbitrary λ > 0, v ∈ dF (x, y)(u 1 ) and

v 2 ∈ d 2 F (x, y, u 1 , v 1 )(u 2 ), we have λ(v -v 1 ) + v 2 ∈ d 2 F (x, y, u 1 , v 1 )(u 2 ). Fix such λ, v, v 2 and a sequence h i → 0+. Let (v i 2
) i∈N be as above and v i → v when i → ∞, be such that,

y + h i v i ∈ F (x + h i u 1 ) ⊂ F (x + h i u 1 + h 2 i u 2 ) + o(h i )B ∀ i ∈ N. (14)
Then, multiplying (11) by 1 -λh i and ( 14) by λh i , taking their sum and using that for i large enough λh i < 1, we have,

y + h i v 1 + h 2 i v i 2 + λv i -λv 1 ∈ F (x + h i u 1 + h 2 i u 2 ) + o(h 2 i ), which allows to conclude that λ(v -v 1 ) + v 2 ∈ d 2 F (x, y, u 1 , v 1 )(u 2 )
. Hence the proof of ( 13) is complete.

Step 3 : The inclusion "⊂" in (9) follows directly from ( 8) and the convexity of dF (x, y)(u 1 ). The converse inclusion "⊃" is due to the fact that 0 ∈ dF (x, y)(u 1 ) -v 1 . The proof is complete.

Lemma 2.12. Let F : R n ; R m be a set-valued map with convex images which is Lipschitz around

x ∈ R n , y ∈ F (x) and v 1 ∈ dF (x, y)(u 1 ), v 2 ∈ T F (x) (y). Then, T (2)
F (x) (y, v 2 ) ⊂ T dF (x,y)(u 1 ) (v 1 + v 2 ).
Proof. By [START_REF] Cominetti | Metric regularity, tangent sets and second order optimality conditions[END_REF] we have v 1 + T F (x) (y) ⊂ dF (x, y)(u 1 ). It follows (see for instance [1, Table 4.3]) that,

T dF (x,y)(u 1 ) (v 1 + v 2 ) ⊃ T v 1 +T F (x) (y) (v 1 + v 2 ) = T T F (x) (y) (v 2 ).
Since by [START_REF] Cominetti | Metric regularity, tangent sets and second order optimality conditions[END_REF]Prop. 3.1] it is known that T

(2)

F (x) (y, v 2 ) ⊂ T T F (x) (y) (v 2 )
, the statement follows.

The graph of a set-valued map

F : R n ; R m is the set Gr (F ) := {(x, y) ∈ R n × R m | y ∈ F (x)}.
Let F : R n ; R m be a set-valued map which is Lipschitz on a neighborhood of some x ∈ R n and let y ∈ F (x). The circatangent derivative CF (x, y) is the set-valued map defined by,

CF (x, y)(u) := Liminf (x , y ) Gr(F ) -→ (x, y) h → 0+ F (x + hu) -y h ∀ u ∈ R n .
It is not difficult to realize that Gr (CF (x, y)) is equal to the Clarke tangent cone to Gr (F ) at (x, y).

Hence CF (x, y) : R n ; R m is a closed convex process, i.e. a set-valued map whose graph is a closed convex cone. For a closed convex process A : R n ; R m , its adjoint process A * : R m → R n is defined by,

A * (p) := {q ∈ R n | q, u ≤ p, v ∀ (u, v) ∈ Gr (A)} .
Below, for a set-valued map [0, 1] × R n (t, x) ; F (t, x) and t 0 ∈ [0, 1] such that F (t 0 , •) is Lipschitz on a neighborhood of some x 0 ∈ R n , we denote the partial derivatives with respect to the second variable by a subscript x. That is d x F (t 0 , x 0 , y 0 ) is equal to the adjacent derivative of F (t 0 , •) at (x 0 , y 0 ) for any y 0 ∈ F (t 0 , x 0 ). Similarly, C x F (t 0 , x 0 , y 0 ) denotes the circatangent derivative of F (t 0 , •) at (x 0 , y 0 ) ∈ Gr (F (t 0 , •)).

Second-order variational inclusion

We consider the following differential inclusion with state constraints:

     ẋ(t) ∈ F (t, x(t)) a.e. in [0, 1] x(0) ∈ K 0 x(t) ∈ K ∀ t ∈ [0, 1], (15)
where the set-valued map F : [0, 1] × R n ; R n and the closed nonempty sets K 0 , K ⊂ R n are given. Denote by F : [0, 1]×R n ; R n the set-valued map defined by F (t, x) := co F (t, x) for all (t, x) ∈ [0, 1]×R n . Moreover we introduce the following sets:

S(x 0 ) := x ∈ W 1,1 ([0, 1]; R n ) ẋ(t) ∈ F (t, x(t)) a.e. and x(0) = x 0 S rel (x 0 ) := x ∈ W 1,1 ([0, 1]; R n ) ẋ(t) ∈ F (t, x(t)) a.e. and x(0) = x 0 K := {x ∈ C([0, 1]; R n ) | x(t) ∈ K ∀ t ∈ [0, 1]} S K (x 0 ) := S(x 0 ) ∩ K S rel K (x 0 ) := S rel (x 0 ) ∩ K. (16)
The following regularity assumptions on the dynamics are imposed throughout this section:

          
F has nonempty, compact images and is locally bounded at every (t, x)

∈ [0, 1] × ∂K; F (•, x) is Lebesgue measurable for every x ∈ R n ; ∃ a 1 ∈ L 1 ([0, 1]; R + ) such that sup v∈ F (t,x) |v| ≤ a 1 (t)(1 + |x|) ∀ (t, x) ∈ [0, 1] × R n ; ∀ R > 0, ∃ k R ∈ L 1 ([0, 1]; R + ) such that F (t, •) is k R (t)-Lipschitz on RB for a.e. t ∈ [0, 1]. (A)
In addition, we need the following inward pointing condition:

           ∀ t ∈ [0, 1], ∀ x ∈ ∂K, ∃ Ω t,x ⊂ [0, 1] of zero Lebesgue measure such that ∀ v ∈ Limsup (s,y)→(t,x) s / ∈Ωt,x F (s, y) with max n∈N 1 K (x) n, v ≥ 0, ∃ w ∈ Liminf (s,y)→(t,x) s / ∈Ωt,x F (s, y) satisfying max n∈N 1 K (x) n, w -v < 0.
(IPC)

Remark 3.1. Assumption (IPC) implies that for every x ∈ K, the interior of C K (x) is nonempty. The above inward pointing condition is needed to handle the case of F merely measurable with respect to time. If moreover F is left absolutely continuous in time, then a simpler condition proposed in [START_REF] Bettiol | L ∞ estimates on trajectories confined to a closed subset[END_REF] can be used instead of (IPC).

Consider a reference trajectory x ∈ S K (x 0 ) where x0 ∈ K 0 . The corresponding set of admissible first-order variations V (1) (x) is the set of absolutely continuous maps

y ∈ W 1,1 ([0, 1]; R n ) satisfying, (i) ẏ(t) ∈ d x F (t, x(t), ẋ(t))(y(t)) for a.e. t ∈ [0, 1]; (ii) y(0) ∈ T K 0 (x 0 ); (iii) y ∈ T K (x); (iv) ∃ a 2 ∈ L 1 ([0, 1]; R + ), ∃ h 0 > 0 such that for all h ∈ [0, h 0 ] and a.e. t ∈ [0, 1], dist F (t,x(t)+hy(t)) ( ẋ(t) + h ẏ(t)) ≤ a 2 (t)h 2 .
For a given y ∈ V (1) (x), we abbreviate (t, x(t), ẋ(t), y(t), ẏ(t)) by [t] and define the set of admissible second-order variations V (2) (x, y) as the set of absolutely continuous maps

w ∈ W 1,1 ([0, 1]; R n ) satisfying, (i) ẇ(t) ∈ d 2 x F [t](w(t)) for a.e. t ∈ [0, 1]; (ii) w(0) ∈ T (2)
K 0 (x(0), y(0));

(iii) w ∈ T (2)
K (x, y).

K (x, y) where the condition

∇b i (x(t)), w(t) + 1 2 b i (x(t))y(t)y(t) ≤ 0 ∀ i ∈ I (1) (x(t), y(t)) ∀ t ∈ [0, 1], (17)
in general, does not imply that w ∈ T

K (x, y). A pointwise characterization of T

K (x, y) is given in [START_REF] Kawasaki | Second order necessary optimality conditions for minimizing a sup-type function[END_REF] for the case when K is the cone of nonnegative continuous functions. Further investigations of this subject can be found in [START_REF] Cominetti | Metric regularity, tangent sets and second order optimality conditions[END_REF][START_REF] Páles | Nonsmooth optimum problems with constraints[END_REF]. It turns out that a convenient pointwise condition is an appropriate strengthening of inequality [START_REF] Hoehener | Second-order optimality conditions for a Bolza problem with mixed constraints[END_REF], see for instance [START_REF] Hoehener | Variational approach to second-order optimality conditions for control problems with pure state constraints[END_REF](TV)].

We are ready to state the main result of this section: Theorem 3.3. Assume (A) and (IPC). Let x ∈ S K (x 0 ) for some x0 ∈ K 0 , y ∈ V (1) (x) and w ∈ V (2) (x, y). Consider any sequences h i → 0+, w 0 i → w(0) such that x(0) + h i y(0) + h 2 i w 0 i ∈ K 0 . Then there exist

x i ∈ S K (x(0) + h i y(0) + h 2 i w 0 i ) such that 1 h 2 i (x i -x -h i y) converge uniformly to w when i → ∞.
Proof. Let x, y, w be as above and fix any sequences h i → 0+, w 0 i → w(0) such that x(0)+h i y(0)+h 2 i w 0 i ∈ K 0 . To simplify the notations we set y 0 := y(0) and w 0 := w(0). Define for all i ∈ N,

x 1 i (t) := x(t) + h i y(t) + h 2 i w(t) + h 2 i (w 0 i -w 0 ).
Note that with this definition

x 1 i (0) = x0 + h i y 0 + h 2 i w 0 i ∈ K 0 and ẋ1 i (t) = ẋ(t) + h i ẏ(t) + h 2 i ẇ(t)
. First, we show that there exist

x 2 i ∈ S rel (x 0 + h i y 0 + h 2 i w 0 i ) such that, 1 h 2 i x 1 i -x 2 i W 1,1 → 0, when i → ∞. (18)
For this aim define, γ i (t) := dist F (t,x 1 i (t)) ( ẋ1 i (t)). Let R > 0 be such that for all t ∈ [0, 1] and i ≥ 1,

|x(t) + h i y(t)| + |x(t) + h i y(t) + h 2 i w(t)| + |x 1 i (t)| ≤ R.
Then it follows from the k R (t)-Lipschitz continuity of F (t, •) that for all i large enough,

γ i (t) ≤ dist F (t,x(t)+h i y(t)+h 2 i w(t)) ( ẋ(t) + h i ẏ(t) + h 2 i ẇ(t)) + k R (t)h 2 i w 0 i -w 0 ≤ dist F (t,x(t)+h i y(t)) ( ẋ(t) + h i ẏ(t)) + h 2 i k R (t) w ∞ + w 0 i -w 0 + | ẇ(t)| ≤ h 2 i k R (t) w ∞ + w 0 i -w 0 + | ẇ(t)| + a 2 (t) . (19)
By [START_REF] Kawasaki | Second order necessary optimality conditions for minimizing a sup-type function[END_REF], the sequence γ i (t)

h 2 i i∈N is integrably bounded. Since w(•) ∈ V (2) (x, y), 1 h 2 i dist F (t,x(t)+h i y(t)+h 2 i w(t)) ( ẋ(t) + h i ẏ(t) + h 2 i ẇ(t)) → 0 for a.e. t ∈ [0, 1]. Hence lim i→∞ 1 h 2 i 1 0 γ i (t)dt = 1 0 lim i→∞ γ i (t) h 2 i dt = 0.
This and Filippov's theorem, see for instance [1, Thm. 10.4.1], imply the existence of x 2 i ∈ S rel (x 0 + h i y 0 + h 2 i w 0 i ) satisfying [START_REF] Hoehener | Variational approach to second-order optimality conditions for control problems with pure state constraints[END_REF]. Next, [START_REF] Frankowska | On relations of the adjoint state to the value function for optimal control problems with state constraints[END_REF]Thm. 3.3] implies that for a constant L > 0 and for every i ∈ N, there exists

x 3 i ∈ S rel K (x 0 + h i y 0 + h 2 i w 0 i ) such that, x 2 i -x 3 i W 1,1 ≤ L max t∈[0,1] dist K (x 2 i (t)).
Note that for all i,

dist K (x 2 i (t)) ≤ dist K (x(t) + h i y(t) + h 2 i w(t)) + h 2 i w 0 i -w 0 + x 2 i (t) -x 1 i (t) , (20)
and for every x ∈ C([0, 1]; R n ) and all t ∈ [0, 1],

dist K (x(t)) = inf k∈K |x(t) -k| ≤ inf κ∈K |x(t) -κ(t)| ≤ dist K (x).
Thus, max t∈[0,1] dist K (x(t) + h i y(t) + h 2 i w(t)) ≤ dist K (x + h i y + h 2 i w). Since w ∈ V (2) (x, y), this implies by [START_REF] Hoehener | Variational approach to second-order optimality conditions for control problems with pure state constraints[END_REF] and (20) that, 1

h 2 i x 2 i -x 3 i W 1,1 → 0, when i → +∞. (21)
Furthermore, by [START_REF] Frankowska | On relations of the adjoint state to the value function for optimal control problems with state constraints[END_REF]Cor. 3.4], for every i ∈ N, there exists x i ∈ S K (x 0 + h i y 0 + h 2 i w 0 i ) such that,

x 3 i -x i ∞ ≤ h 3 i . (22)
Finally, by the very definition of x 1 i , we find that,

x + h i y + h 2 i w -x i ∞ ≤ h 2 i w 0 i -w 0 + x 1 i -x 2 i W 1,1 + x 2 i -x 3 i W 1,1 + x 3 i -x i ∞ .
Thus, from ( 18), ( 21) and ( 22) it follows that,

lim i→∞ x + h i y + h 2 i w -x i ∞ h 2 i = 0.

Second-order optimality conditions: differential inclusions

We study here necessary optimality conditions for a Mayer optimization problem involving the differential inclusion [START_REF] Galbraith | Lipschitz continuity of optimal controls for state constrained problems[END_REF]. Throughout this section we associate with any x ∈ S K (x 0 ) the linearized differential inclusion: ẏ(t) ∈ d x F (t, x(t), ẋ(t))(y(t)) a.e. y(0) ∈ T K 0 (x(0)). ( 23)

Our first goal is to define subsets of the sets of admissible variations that are convenient for the expression of necessary optimality conditions. For every t ∈ [0, 1] such that ẋ(t) ∈ F (t, x(t)) define the closed convex process

A(t) := C x F (t, x(t), ẋ(t))
and the closed convex cone

L(t) := T F (t,x(t)) ( ẋ(t)).
Moreover, for every trajectory y(•) of ( 23

For a fixed solution y of ( 23), we introduce the set-valued map F I (y; •) : [0, 1] ; R n given by

F I (y; t) := Liminf h→0+ F (t, x(t) + hy(t)) -h ẏ(t) -ẋ(t) h 2 ∀ t ∈ [0, 1],
and define the second-order approximation of the differential inclusion from (15) as follows: ẇ(t) ∈ A(t)w(t) + F I (y; t) + E(y; t) a.e.

w(0) ∈ T

K 0 (x(0), y(0)).

(

) 25
This definition is motivated by [START_REF] Delfour | Shapes and Geometries: Analysis, Differential Calculus and Optimization[END_REF] and the following fact:

Proposition 4.1. Assume (A) and let x ∈ S(x 0 ), y ∈ W 1,1 ([0, 1]; R n ) be a solution of [START_REF] Rampazzo | A theorem on existence of neighboring trajectories satisfying a state constraint, with applications to optimal control[END_REF]. Then for a.e. t ∈ [0, 1],

A(t)w + F I (y; t) ⊂ d 2 x F [t](w) ∀ w ∈ R n . Proof. Let R := x ∞ + 1, k R ∈ L 1 ([0, 1]; R + ) be as in (A) and t ∈ [0, 1] be such that F (t, •) is k R (t)-Lipschitz on RB, ẋ(t) ∈ F (t, x(t)), ẏ(t) ∈ d x F (t, x(t), ẋ(t))(y(t)).
If A(t)w or F I (y; t) is empty, then there is nothing to prove. Fix w ∈ R n , v ∈ F I (y; t), α ∈ A(t)w and a sequence h i → 0+. By the very definition of F I (y; t), there exists a sequence v i → v such that,

ẋ(t) + h i ẏ(t) + h 2 i v i ∈ F (t, x(t) + h i y(t)) ∀ i ∈ N.
On the other hand, since Gr (A(t)) is equal to the Clarke tangent cone to Gr (F (t, •)) at (x(t), ẋ(t)), there exist sequences (w i , α i ) → (w, α) satisfying,

Gr (F (t, •)) x(t) + h i y(t), ẋ(t) + h i ẏ(t) + h 2 i v i + h 2 i (w i , α i ) = x + h i y(t) + h 2 i w i , ẋ(t) + h i ẏ(t) + h 2 i (v i + α i ) ,
which implies, by the Lipschitz continuity of

F (t, •), that v + α ∈ d 2 x F [t](w).
The proof is complete.

Thanks to Propositions 2.11 and 4.1, for any y ∈ V (1) (x) we can define the following subset of second-order admissible variations: V

I (x, y) := w ∈ V (2) (x, y) w solves [START_REF] Tamzali-Lafond | Variational inclusions under state constraints[END_REF] .

Consider now the Mayer problem

Minimize {ϕ(x( 1)

) | x(•) ∈ S K (x 0 ), x 0 ∈ K 0 } , (26)
where ϕ : R n → R is a given twice differentiable function.

A trajectory x(•) ∈ S K (K 0 ) is called a strong local minimizer of the above Mayer problem if there exists ε > 0 such that for every

x(•) ∈ S K (K 0 ) satisfying x -x ∞ < ε we have ϕ(x(1)) ≤ ϕ(x(1)).
Let us start by recalling the first-order necessary optimality conditions of [START_REF] Cernea | A connection between the maximum principle and dynamic programming for constrained optimal control problems[END_REF]Corollary 3.8] in the form of a maximum principle. We need the following additional assumption:

∃ ≥ 0 such that A(t) : R n ; R n is -Lipschitz for a.e. t ∈ [0, 1]. ( A)
Remark 4.2. Assumption ( A) concerns a reference trajectory x. It was imposed in [START_REF] Cernea | A connection between the maximum principle and dynamic programming for constrained optimal control problems[END_REF] to apply the duality theory of convex analysis to closed convex processes. In the next section, when dealing with control systems, we do not need such assumption, because in this case a more direct approach not involving abstract duality theorems is used.

Recall that the domain of A(t) is equal to R n if and ony if A(t) is Lipschitz with a constant c(t) ≥ 0. Assumption ( A) requires c(•) to be essentially bounded.

If for almost every t ∈ [0, 1], the set Gr(F (t, •)) is sleek and k R (•) in (A) is essentially bounded for every R > 0, then ( A) is satisfied for any reference trajectory x.

Theorem 4.3 (Maximum Principle). Let x be a strong local minimizer of Problem [START_REF] Vinter | Optimal Control[END_REF] and for all t ∈ [0, 1], int C K (x(t)) = ∅. Assume ( A) and that (A) holds true with a bounded a 1 (•).

If int C K 0 (x(0))∩ int C K (x(0)) = ∅, then there exist λ ∈ {0, 1}, ψ ∈ N BV ([0, 1]; R n ) and p ∈ W 1,1 ([0, 1]; R n ) such that λ + ψ T V = 0 and (i) ṗ(t) ∈ A(t) * (-p(t) -ψ(t)) a.e. in [0, 1] (ii) p(0) ∈ λ (N K 0 (x(0)) + N K (x(0))) (iii) p(1) = -λ∇ϕ(x(1)) -ψ(1) (iv) p(t) + ψ(t), ẋ(t) = max z∈F (t,x(t)) p(t) + ψ(t), z a.e. in [0, 1]. Furthermore, ψ(0+) ∈ N K (x(0)), ψ(t) -ψ(t-) ∈ N K (x(t)), ψ(t) = [0,t] ν(s)dµ(s) ∀ t ∈ ]0, 1],
for a non-negative (scalar) Borel measure µ on [0, 1] and a Borel measurable mapping ν :

[0, 1] → R n satisfying ν(s) ∈ N K (x(s)) ∩ B µ-a.e.
Moreover, the following non degeneracy conditions hold true

λ + sup t∈]0,1[ |p(t) + ψ(t)| = 0 and λ + var(ψ, ]0, 1]) = 0,
where var(ψ, ]0, 1]) denotes the total variation of ψ on ]0, 1]. Finally, if there exists a solution to the constrained differential inclusion

     ẏ(t) ∈ A(t)y(t) + L(t) a.e. y(t) ∈ int C K (x(t)) ∀ t ∈ [0, 1] y(0) ∈ int C K 0 (x(0)), (27)
then the above holds true with λ = 1.

An extremal is a tuple (x, p, ψ, µ, ν, λ), where x is a feasible trajectory of [START_REF] Galbraith | Lipschitz continuity of optimal controls for state constrained problems[END_REF]

p(t) + ψ(t), w ≤ 0 ∀ w ∈ T (2)
F (t,x(t)) ( ẋ(t), v). ( 29
)
Proof. Indeed, let t ∈ [0, 1] be such that (iv) of Theorem 4.3 holds true and w ∈ T

(2)

F (t,x(t)) ( ẋ(t), v) for some v ∈ L(t) satisfying p(t) + ψ(t), v = 0.
By the very definition of the second-order adjacent set, for every h > 0 there exists w h such that w h → w when h → 0+ and ẋ(t) + hv + h 2 w h ∈ F (t, x(t)) for all h. Since p(t) + ψ(t), v = 0 it follows from (28) that h 2 p(t) + ψ(t), w h ≤ 0. Dividing by h 2 and passing to the limit, we get (29).

Consider the following linearization of the differential inclusion in (15): ẏ(t) ∈ A(t)y(t) + F (t, x(t)) -ẋ(t) a.e. y(0) ∈ T K 0 (x(0)).

(30)

Remark 4.5. It is clear that Gr (A(t)) ⊂ Gr (d x F (t, x(t), ẋ(t))). Hence, by [START_REF] Cominetti | Metric regularity, tangent sets and second order optimality conditions[END_REF], for any y(•) solving (30) we have ẏ(t) ∈ d x F (t, x(t), ẋ(t))(y(t)) a.e.

Remark 4.6. Let (x, p, ψ, µ, ν) be a normal extremal. Consider y ∈ V (1) (x) satisfying (30) with y(0) ∈ C K 0 (x(0)), y(t) ∈ C K (x(t)) for all t ∈ [0, 1] and integrable selections γ(t) ∈ A(t)y(t), v(t) ∈ F (t, x(t)) such that ẏ(t) = γ(t) + v(t) -ẋ(t) a.e. Then we have,

∇ϕ(x(1)), y(1) = -p(1) -ψ(1), y (1)
= -ψ(1), y(1) + -p(0), y(0)

+ 1 0 -p(t), ẏ(t) + -ṗ(t), y(t) dt ≥ -ψ(1), y (1)
+ [0,1] ν(t), y(t) dµ(t) + 1 0 -p(t), ẏ(t) + -ṗ(t), y(t) dt = 1 0 -p(t) -ψ(t), ẏ(t) + -ṗ(t), y(t) dt ≥ 1 0 -p(t) -ψ(t), ẏ(t) dt + 1 0 p(t) + ψ(t), γ(t) dt = 1 0 -p(t) -ψ(t), v(t) -ẋ(t) dt ≥ 0.
Therefore if ∇ϕ(x(1)), y(1) = 0, then (iv) of Theorem 4.3 yields p(t) + ψ(t), v(t) -ẋ(t) = 0 a.e. This and Corollary 4.4 imply that for almost every t ∈ [0, 1],

p(t) + ψ(t), w ≤ 0 ∀ w ∈ T (2)
F (t,x(t)) ( ẋ(t), v(t) -ẋ(t)). (31)
Consider the sets

Λ(t) := {v ∈ F (t, x(t)) | p(t) + ψ(t), ẋ(t) = p(t) + ψ(t), v } ,
and the constrained differential inclusion

     ẏ(t) ∈ A(t)y(t) + v(t) -ẋ(t), v(t) ∈ F (t, x(t)) a.e. y(0) ∈ C K 0 (x(0)) y(t) ∈ C K (x(t)) ∀ t ∈ [0, 1]. (32)
Therefore we have derived the following alternative: If F (t, x(t)) = { ẋ(t)} on a set of positive measure, then either for every y(•) ∈ V (1) (x) satisfying (32) with v(•) = ẋ(•) on a set of positive measure we have ∇ϕ(x(1)), y(1) > 0, or Λ(t) = { ẋ(t)} on a set of positive measure. In this later case, such normal extremals can then be considered as singular.

Using the second-order variational equation from Section 3, we find the following necessary optimality conditions for problem (26):

Theorem 4.7. Assume (A), (IPC) and let x be a strong local minimizer of problem (

≥ 0 ∀ y ∈ V (1) (x). 26). Then, ∇ϕ(x(1)), y(1)
Moreover, for all y ∈ V (1) (x) such that ∇ϕ(x(1)), y(1) = 0, we have

∇ϕ(x(1)), w(1) + 1 2 ϕ (x(1))y(1)y(1) ≥ 0 ∀ w ∈ V (2) (x, y). (34)
Proof. To show (33), let y ∈ V (1) (x) and fix a sequence h i → 0+. Since y(0) ∈ T K 0 (x(0)), there exists a sequence y 0 i → y(0) such that x(0) + h i y 0 i ∈ K 0 for all i. Thus, by [START_REF] Frankowska | On relations of the adjoint state to the value function for optimal control problems with state constraints[END_REF]Thm. 4.3], we can find a sequence (y i ) i∈N converging uniformly to y, such that x + h i y i ∈ S K (x(0) + h i y 0 i ) for all i. Since x is a strong local minimizer, for all large i,

0 ≤ ϕ(x(1) + h i y i (1)) -ϕ(x(1)) = h i ∇ϕ(x(1)), y(1) + o(h i ),
where o(h i )/h i → 0+ when i → ∞. Dividing by h i and passing to the limit, we find (33). Next, let y ∈ V (1) (x) be such that ∇ϕ(x(1)), y(1) = 0, w ∈ V (2) (x, y) and h i → 0+. By Theorem 3.3 there exists a sequence w i → w uniformly, when i → ∞ such that x + h i y + h 2 i w i ∈ S K (K 0 ) for all i. Since x is a strong local minimizer, it follows that for all i ∈ N large enough,

0 ≤ ϕ(x(1) + h i y(1) + h 2 i w i (1)) -ϕ(x(1)) = h 2 i ∇ϕ(x(1)), w(1) + h 2 i 2 ϕ (x(1))y(1)y(1) + o(h 2 i ),
where o(h 2 i )/h 2 i → 0, when i → ∞. Dividing by h 2 i and passing to the limit, we get (34).

Consider the following "second-order linearization" of (15):

     ẇ(t) ∈ A(t)w(t) + E(y; t) for a.e. t ∈ [0, 1] w(0) ∈ C K 0 (x(0)) w(t) ∈ C K (x(t)) for all t ∈ [0, 1]. (35)
Theorem 4.8 (Second Order Maximum Principle). Let x be a strong local minimizer of Problem ( 26) and ( A), (IPC), (A) hold true with a bounded a 1 (•). Assume that y ∈ V (1) (x) is such that V

I (x, y) = ∅ and ∇ϕ(x(1)), y(1) = 0. Then for every solution w of (35) we have ∇ϕ(x(1)), w(1) ≥ 0.

Furthermore, if int C K 0 (x(0)) ∩ int C K (x(0)) = ∅, then there exist λ ∈ {0, 1}, ψ ∈ N BV ([0, 1]; R n ) and p ∈ W 1,1 ([0, 1]; R n ) satisfying all the conclusions of Theorem 4.3 such that in addition max v∈dxF (t,x(t), ẋ(t))(y(t)) p(t) + ψ(t), v = p(t) + ψ(t), ẏ(t) a.e. in [0, 1]. Finally, if the constrained differential inclusion      ż(t) ∈ A(t)z(t) + E(y; t) a.e. z(t) ∈ int C K (x(t)) ∀ t ∈ [0, 1] z(0) ∈ int C K 0 (x(0)) (36)
has a solution, then the above holds true with λ = 1.

Proof. Let w ∈ V (2)
I (x, y) and w be a solution of (35). Since A(t) is a closed convex process and E(y; t) is a closed convex cone for a.e. t ∈ [0, 1], it follows from Lemmas 2.4, 2.5 and Propositions 2.11, 4.1 that w + w ∈ V

(2) I (x, y). Therefore, by Theorem 4.7,

∇ϕ(x(1)), w(1) + ∇ϕ(x(1)), w(1) + 1 2 ϕ (x (1))y(1)y(1) ≥ 0. (37)
The first conclusion follows from the fact that the set of solutions to (35) is a cone and the last two terms of the above inequality do not depend on w. Therefore w ≡ 0 is an optimal solution of the problem Minimize ∇ϕ(x(1)), w(1) , over solutions w of (35). For all t ∈ [0, 1] consider the closed convex process A 1 (t) : R n ; R n defined by

A 1 (t)x = A(t)x + E(y; t) ∀ x ∈ R n .
The same arguments as in [6, pp. 687, 693-697] applied to the closed convex process A 1 (t) instead of B(t, •) introduced in [6, p. 693] and (24) lead to the desired result.

5 Second-order optimality conditions: control systems

We investigate here the following constrained control system:

     ẋ(t) = f (t, x(t), u(t)), u(t) ∈ U (t) for a.e. t ∈ [0, 1], x(0) ∈ K 0 x(t) ∈ K ∀ t ∈ [0, 1], (38)
where f : [0, 1] × R n × R m → R n , K 0 , K ⊂ R n are nonempty closed sets and U : [0, 1] ; R m is a set-valued map. Throughout this section (x, ū) denotes a trajectory control pair of the control system (38). In order to simplify the notations, we will abbreviate (t, x(t), ū(t)) by [t], so for instance f [t] = f (t, x(t), ū(t)). We assume that Given a trajectory control pair (x, ū), in some results below we assume that f x (t, •, •) exists and is continuous on a neighborhood of (x(t), ū(t)) for almost all t ∈ [0, 1].

(H1) (a) ∀ x ∈ R n , ∀ u ∈ R m , f (•, x,
(39)

One readily checks that (H1) implies that the set-valued map (t, x) ; f (t, x, U (t)) satisfies (A). Thus (38) is a special case of [START_REF] Galbraith | Lipschitz continuity of optimal controls for state constrained problems[END_REF]. The set-valued map x 0 ; S K (x 0 ), defined in [START_REF] Gilbert | Second-order necessary conditions in optimal control: Accessoryproblem results without normality conditions[END_REF], is in this section understood with respect to F (t, x) := f (t, x, U (t)). As before, consider the set-valued map

(t, x) ; F (t, x) := co {f (t, x, u) | u ∈ U (t)} .
Let us now fix a solution y of ( 23) and introduce the sets F C (y; t) ⊂ R n defined for all t ∈ [0, 1] such that ẏ(t) does exist by,

F C (y; t) :={v ∈ R n | ∀ h > 0, ∃ u h ∈ U (t), v h ∈ R n , (u h , v h ) → (ū(t), v) when h → 0+ such that f (t, x(t) + hy(t), u h ) = f [t] + h ẏ(t) + h 2 v h },
and consider the following second-order approximation of the control system from (38):

ẇ(t) ∈ f x [t]w(t) + F C (y; t) + E(y; t) for a.e. t ∈ [0, 1] w(0) ∈ T (2)
K 0 (x(0), y(0)).

(

) 40
This definition is motivated by the following observation:

Proposition 5.1. Let (x, ū) be a trajectory control pair of the control system (38) and y be a solution of [START_REF] Rampazzo | A theorem on existence of neighboring trajectories satisfying a state constraint, with applications to optimal control[END_REF]. If (39) holds true, then for almost all t ∈ [0, 1] and all w ∈ R n ,

f x [t]w + F C (y; t) + E(y; t) ⊂ d 2
x F [t](w). Proof. By Proposition 2.11 it is enough to show that for a.e. t ∈ [0, 1] and for all w ∈ R n we have

f x [t]w + F C (y; t) ⊂ d 2
x F [t](w). Fix t ∈ [0, 1] such that ẋ(t) = f (t, x(t), ū(t)) and ẏ(t) ∈ d x F (t, x(t), ẋ(t))(y(t)). Let w ∈ R n , v ∈ F C (y; t) and h i → 0+. By the very definition of F C (y; t), there exist u i

U (t) -→ ū(t) and v i → v such that, f (t, x(t) + h i y(t), u i ) = f [t] + h i ẏ(t) + h 2 i v i ∀ i ∈ N.
Moreover, using the continuity of f x (t, •, •) on a neighborhood of (x(t), ū(t)), for all sufficiently large i,

F (t, x(t) + h i y(t) + h 2 i w) f (t, x(t) + h i y(t) + h 2 i w, u i ) = f (t, x(t) + h i y(t), u i ) + 1 0 f x (t, x(t) + h i y(t) + θh 2 i w, u i )h 2 i wdθ = f (t, x(t) + h i y(t), u i ) + f x [t]h 2 i w + 1 0 f x (t, x(t) + h i y(t) + θh 2 i w, u i ) -f x [t] h 2 i wdθ = f [t] + h i ẏ(t) + h 2 i (v i + f x [t]w) + o(h 2 i ),
where o(h 2 i )/h 2 i → 0 when i → ∞, completing the proof.

Consider the following classical linearization of control system (38):

ẏ(t) = f x [t]y(t) + v(t) -ẋ(t), v(t) ∈ F (t,
     ẏ(t) = f x [t]y(t) + v(t) -ẋ(t), v(t) ∈ F (t, x(t)) for a.e. t ∈ [0, 1] y(t) ∈ T K (x(t)) ∀ t ∈ [0, 1] y(0) ∈ T K 0 (x(0)) (44) satisfies y ∈ V (1) (x).
For every y ∈ V (1) (x) define the following subset of the set of admissible second-order variations:

V (2) C (x, y) := w ∈ V (2) (x, y) w is solution of (40) .
Let us consider the following Mayer optimal control problem:

Minimize {ϕ(x(1)) | x(•) ∈ S K (x 0 ), x 0 ∈ K 0 } , (45)
where ϕ : R n → R is a given twice differentiable function. Next, we recall the celebrated maximum principle, see for instance [26, Thm. 9.5.1]:

: [0, 1] → R n satisfying, ν(t) ∈ N K (x(t)) ∩ B µ-a.e., (46)
such that for ψ : [0, 1] → R n defined by ψ(t) := [0,t] ν(s)dµ(s) if t ∈ ]0, 1] and ψ(0) = 0 we have (p, ψ, λ) = 0, A tuple (x, ū, p, ψ, µ, ν, λ) such that (x, ū) is a trajectory control pair of (38) and (p, ψ, µ, ν, λ) is as in the maximum principle, is called an extremal. An extremal is normal if λ = 1. There are several results on normality of the maximum principle for optimal control problems, see for instance [START_REF] Bettiol | Normality of the maximum principle for nonconvex constrained Bolza problems[END_REF][START_REF] Fontes | Normal forms of necessary conditions for dynamic optimization problems with pathwise inequality constraints[END_REF][START_REF] Frankowska | Regularity of minimizers and of adjoint states in optimal control under state constraints[END_REF][START_REF] Frankowska | Normality of the maximum principle for absolutely continuous solutions to Bolza problems under state constraints[END_REF][START_REF] Frankowska | On relations of the adjoint state to the value function for optimal control problems with state constraints[END_REF][START_REF] Frankowska | Inward pointing trajectories, normality of the maximum principle and the non occurence of the Lavrentieff phenomenon in optimal control under state constraints[END_REF][START_REF] Galbraith | Lipschitz continuity of optimal controls for state constrained problems[END_REF][START_REF] Rampazzo | A theorem on existence of neighboring trajectories satisfying a state constraint, with applications to optimal control[END_REF] and the references therein. We provide next second-order necessary optimality conditions for problem (45):

(i) -ṗ(t) = f x (t, x(t), ū(t)) * (p(t) + ψ(t)) a.e. (ii) p(0) ∈ N K 0 (x(0)) (iii) -p(1) = λ∇ϕ(x(1)) + ψ(1) (
Theorem 5.7. Assume (H1) and (IPC) and let (x, ū) be a strong local minimizer. Then ∇ϕ(x(1)), y(1) ≥ 0 ∀ y ∈ V (1) (x).

If the maximum principle of Theorem 5.5 holds true with λ = 1 and some (p, ψ, µ, ν) and y ∈ V (1) (x) is such that ∇ϕ(x(1)), y(1) = 0, then for every w ∈ V

C (x, y), 1 2 ϕ (x(1))y(1)y(1) -p(0), w(0) -

[0,1] ν(t), w(t) dµ(t) - 1 0 p(t) + ψ(t), v(t) dt ≥ 0, where v(t) := ẇ(t) -f x [t]w(t) ∈ F C (y; t) + E(y; t) a.e.
Proof. Theorem 4.7 implies the first statement. Let y ∈ V (1) (x) be such that ∇ϕ(x(1)), y(1) = 0. Then, by Theorem 4.7, for all w ∈ V (2) (x, y) we have

∇ϕ(x(1)), w(1) + 1 2 ϕ (x(1))y(1)y(1) ≥ 0. ( 47
) If w ∈ V (2)
C (x, y), then ẇ(t) = f x [t]w(t) + v(t) for some v(t) ∈ F C (y; t) + E(y; t) and ∇ϕ(x(1)), w(1) = -p(1) -ψ(1), w(1)

= 1 0 -p(t) -ψ(t), ẇ(t) + -ṗ(t), w(t) dt - [0,1] ν(t), w(t) dµ(t) -p(0), w(0) = - 1 0 p(t) + ψ(t), v(t) dt - [0,1] ν(t), w(t) dµ(t) -p(0), w(0) .
This and (47) complete the proof.

Proof. Let v(t) ∈ F (t, x(t)) be such that ẏ(t) = f x [t]y(t) + v(t) -ẋ(t) a.e. Fix t ∈ [0, 1] such that (53) is satisfied. Consider an integer k > 0, u i ∈ U (t) and c i ≥ 0, i = 1, ..., k such that k i=1 c i = 1 and f (t, x(t), ū(t)) = k i=1 c i f (t, x(t), u i ). Then k i=1 c i f x (t, x(t), u i )y(t) ∈ d x F (t, x(t), ẋ(t))(y(t))
and therefore

k i=1 c i f x (t, x(t), u i )y(t) + v(t) -ẋ(t) ∈ d x F (t, x(t), ẋ(t))(y(t)).
Consequently, for a.e. t ∈ [0, 1],

p(t) + ψ(t), k i=1 c i f x (t, x(t), u i )y(t) ≤ p(t) + ψ(t), f x [t]y(t) .
Remark 5.12. The above corollary can be stated in a much more general way. Namely let p, ψ be as in the conclusions of Theorem 5.10 and t be such that (53) is satisfied. Consider any mappings c i : R n → R + , i = 1, ..., k that are continuously differentiable on a neighborhood of x(t) and such that k i=1 c i (x) = 1 for every x sufficiently close to x(t). Then for all u i ∈ U (t) satisfying f [t] = k i=1 c i (x(t))f (t, x(t), u i ) the following inequality holds true:

p(t) + ψ(t), f x [t]y(t) ≥ p(t) + ψ(t), k i=1 c i (x(t))f x (t, x(t), u i )y(t) + k i=1
(c i (x(t))y(t))f (t, x(t), u i ) .

Optimality conditions in the presence of endpoint constraints

In this section we impose an additional endpoint constraint:

x(1) ∈ K 1 , (55)
where K 1 ⊂ R n is nonempty and closed. Consider again the differential inclusion [START_REF] Galbraith | Lipschitz continuity of optimal controls for state constrained problems[END_REF]. As before, F (t, x) := co F (t, x) for all (t, x) ∈ [0, 1] × R n . Theorem 6.1. Assume (A), (IPC) and let x ∈ S K (x 0 ) for some x0 ∈ K 0 satisfy x(1) ∈ K 1 . Suppose that y ∈ V (1) (x) and w ∈ V (2) (x, y) are such that w(1) ∈ D 2 K 1 (x(1), y(1)). Consider any sequences h i → 0+, w 0 i → w(0) with x(0) + h i y(0) + h 2 i w 0 i ∈ K 0 . Then there exist x i ∈ S K (x(0) + h i y(0) + h 2 i w 0 i ) satisfying x i (1) ∈ K 1 and such that 1 h 2 i (x i -x -h i y) converge uniformly to w when i → ∞.

Proof. Let x i be as in Theorem 3.3. It is enough to observe that since w(1) ∈ D 2 K 1 (x(1), y(1)) and For any y ∈ V (1) (x) define the following subset of second-order admissible variations: V

I (x, y, K 1 ) := w ∈ V

I (x, y) w(1) ∈ D 2 K 1 (x(1), y(1)) .

Consider now the Mayer problem

Minimize {ϕ(x(1)) | x(•) ∈ S K (x 0 ), x 0 ∈ K 0 , x(1)

∈ K 1 } , (56)
where ϕ : R n → R is a given twice differentiable function. Let us start by recalling the first-order necessary optimality conditions from [START_REF] Cernea | A connection between the maximum principle and dynamic programming for constrained optimal control problems[END_REF]Corollary 3.8] in this more general case.

Moreover, for all y ∈ V (1) (x) such that ∇ϕ(x(1)), y(1) = 0, we have ∇ϕ(x(1)), w(1) + 1 2 ϕ (x(1))y( 1)y(1) ≥ 0 ∀ w ∈ V (2) (x, y) satisfying w(1) ∈ D 2 K 1 (x(1), y(1)). (58)

As in Section 4, we can prove the following second-order maximum principle.

Theorem 6.6. Let x be a strong local minimizer of problem (56), ( A), (IPC), (A) hold with a bounded a 1 (•) and int C K 1 (x(1)) = ∅. Let y ∈ V (1) (x) be such that ∇ϕ(x(1)), y(1) = 0 and V

I (x, y, K 1 ) = ∅. Then for every solution w of (35) such that w(1) ∈ int C K 1 (x(1)), we have ∇ϕ(x(1)), w(1) ≥ 0. Furthermore, if int C K 0 (x(0)) ∩ int C K (x(0)) = ∅, then there exist λ, ψ, p as in the conclusions of Theorem 6.2 such that in addition max v∈dxF (t,x(t), ẋ(t))(y(t)) p(t) + ψ(t), v = p(t) + ψ(t), ẏ(t) a.e. in [0, 1].

Finally, if the constrained differential inclusion (36) has a solution z satisfying z(1) ∈ int C K 1 (x(1)), then the above holds true with λ = 1.

Proof. Let w ∈ V

(2) I (x, y, K 1 ) and w be a solution of (35) such that w(1) ∈ int C K 1 (x(1)). By Lemma 2.8, w(1) + w(1) ∈ D 2 K 1 (x(1), y(1)). Arguments similar to those of the proof of Theorem 4.8 imply the result.

Consider the constrained control system (38) under an additional endpoint constraint (55) and define the set-valued maps F , F as in Section 5. For a solution y of (23) let the sets F C (y; t) ⊂ R n be as in Section 5. For every y ∈ V (1) (x) we introduce the following subset of the set of admissible second-order variations: V

C (x, y, K 1 ) := w ∈ V

C (x, y) w(1) ∈ D 2 K 1 (x(1), y(1)) . We investigate next the Mayer optimal control problem (56) where x 0 ; S K (x 0 ) is understood with respect to the control system (38).

1 h 2 i(x i ( 1 )

 121 -x(1) -h i y(1)) converge to w(1) for i → ∞, we have x i (1) ∈ K 1 , for i sufficiently large.

Theorem 6 . 2 .Corollary 6 . 3 .

 6263 Let x be a strong local minimizer of Problem (56) and for all t ∈ [0, 1], int C K (x(t)) = ∅. Assume ( A) and that (A) holds true with a bounded a 1 (•).If int C K 0 (x(0)) ∩ int C K (x(0)) = ∅ and int C K 1 (x(1)) = ∅, then there exist λ, ψ, p as in Theorem 4.3 with (iii) replaced byp(1) ∈ -λ∇ϕ(x(1)) -ψ(1) -N K 1 (x(1)). Moreover, if C K 1 (x(1)) ∩ int C K (x(1)) = ∅, then λ + sup t∈]0,1[ |p(t) + ψ(t)| = 0.Finally, if there exists a solution y of (27) satisfying y(1) ∈ int C K 1 (x(1)), then the above holds true with λ = 1.The very same proof as the one of Corollary 4.4 implies the following result. Under all the assumptions of Theorem 6.2 consider any λ, p, ψ as in its conclusions. Then for almost every t ∈ [0, 1] and all v ∈ R m satisfying p(t) + ψ(t), v = 0 we have p(t) + ψ(t), w ≤ 0 ∀ w ∈ T

FRemark 6 . 4 .

 64 (t,x(t)) ( ẋ(t), v). Let (x, p, ψ, µ, ν) be a normal extremal. Consider y ∈ V(1) (x) satisfying (30) withy(0) ∈ C K 0 (x(0)), y(1) ∈ C K 1 (x(1)), y(t) ∈ C K (x(t)) for all t ∈ [0, 1].Then conclusions similar to those of Remark 4.6 hold true.Using Theorem 6.1 and the same proof strategy as for Theorem 4.7 we find the following necessary optimality conditions for problem (56): Theorem 6.5. Assume (A), (IPC) and let x be a strong local minimizer of problem (56). Then, ∇ϕ(x(1)), y(1) ≥ 0 ∀ y ∈ V(1) (x) satisfying y(1) ∈ D K 1 (x(1)).

  Remark 3.2. Let K be as in Example 2.3. Then it is known that y ∈ T K (x) if and only if for all t ∈ [0, 1], ∇b

i (x(t)), y(t) ≤ 0 for all i ∈ I(x(t)) (the set of active indices at x(t)), see for instance

[START_REF] Frankowska | On relations of the adjoint state to the value function for optimal control problems with state constraints[END_REF] Remark 4.2]

. The situation is more complicated for T

  and p, ψ, µ, ν, λ are as in the maximum principle (Theorem 4.3). An extremal is normal if λ = 1. Note that (iv) of Theorem 4.3 is equivalent to p(t) + ψ(t), z -ẋ(t) ≤ 0 for all z ∈ F (t, x(t)) and a.e.

	t ∈ [0, 1]. In other words,
	p(t) + ψ(t) ∈ N F (t,x(t)) ( ẋ(t))	for a.e. t ∈ [0, 1].	(28)

Corollary 4.4

. Under all the assumptions of Theorem 4.3 consider λ, p, ψ as in its conclusions. Then for almost every t ∈ [0, 1] and all v ∈ L(t) satisfying p(t) + ψ(t), v = 0 we have

  u) is measurable and for almost all t ∈ [0, 1], f (t, •, •) is continuous, and f (t, x, U (t)) is closed for all t, x. Moreover, (t, x) ;f (t, x, U (t)) is locally bounded on [0, 1] × ∂K; (b) ∀ R > 0, there exists k R ∈ L 1 ([0, 1]; R + ) such that for a.e. t ∈ [0, 1], f (t, •, u) is k R (t)-Lipschitzon RB for all u ∈ U (t); (c) There exists a 1 ∈ L 1 ([0, 1]; R + ) such that for a.e. t ∈ [0, 1] and all x ∈ R n , The set-valued map U : [0, 1] ; R m is measurable with closed, nonempty images.

	sup	|f (t, x, u)| ≤ a 1 (t)(1 + |x|);
	u∈U (t)
	(d)

  Assume (H1) and that for some ε > 0, a 0 ∈ L 1 ([0, 1]; R + ) and for a.e. t ∈ [0, 1], f x (t, •, ū(t)) is Lipschitz on B(x(t), ε) with Lipschitz constant a 0 (t). Then for any solution y of (41), there exists h 0 > 0 such that for R := x ∞ + y ∞ and for every h ∈ [0, h 0 ] the following inequality holds true:distF (t,x(t)+hy(t)) ( ẋ(t) + h ẏ(t)) ≤ (2k R (t)R + a 0 (t)R 2 )h 2 a.e. Moreover, if for all t ∈ [0, 1], y(t) ∈ C K (x(t)) and int C K (x(t)) = ∅, then y ∈ V (1) (x).Proof. There exists 0 < h 0 < 1 such that for a.e. t ∈ [0, 1] and for all h ∈ [0, h 0 ], ẋ(t) + hf x [t]y(t) ∈ f (t, x(t) + hy(t), ū(t)) + a 0 (t)h 2 |y(t)| 2 B ⊂ F (t, x(t) + hy(t)) + a 0 (t)h 2 R 2 B (42)

	y(0) ∈ T K 0 (x(0)).	x(t))	for a.e. t ∈ [0, 1]	(41)
	Remark 5.2. It is not difficult to check that for a.e. t ∈ [0, 1], f x [t]y(t) ∈ d x F (t, x(t), ẋ(t))(y(t)). Thus
	by (7), any solution y ∈ W 1,1 ([0, 1]; R n ) of (41) satisfies ẏ(t) ∈ d x F (t, x(t), ẋ(t))(y(t)) a.e.
	Proposition 5.3. and
	ẋ(t)+v(t)-ẋ(t) ∈ F (t, x(t)) ⊂ F (t, x(t)+hy(t))+hk R (t)|y(t)|B ⊂ F (t, x(t)+hy(t))+hk R (t)RB. (43)

Multiplying (42) by 1 -h and (43) by h, adding them and using the convexity of F (t, x(t) + hy(t)), we obtain

ẋ(t) + hf x [t]y(t) + h(v(t) -ẋ(t)) ∈ h 2 f x [t]y(t) + F (t, x(t) + hy(t)) + a 0 (t)h 2 R 2 B + k R (t)h 2 RB,

and the first statement of our proposition follows. Lemma 2.6 completes the proof.

Corollary 5.4. Under all the assumptions of Proposition 5.3, suppose that K is sleek and that int T K (x(t)) = ∅ for every t ∈ [0, 1]. Then every solution y of

  Theorem 5.5 (Maximum Principle). Assume (H1) and let (x, ū) be a strong local minimizer of problem (45) such that int C K (x(t)) = ∅ for every t ∈ [0, 1]. If (39) holds true, then there exist p ∈ W 1,1 ([0, 1]; R n ), λ ∈ {0, 1}, a non-negative Borel measure µ on [0, 1] and a Borel measurable ν

  iv) p(t) + ψ(t), f (t, x(t), ū(t)) = max u∈U (t) p(t) + ψ(t), f (t, x(t), u) a.e. Remark 5.6. Note that in [26, Thm. 9.5.1] integrals are taken over [0, t[ in the definition of ψ. However, since ψ is of bounded total variation, it has only a countable number of jumps, thus (i) and (iv) remain both valid a.e. Moreover, (46) follows from [26, Thm. 9.5.1] since K = {ξ ∈ R n | dist K (ξ) ≤ 0} and by[START_REF] Vinter | Optimal Control[END_REF] Prop. 4.7.6, Thm. 4.8.5] we always have ∂dist K (ξ) ⊂ N K (ξ) for all ξ ∈ K. Here ∂ denotes the Clarke subdifferential.

* Partial support by the European Commission (FP7-PEOPLE-2010-ITN, Grant Agreement no. 264735-SADCO) is gratefully acknowledged.

Remark 5.8. Let (x, ū, p, ψ, µ, ν) be a normal extremal, y ∈ V (1) (x) satisfy (41), y(t) ∈ C K (x(t)) for all t ∈ [0, 1] and y(0) ∈ C K 0 (x(0)). Then ẏ(t) = f x [t]y(t) + v(t) -ẋ(t) for some integrable selection v(t) ∈ F (t, x(t)) a.e. By (iv) of Theorem 5.5 we have that for a.e. t ∈ [0, 1],

which implies that p(t) + ψ(t) ∈ L(t) -for almost every t. This in turn yields, 0 ≤ ν(t), y(t) dµ(t) -ψ(1), y

≤ -p(1) -ψ(1), y(1) + p(0), y(0) ≤ ∇ϕ(x(1)), y(1) .

(49)

If ∇ϕ(x(1)), y(1) = 0, then it follows from (49) and (iv) of Theorem 5.5 that p(t) + ψ(t), v(t) -f [t] = 0 a.e. In the same way as in Remark 4.6 we deduce that in this case for almost every t ∈ [0, 1],

Consider the sets

and the constrained control system

We deduce that if f (t, x(t), U (t)) = f (t, x(t), ū(t)) on a set of positive measure, then the following alternative holds true: either for every y(•) ∈ V (1) (x) satisfying (50) with f (•, x(•), u(•)) = f [•] on a set of positive measure we have ∇ϕ(x(1)), y(1) > 0, or f (t, x(t), U (t)) = {f (t, x(t), ū(t))} on a set of positive measure. Consequently, also U (t) = {ū(t)} on a set of positive measure. We can consider then such normal extremals as singular.

Example 5.9. Let f (t, •, •) be twice differentiable for all t ∈ [0, 1] and f (t, •, •), f (t, •, •) denote the derivative, respectively the Hessian of the map (x, u) → f (t, x, u). Assume that for some ε > 0,

and assume that there exists v

U (t) (x(t), u(t)) a.e. Then [START_REF] Hoehener | Variational approach to second-order optimality conditions for control problems with pure state constraints[END_REF]Prop. 4.2] states that for every h > 0 there exists vh

By the Taylor formula we find that

Hence the second-order approximation obtained in [START_REF] Hoehener | Variational approach to second-order optimality conditions for control problems with pure state constraints[END_REF] is a special case of the second-order approximation (40) introduced in this section. Consequently, in the case of strong local minimizers, the second-order necessary optimality conditions of Theorem 5.7 generalize those of [START_REF] Hoehener | Variational approach to second-order optimality conditions for control problems with pure state constraints[END_REF]Thm. 3.5].

Next we deduce from the above results a pointwise second-order necessary condition for optimality.

Consider the following "second-order linearization" of (38):

Theorem 5.10 (Second-Order Maximum Principle). Let (x, ū) be a strong local minimizer of problem (45) and (H1), (IPC), (39) hold true. If y ∈ V (1) (x) is such that ∇ϕ(x(1)), y(1) = 0 and V

C (x, y) = ∅, then for every solution w of (52) we have ∇ϕ(x(1)), w(1) ≥ 0. Furthermore, there exist λ ∈ {0, 1}, ψ ∈ N BV ([0, 1]; R n ) and p ∈ W 1,1 ([0, 1]; R n ) satisfying all the conclusions of Theorem 5.5 such that in addition max v∈dxF (t,x(t), ẋ(t))(y(t))

Finally, if the following linear system under state constraint has a solution:

then the above holds true with λ = 1.

C (x, y) and w be a solution of (52). Then it follows from Lemmas 2.4, 2.5 and Proposition 5.1 that w + w ∈ V

(2) C (x, y). Therefore, by Theorem 4.7 we have (37). In the same way as in the proof of Theorem 4.8 we deduce that ∇ϕ(x(1)), w(1) ≥ 0 for every solution w of (52). To complete the proof it is enough to apply the same arguments as in [25, pp. 358-361] with V (t) defined in [25, pp. 356] replaced by E(y; t). Corollary 5.11. Let (x, ū) be a strong local minimizer of Problem (45) and (H1), (IPC), (39) hold true. Let a solution y of (41) satisfy y ∈ V (1) (x), ∇ϕ(x(1)), y(1) = 0, V

C (x, y) = ∅ and consider p, ψ as in the conclusions of Theorem 5.10. Then for a.e. t ∈ [0, 1],

Theorem 6.7 ( [START_REF] Vinter | Optimal Control[END_REF]). Let (x, ū) be a strong local minimizer of problem (56) such that int C K (x(t)) = ∅ for every t ∈ [0, 1]. If (H1) and (39) hold true, then there exist λ, ψ, p, µ, ν as in Theorem 5.5, with (iii) replaced by -p(1) ∈ λ∇ϕ(x(1)) + ψ(1) + N K 1 (x(1)).

The second-order necessary optimality conditions for problem (56) are similar to those derived in Section 5.

Theorem 6.8. Let (x, ū) be a strong local minimizer of problem ( 56) and (H1), (IPC) be satisfied. Then, ∇ϕ(x(1)), y(1) ≥ 0 ∀ y ∈ V (1) (x) satisfying y(1) ∈ D K 1 (x(1)).

Moreover, if the maximum principle of Theorem 6.7 holds true with λ = 1 and some (p, ψ, µ, ν) and y ∈ V (1) (x) is such that ∇ϕ(x(1)), y(1) = 0, then for every w ∈ V

(2)

C (x, y, K 1 ) = ∅, then for every solution w of (52) satisfying w(1) ∈ int C K 1 (x(1)), we have ∇ϕ(x(1)), w(1) ≥ 0. Furthermore, there exist λ, ψ, p as in Theorem 6.7 such that in addition max v∈dxF (t,x(t), ẋ(t))(y(t)) p(t) + ψ(t), v = p(t) + ψ(t), ẏ(t) a.e. in [0, 1].

Finally, if (54) has a solution z with z(1) ∈ int C K 1 (x(1)), then the above holds true with λ = 1.

Corollary 6.10. Let (x, ū) be a strong local minimizer of problem (56) and (H1), (IPC), (39) hold true. If y is a solution of (41) such that y ∈ V (1) (x), ∇ϕ(x(1)), y(1) = 0, V

C (x, y, K 1 ) = ∅ and λ, p, ψ are as in the conclusions of Theorem 6.9. Then for a.e. t ∈ [0, 1], p(t) + ψ(t), f x [t]y(t) = max{ p(t) + ψ(t), k i=1 c i f x (t, x(t), u i )y(t) | c i ≥ 0, k i=1 c i = 1, u i ∈ U (t), f [t] = k i=1 c i f (t, x(t), u i )}.