M Carlavan
email: carlavan@i3s.unice.fr

L Blanc-Féraud

M Antonini

C Thiebaut
email: carole.thiebaut@cnes.fr

C Latry

Y Bobichon
email: yves.bobichon@thalesaleniaspace.com

GLOBAL RATE-DISTORTION OPTIMIZATION OF SATELLITE IMAGING CHAINS

INTRODUCTION

The joint compression/restoration optimization of a satellite imaging chain is a challenging problem which has been little investigated so far. Some works have been done in designing an optimal coding/decoding structure which takes into account the characteristics of the imaging chain [START_REF] Parisot | Optimization of the joint coding/decoding structure[END_REF], but, to the best of our knowledge, the study of the global system optimization has not devoted much work; so that each process is usually optimized separately.

Images acquired by imaging systems are most of the time degraded by blur and noise, which mainly comes from the imperfections of the optical instrument and the electronic chain. It is well-known that noise decreases the performances of coding schemes as it reduces the correlation between pixels [START_REF] Lo | Noise impact on error-free image compression[END_REF]. This problem is commonly referred as the noisy source coding problem [START_REF] Dobrushin | Information transmission with additional noise[END_REF]. Many works have been devoted to address this issue [START_REF] Ephraim | A unified approach for encoding clean and noisy sources by means of waveform and autoregressive model vector quantization[END_REF][START_REF] Wolf | Transmission of noisy information to a noisy receiver with minimum distortion[END_REF][START_REF] Fischer | Estimation and noisy source coding[END_REF] and [START_REF] Al-Shaykh | Lossy compression of noisy images[END_REF]. The majority of the mentioned works are based on the study of the global distortion optimization initially proposed by [START_REF] Wolf | Transmission of noisy information to a noisy receiver with minimum distortion[END_REF]. This study states that the global distortion, if measured by the Mean Square Error (MSE), can be treated as two separated problems. First, the original source image should be optimally, in the minimum MSE sense, estimated from the degraded data and this estimate should then be optimally coded [START_REF] Wolf | Transmission of noisy information to a noisy receiver with minimum distortion[END_REF].

However, an optimal estimation of the original image from the acquired one is usually not available. A suboptimal solution consists then in restoring the acquired image, before coding, and to encode this restored image [START_REF] Al-Shaykh | Lossy compression of noisy images[END_REF]. Although we will also consider the imaging chain where the restoration is performed on-board before coding, critical applications such as satellite imaging may not afford to insert this supplementary step due to the high limitation of on-board resources. Therefore, one has to focus on the current imaging chain which restores the image after coding, and the global distortion of this chain needs then to be modeled and optimized. This is the focus of this paper.

We are considering here the problem of optimal coding and restoration of a degraded image. We show that, under certain hypotheses that we will describe, a closed-form expression of the global distortion can be obtained. We propose then to optimize this distortion, with respect to the coding and the restoration parameters, to reach the minimal global distortion. The originality of the proposed approach relies on the fact that we propose a global optimization which takes into account all the parameters of the imaging chain. More precisely, we emphasize the necessity to take into account the restoration step in the rate-distortion allocation of the coder. And, as we will see, this requirement is confirmed by the results which display a significant improvement in comparison to the classical method which executes coding and restoration independently. The analysis of the global distortion allows us to also address the question of the position of the restoration in the chain (on-board versus on-ground) by comparing numerically the minimal distortion obtained in each case.

The paper is organized as follows. In section 2, we present the studied imaging chain and introduce hypotheses and notations. We detail in section 3 the proposed approach and we show how to get a closed-form expression of the global distortion for the studied case. We detail the optimization of this distortion in section 4 and we present the algorithm to get the optimal parameters of the chain. We show results, visually and in a rate-distortion sense, of the proposed optimization algorithm on a remote sensing image, in section 5. We conclude in section 6 and present perspectives for future works.

HYPOTHESES AND NOTATIONS

In the following, the operators applied to the image are denoted with a bold uppercase letter. The non-bold uppercase letters represent random variables whose realizations are denoted by a lowercase letter. With this notation, x is a realization of the random variable X. (X) i denotes the i-nth element of the random variable X. These variables are multidimensional x ∈ R N where N is the number of pixels. W x is a random variable associated to the wavelet transform of x and we note W x,j , j ∈ {0, . . . , J -1} (J being the number of subbands) the j-nth subband of the random variable W x . We have w x,j ∈ R Nj where N j is the size of the subband. Finally, we suppose that a wavelet subband w x,j follows a generalized centered Gaussian distribution law of parameter α wx,j and variance σ 2 wx,j [START_REF] Antonini | Image coding using wavelet transform[END_REF]. The parameters σ 2 wx,j and α wx,j of the distribution law will be estimated using the kurtosis-based technique proposed in [START_REF] Kasner | Universal trellis coded quantization[END_REF]. Note that the same assumption will be applied to all wavelet transforms in the chain. In this paper, we study the imaging chains shown figure 3 and4. The imaging chain represented figure 3 is the usual satellite imaging chain which performs the restoration on-ground after the transmission of the image. In contrast with this chain, we also study the imaging chain depicted figure 4 which restores the image on-board before the coding step.

We consider the special case of coding techniques based on wavelet transforms [START_REF] Antonini | Image coding using wavelet transform[END_REF]. The coding step is then approximately decomposed in a non-redundant wavelet transform followed by a scalar subband quantizer. We also consider that the denoising step is performed in the same wavelet basis than the coding. This choice may however need further explanations. Usually, an efficient wavelet transform for image denoising strongly differs from a wavelet transform suited for image coding. Image denoising techniques actually require redundant wavelet transforms to represent the characteristics of an image such as contours and oriented details while increasing the number of coefficients in image compression may be problematic [START_REF] Chappelier | Oriented wavelet transform for image compression and denoising[END_REF]. Hence, a non-redundant wavelet transform used for image compression leads most of the time to poor denoising results. We are however very confident that using the same basis for both coding and denoising may provide an optimized coding-denoising structure gathered in a single fast and low-ressources algorithm. Extending the current work to complex denoising schemes such as [START_REF] Donoho | De-noising by soft-thresholding[END_REF] is a difficult task that will be addressed in future works.

Based on these considerations, the studied imaging chains are represented in detail on figures 3 and 4. Due to the space limitation, we only detail the proposed method for the on-ground chain shown figure 3. However, the extension of this work to the on-board chain displayed figure 4 is trivial and we will directly present the obtained results. In the chain displayed figure 3, we consider the instrumental noise z to be independent, identically distributed and to follow a centered normal distribution with variance σ 2 z . W is the Cohen-Daubechies-Feauveau (CDF) 9/7 wavelet transform [START_REF] Cohen | Biorthogonal bases of compactly supported wavelets[END_REF], W its inverse. Each quantized subband w ỹ,j will be coded using an entropy encoder. As this operation does not introduce any

where λ j > 0 is a regularizing parameter. The restoration algorithm (1) has a closed-form solution which writes

w x,j = w ỹ,j 1 + λ j . (2)
We are aware of the simplicity of the considered algorithm, it appears however that the linearity of the restoration algorithm R is required if one wants to write the global distortion in closed-form. As mentioned previously, much work need to be addressed to consider state-of-art denoising algorithms.

The quantizer Q is an infinite scalar subband quantizer of step ∆ j > 0 and is modeled as

Q(w y,j ) = ∆ j w y,j ∆ j + 1 2 , (3)
where ⌊ ⌋ is the floor function which returns the greatest integer less than or equal to its argument. We now detail the basis of the proposed method to compute a closed-form expression of the global distortion. Let w b,j be the coding error of the subband j w b,j = Q(w y,j )w y,j .

We have w ỹ,j = w y,j + w b,j = w x,j + w ǫ,j ,

where w ǫ,j = w z,j + w b,j . The main hypothesis of the proposed method is to consider the moments of the term w ǫ,j to be independent to the ones of w x,j . This hypothesis is mainly based on the fact that the quantizing part of the scheme figure 3 can be seen as a non-substractive dithering system where the Gaussian instrumental noise acts as a dithering noise. And, as mentioned in [START_REF] Wannamaker | A theory of nonsubtractive dither[END_REF], a non-substractive dithering system allows the moments of the global error (that is the sum of the coding and instrumental errors) to be decorrelated to the moments of the coding source. More precisely, we have [START_REF] Wannamaker | A theory of nonsubtractive dither[END_REF]

E [W ǫ,j W x,j ] = E [W ǫ,j ] E [W x,j ] , (6)
E [W ǫ,j ] = 0, (7)
E W ǫ,j 2 = N j σ 2 wz,j + N j ∆ 2 j 12 , (8)
where σ wz,j is the standard deviation of the distribution law of the wavelet transform w z,j . From [START_REF] Vanderkooy | Dither in digital audio[END_REF] we know that a Gaussian noise effectively owns the properties of a dither noise if the standard deviation of its distribution law is large enough. In the present case, this dithering hypothesis will be verified if the following statement is true

σ wz,j > ∆ j 2 . ( 9
)
As the standard deviation of instrumental noise is usually low in imaging systems, the condition (9) assumes that the proposed approach will be valid only for aymptotic coding rates, i.e. high coding rates. We will however develop our method to consider all coding rates.

GLOBAL RATE-DISTORTION ANALYSIS

As mentioned in the section 2, the studied imaging chain depends on two sets of parameters: The regularizing parameters λ j in (2) and the quantizing steps ∆ j in (3), for each j ∈ {0, . . . , J -1}. The global rate-distortion optimization problem consists in finding the optimal parameters λ * j and ∆ * j which minimize the global distortion D under the constraint that the coding rate R does not exceed the target rate R c . This can be formalized as the following unconstrained problem

Find λ * j , ∆ * j = arg min D(λ j , ∆ j ) + τ (R(λ j , ∆ j ) -R c ) subject to λ j > 0, ∆ j > 0, τ > 0 , ( 10
)
where τ is a Lagrange multiplier. To solve the global distortion optimization problem [START_REF] Chappelier | Oriented wavelet transform for image compression and denoising[END_REF], we then need to express the global distortion D and the global coding rate R as a function of the regularizing parameters λ j and the quantizing steps ∆ j . We start from the fact that the (mean) global distortion writes

D = 1 N E X -X 2 , ( 11
)
where X is the random variable associated to the output final image x. Thanks to the orthogonality of the wavelet subbands, the global distortion can also be formulated as

D = 1 N J-1 j=0 π j E W x,j -W x,j 2 , ( 12
)
where π j are weighting coefficients as the considered wavelet transform is bi-orthogonal [START_REF] Usevitch | Optimal bit allocation for biorthogonal wavelet coding[END_REF]. In the case of the studied imaging chain displayed figure 3, the final image can be expressed as a function of the source and the global error. Using equations ( 2) and ( 5), it writes

w x,j = w x,j 1 + λ j + w ǫ,j 1 + λ j . (13)
From equations ( 12), ( 13) and using the moments decorrelation hypothesis (6), we deduce the global distortion

D = 1 N J-1 j=0 π j λ 2 j (1 + λ j ) 2 E W x,j 2 + π j (1 + λ j ) 2 E W ǫ,j 2 . (14)
Finally, the global distortion ( 14) can be further developed using the results (8)

D = J-1 j=0 π j a j λ 2 j (1 + λ j ) 2 σ 2 wx,j + π j a j (1 + λ j ) 2 σ 2 wz,j + π j a j (1 + λ j ) 2 ∆ 2 j 12 , (15)
where a j = Nj N is the weight of the subband j in the whole image. Note that the global distortion [START_REF] Usevitch | Optimal bit allocation for biorthogonal wavelet coding[END_REF] requires the knowledge of the variance of each subband of the original image σ 2 wx,j . This variance is generally unknown but can be approximatively computed during the rate-allocation of the coder from the observed subband variance σ 2 wy,j , as the variance of the instrumental noise is assumed to be known. The second part of the problem [START_REF] Chappelier | Oriented wavelet transform for image compression and denoising[END_REF] requires the expression of the global coding rate R. This rate can be expressed as the weighted sum of the rate in each subband

R j R = J-1 j=0 a j R j (∆ j ). (16)
As mentioned in the hypotheses section, we assume that each quantized subband is encoded using an entropy encoder. Then, in a general non-asymptotic case, the coding rate of a subband j can be estimated by its entropy [START_REF] Shannon | A mathematical theory of communication[END_REF] R j (∆ j ) = -+∞ m=-∞ P wy,j (m, ∆ j ) log 2 P wy,j (m, ∆ j ) ,

where P wy,j (m, ∆ j ) is the probability of a coefficient of the quantized subband w ỹ,j to be equal to the symbol m. The same analysis can be applied to the on-board chain figure 4 to modelize the global distortion. It writes

D = J-1 j=0 π j a j λ 2 j (1 + λ j ) 2 σ 2 wx,j + π j a j (1 + λ j ) 2 σ 2 wz,j + π j a j ∆ 2 j 12 . ( 18
)
We detail in the next part how to solve the optimization problem of the on-ground global rate-distortion.

GLOBAL RATE-DISTORTION OPTIMIZATION

Using equations ( 15) and ( 16), the optimization problem [START_REF] Chappelier | Oriented wavelet transform for image compression and denoising[END_REF] becomes Find ∆ * j , λ * j = arg min φ(∆ j , λ j , τ ) subject to λ j > 0, ∆ j > 0, τ > 0

. ( 19
)
where

φ(∆ j , λ j , τ ) = J-1 j=0 π j a j λ 2 j (1 + λ j ) 2 σ 2 wx,j + π j a j (1 + λ j ) 2 σ 2 wz,j + π j a j ∆ 2 j 12(1 + λ j ) 2 + τ   J-1 j=0 a j R j (∆ j ) -R c   . (20)
The theoretical study of existence and uniqueness of solutions of problem ( 19) is an open problem and we explicitly assume that a minimum of problem [START_REF] Gish | Asymptotically efficient quantizing[END_REF] exists and is unique. We propose a numerical algorithm to find this minimum. This algorithm is based on the resolution of the simultaneous equations obtained from the Karush-Kuhn-Tucker (KKT) conditions [START_REF] Kuhn | Nonlinear programming[END_REF] of problem [START_REF] Gish | Asymptotically efficient quantizing[END_REF]. The KKT conditions of problem [START_REF] Gish | Asymptotically efficient quantizing[END_REF] write

         ∂φ(∆ * j ,λ * j ,τ * ) ∂∆j = πj ∆ * j 6(1+λ * j ) 2 + τ * ∂Rj ∂∆j (∆ * j ) = 0 ∂φ(∆ * j ,λ * j ,τ * ) ∂τ = J-1 j=0 a j R j (∆ * j ) -R c = 0 ∂φ(∆ * j ,λ * j ,τ * ) ∂λj = 12λ * j σ 2 w x,j -12σ 2 w z,j -∆ * j 2 6(1+λ * j ) 3 = 0 (21) with ∂R j ∂∆ j (∆ j ) = - 1 log(2) +∞ m=-∞ 1 + log P wy,j (m, ∆ j ) × p wy,j m∆ j + ∆ j 2 m + 1 2 -p wy,j m∆ j - ∆ j 2 m - 1 2 . ( 22
)
We deduce from (21) that the optimization problem (19) admits a solution (λ * j , τ * , ∆ * j ) which verifies [START_REF] Carlavan | An optimized structure for joint noisy image coding/denoising[END_REF] λ

* j = σ 2 wz,j σ 2 wx,j + ∆ * j 2 12σ 2 wx,j , (23)
π j ∆ * j 6(1 + λ j ) 2 + τ * ∂R j ∂∆ j (∆ * j ) = 0, (24)
J-1 j=0

a j R j (∆ * j ) = R c . (25)
As shown by equations ( 24) and (25), the optimal quantizing step ∆ * j and the optimal Lagrange multiplier τ * can not be expressed in closed-form. These parameters can however be computed using any root-finding algorithm [START_REF] Carlavan | An optimized structure for joint noisy image coding/denoising[END_REF]. The case of the low frequency subband (j = J -1) will be treated differently as we do not quantize it. We will set ∆ * J-1 = 1 and will deduce λ * J-1 from (23). The overall optimization procedure for solving problem [START_REF] Chappelier | Oriented wavelet transform for image compression and denoising[END_REF] is given in the algorithm 1. The algorithm 1 intends to be quite general and we let the choice of the root-finding algorithm to the user (we note ρ the precision of this algorithm).

RESULTS

We simulate the optimization algorithm 1 on the high-dynamic range remote sensing image displayed figure 5. For this simulation, the reference image has been noised with an additive centered Gaussian noise with a standard deviation equal to 10. For each target rate, we simulate the imaging chains figures 3 and 4 with the common choice of selecting the quantizing steps and the regularizing parameters such that the coding and the restoration errors are independently minimized. The coding error minimization has been achieved using the rate-distortion allocation based model proposed in [START_REF] Parisot | Optimization of the joint coding/decoding structure[END_REF]. As for the restoration error, it has

Fig. 1 .

 1 Fig. 1. On-ground satellite imaging chain.

Fig. 2 .

 2 Fig. 2. On-board satellite imaging chain.

Fig. 3 .Fig. 4 . 2 2 + λ j w 2 2

 3422 Fig. 3. On-ground considered imaging chain

Fig. 6 .

 6 Fig. 6. Visual comparison of reconstruction results. From left to right: Reference image, observed image, image reconstructed using the separated optimization of the on-ground distortion, image reconstructed using the joint optimization of the on-ground distortion model, image reconstructed using the separated optimization of the on-board distortion and image reconstructed using the joint optimization of the on-board distortion model. The coding rate is 2.5 bits/pixel. The image range has been extended to point up the image reconstruction artifacts.

Algorithm 1 Global rate-distortion optimization algorithm

Set τ = 1 and ρ = 0.1. while J-1 j=0 a j R j -R c > ρ do for j from 0 to J -2 do Set ∆ j = 1.

Compute the value of the regularizing parameter λ j from (23).

while πj ∆j 6(1+λj ) 2 + τ ∂Rj ∂∆j (∆ j ) > ρ do Increase the value of ∆ j . Compute the value of the regularizing parameter λ j from (23). end while end for Set ∆ J-1 = 1. Compute the regularizing paramater λ J-1 from (23). if

Increase the value of τ . end if end while Output the optimal regularizing parameters λ * j and the optimal quantizing steps ∆ * j .

been minimized using an exhaustive search of the optimal regularizing parameters. Once the final image has been reconstructed using these parameters, we compute numerically the global distortion

where x is the reference image (assumed to be known in our numerical experiments) and x is the final image. The proposed model of distortion [START_REF] Usevitch | Optimal bit allocation for biorthogonal wavelet coding[END_REF] has then been computed with the same values of parameters to ensure its validity. And finally, we use the proposed algorithm 1 to compute the joint optimization. The obtained parameters have then been inserted into the theoretical model of distortion [START_REF] Usevitch | Optimal bit allocation for biorthogonal wavelet coding[END_REF] to compute the minimal distortion. The same comparison has been applied to the on-board imaging chain displayed figure 4.

The resulting rate-distortion curves are given figure 5. We see that the proposed model approximates well the true distortion if the coding rate is around 3 bits/pixel and more, see figure 5. It is however not valid below this rate. This can explained by the fact that below this rate, the condition ( 9) is not respected anymore and that the global error can not consequently be considered decorrelated to the source. This bound may be improved by considering a more realistic model for the variance of the global error (8) rather than the classical ∆ 2 j 12 term used for asymptotic rate hypotheses [START_REF] Gish | Asymptotically efficient quantizing[END_REF]. Even if it does not solve the correlation issue, refining the model may allow to fit the true distortion for lower rates. This will be the subject of future works. Regarding to the on-board/on-ground comparison, it seems from the results that the on-ground chain always gives better results, at least for the studied case. It should be noted however that, in the case of the on-board chain, the coding noise is not processed. An "hybrid" chain, with one restoration on-board and one on-ground, may then give interesting results. This will also be the suject of future works.

Visual results for the target rate of 2.5 bits/pixel are given figure 6 for different parts of the image. We do not focus on the quality of the reconstructed images regarding to the reference one as the considered chain is excessively simple. Clearly, the presence of artifacts on the reconstructed image is due to the simple hypothesis that we made on the restoration algorithm (1). On the contrary, we are more concerned on the improvement of the image quality of the optimized chain with respect to the non-optimized one. We can see that the global optimization of the chain always lead to a reconstructed image witch contains less blurry edges or ringing artifacts. This is particularly visible on the boundaries of objects (cars for example). As mentioned in the introduction of this paper, the obtained results clearly point that optimizing coding and restoration separately is suboptimal. One needs instead to address the problem of imaging chain design in its globality; the proposed method and the obtained results are encouraging in this sense. A lot of works is however required to extend the proposed method to lower coding rates and to more complex restoration techniques.

CONCLUSIONS

In this paper we considered the problem of satellite imaging chain optimization. Most of the time, the coding and the restoration parameters are selected independently such that the coding and the restoration error are respectively minimized. As shown in this paper, this parameters selection technique leads however to a suboptimal distortion. It appears then crucial to address the problem of joint coding/denoising in its globality. We proposed here a technique to modelize the global distortion and we presented an algorithm to get the optimal coding and denoising parameters. We simulated this optimization technique on a highdynamic range remote sensing image. We concluded that our joint coding/denoising optimization approach really improves the quality of the reconstructed final image, in comparison to the image obtained using the classical parameters selection technique. Further works will be focussed on the extension of the proposed model to lower coding rates and to advanced restoration methods.


Francesca C Chittaro

Frédéric Jean

Paolo Mason

P Mason Udc ????

On the inverse optimal control problems of the human locomotion: stability and robustness of the minimizers

In recent papers models of the human locomotion by means of an optimal control problem have been proposed. In this paradigm, the trajectories are assumed to be solutions of an optimal control problem whose cost has to be determined. The purpose of the present paper is to analyze the class of optimal control problems defined in this way. We prove strong convergence result for their solutions on the one hand for perturbations of the initial and final points (stability), and on the other hand for perturbations of the cost (robustness).

Introduction

An important question in the study of human motor control is to determine which law governs a particular body movement, such as arm pointing motions or goal-oriented locomotion. A nowadays widely accepted paradigm in neurophysiology is that, among all possible movements, the accomplished ones satisfy suitable optimality criteria (see [START_REF] Todorov | Optimal control theory[END_REF] for a review). Once a dynamical model of the movements under consideration is given, one is then led to solve an inverse optimal control problem: given recorded experimental data, infer a cost function such that the recorded movements are solutions of the associated optimal control problem.

In the theory of linear-quadratic control, the question of which quadratic cost is minimized in order to control a linear system along certain trajectories was already raised by R. Kalman [START_REF] Kalman | When is a linear control system optimal?[END_REF]. Some methods allowed deducing cost functions from optimal behavior in system and control theory (linear matrix inequalities [START_REF] Boyd | Linear matrix inequalities in system and control theory[END_REF]) and in Markov decision processes (inverse reinforcement learning [START_REF] Ng | Algorithms for inverse reinforcement learning[END_REF]). A new and promising approach has been developed in [START_REF] Berret | A biomechanical inactivation principle[END_REF][START_REF] Darlot | The inactivation principle: Mathematical solutions minimizing the absolute work and biological implications for the planning of arm movements[END_REF] for the pointing movements of the arm. In that approach, based on Thom transversality theory, the cost structure is deduced from qualitative properties highlighted by the experimental data. However all these methods have been conceived for very specific systems and they are not suitable for general inverse optimal control problems. This paper focuses on the inverse optimal control problem associated with the goal-oriented human locomotion. In the approach initiated in [START_REF] Arechavaleta | Optimizing principles underlying the shape of trajectories in goal oriented locomotion for humans[END_REF][START_REF] Arechavaleta | On the nonholonomic nature of human locomotion[END_REF][START_REF] Arechavaleta | An optimality principle governing human walking[END_REF], goal-oriented human locomotion is Fig. 1. Goal-oriented human locomotion. understood as the motion in the 3-D space of both its position (x, y) and its orientation θ of a person walking from an initial point (x 0 , y 0 , θ 0 ) to a final point (x 1 , y 1 , θ 1 ) (see Figure 1).

We take advantage here of the analysis carried out in a previous work [START_REF] Chitour | Optimal control models of the goal-oriented human locomotion[END_REF] (see also [START_REF] Bayen | Asymptotic analysis of an optimal control problem connected to the human locomotion[END_REF]) in modeling the human locomotion by means of optimal control problems. We assume that every admissible locomotion trajectory is solution of a control system that describes the dynamics, exploiting the nonholonomic nature of human locomotion [START_REF] Arechavaleta | On the nonholonomic nature of human locomotion[END_REF]. Among these trajectories, the chosen one minimizes a cost that has to be sought in a particular class L of functionals. Thus, with each cost function in L, one can associate the set of solutions of the corresponding optimal control problem. The inverse optimal control problem consists in determining an inverse of this mapping. From a practical point of view, it amounts to find in L a cost whose minimizing trajectories fit accurately experimental data (see Figure 2). To this purpose Mombaur et al. [START_REF] Mombaur | From human to humanoid locomotionan inverse optimal control approach[END_REF] proposed a purely numerical approach based on parameter identification that is used to implement human-like motion on humanoid robots. This method furnishes a plausible solution but does not give further insight into the considered inverse optimal control problem.

The purpose of the present paper is to analyze the direct problem, and in particular to understand the behavior of minimizing trajectories in function of the cost. This qualitative analysis represents a step forward a global qualitative and numerical analysis of the inverse optimal control problem. We will show that the solutions of the optimal control problem corresponding to costs in L depend continuously on the one hand on the initial and final points (stability), and on the other hand on the cost (robustness). The stability of the minimizers, as well as their existence, are properties that arise naturally in the modeling of a biological motion. The fact that they are satisfied here only enhance the relevance of our model. The robustness has more consequences for the associated inverse optimal control problem. Due to the experimental nature of the recorded data, it implies that if a cost is solution of the inverse control problem, then its small enough perturbations are solutions too. Thus the class of possible costs can be reduced to any dense subclass and, in order to further reduce the class of costs in the spirit of [START_REF] Berret | A biomechanical inactivation principle[END_REF][START_REF] Darlot | The inactivation principle: Mathematical solutions minimizing the absolute work and biological implications for the planning of arm movements[END_REF], we can look for properties of the minimizers that are only approximately satisfied. We intend to exploit this property in forthcoming studies. The optimal control problems modeling the locomotion are described in Section 2. Two types of models are given: in the first one the control is the angular acceleration θ, in the second one it is the angular velocity θ. We also give in this section the equations resulting by the application of the Pontryagin maximum principle.

The existence and stability results are given in Section 3. We first prove the existence of minimizers and the continuity of the value function. Then we show that, for a converging sequence of boundary conditions, the corresponding minimizers and their adjoint vectors uniformly converge.

Section 4 is dedicated to the proof of robustness. We first show that, if a sequence of costs depending on θ converges to a cost also depending on θ, then the minimizers with given terminal conditions and their adjoint vectors uniformly converge. We then prove the same result when the limit of the sequence of costs does not depend on θ but only on θ.

The models and the adjoint equations

2.1. The models of human locomotion. We present in this section the model of the human locomotion developed in [START_REF] Chitour | Optimal control models of the goal-oriented human locomotion[END_REF]. The dynamics are described by control systems of the form

     ẋ = cos θ, ẏ = sin θ, θ (k) = u, (1)
where k > 0 is an integer. System (1) captures the nonholonomic nature of the human locomotion pointed out in [START_REF] Arechavaleta | On the nonholonomic nature of human locomotion[END_REF], which is valid when the subject walks toward a far enough target point. Note also that the paths (x, y) are parameterized by arc-length. The locomotion trajectories between two arbitrary configurations (x 0 , y 0 , θ 0 ) and (x 1 , y 1 , θ 1 ) appear as the solutions of an optimal control problem of the following form: minimize the cost J = T 0 L( θ, . . . , θ (k-1) , u)dt among all admissible controls u(•) steering system (1) from q(0) = (x 0 , y 0 , θ 0 ) to q(T ) = (x 1 , y 1 , θ 1 ). Notice that the final time T and the boundary values of θ (m) , m = 1, . . . , k -1, are free.

The inverse optimal control problem consists in finding the proper cost function L. Of course it is necessary to precise the class L of functionals in which L has to be sought. This class contains different types of functionals depending on the integer k characterizing system [START_REF] Agrachev | Control theory from the geometric viewpoint[END_REF].

The first task is to understand how the integer k affects the structure of the corresponding set of optimal trajectories.

Keeping in mind that the cost should model a human behavior, and that the angular acceleration θ is equivalent to a third-order derivative (so-called jerk ) of the planar path, we make the rather natural assumption that k ≤ 2 in the sequel (see also [START_REF] Arechavaleta | Optimizing principles underlying the shape of trajectories in goal oriented locomotion for humans[END_REF][START_REF] Chitour | Optimal control models of the goal-oriented human locomotion[END_REF]). Thus in the following the class L will be divided into two sub-classes L 1 and L 2 that depend on whether the highest derivative of the angle θ explicitely appearing in the cost is the first or the second one.

We first define the class L 2 as the set of functions L : R 2 → R which satisfy the following three assumptions:

(H1) L(•, •) is non-negative, C 2 in
the pair, and satisfies L(0, 0) = 0. Moreover L(r, 0) ≤ L(r, s) for every (r, s) ∈ R 2 ; (H2) L(•, •) is strictly convex with respect to the second variable, with ∂ 2 L ∂s 2 (r, s) > 0 for every (r, s); (H3) there exist two constants C, R > 0 and p > 1 such that

L(r, s) ≥ C|s| p ∀ r ∈ R, ∀|s| > R. Moreover, there exist two L ∞ loc functions γ, δ : R → R, γ > 0 such that ∂L ∂r (r, s) ≤ γ(r)|s| p + δ(r). (2)
The integral cost associated with a function L ∈ L 2 is

J L (T, κ 0 , u) = T 0 1 + L(κ(t), u(t)) dt,
and has to be evaluated along the trajectory

Q(•) = (x(•), y(•), θ(•), κ(•)) of the control system            ẋ = cos θ ẏ = sin θ θ = κ κ = u , (3)
with initial condition (0, 0, 0, κ 0 ) and corresponding to the control function u(•). The corresponding optimal control problem is

P 2 (L) Minimize J L (T, κ 0 , u) among all controls u ∈ L p ([0, T ]
) and all values κ(0) = κ 0 such that the corresponding trajectories Q(•) satisfy (3) with boundary conditions q 0 = (x(0), y(0), θ(0)) = (0, 0, 0) and q 1 = (x(T ), y(T ), θ(T )) = (x 1 , y 1 , θ 1 ). The final time T and the initial and final values of the curvature, that is κ(0) = κ 0 and κ(T ), are free. Note that the model is valid only when the target (x 1 , y 1 ) is far enough from the initial position (0, 0), so we will always assume (x 1 , y 1 ) = 0.

Remark 1. In [START_REF] Chitour | Optimal control models of the goal-oriented human locomotion[END_REF], for the need of the analysis, the cost functions L are assumed to depend separately on κ and u. We remove this unnecessary assumption here. Hypothesis (H3) then writes in a slightly different -and more general -form.

The second class of costs we consider is the one in which θ and higher derivatives of θ do not appear. We define this class L 1 as the set of functions ℓ : R → R which satisfy the following three assumptions:

(H1 ′ ) ℓ(•) is a non-negative C 2 function satisfying ℓ(0) = 0; (H2 ′ ) ℓ(•) is strictly convex, with ℓ ′′ (r) > 0 for every r; (H3 ′ ) there exist two constants C, R > 0 and p > 1 such that ℓ(r) ≥ C|r| p ∀|r| > R.
The integral cost associated with a function ℓ ∈ L 1 is

J ℓ (T, κ) = T 0 1 + ℓ(κ(t)) dt,
and has to be evaluated along the trajectories q(•) = (x(•), y(•), θ(•)) of the control system

     ẋ = cos θ ẏ = sin θ θ = κ , ( 4
)
where κ is the control. The corresponding optimal control problem is

P 1 (ℓ) Minimize J ℓ (T, κ) among all controls κ(•) ∈ L p ([0, T ]
) such that the corresponding trajectories q(•) satisfy (4) with free final time T and boundary conditions q 0 = (x(0), y(0), θ(0)) = (0, 0, 0) and q 1 = (x(T ), y(T ), θ(T )) = (x 1 , y 1 , θ 1 ).

Remark 2. With a little abuse of notations, when speaking about solutions of P 2 (L) and P 1 (ℓ) we mean the corresponding optimal trajectories.

Finally the class L is defined as the union L 1 ∪ L 2 . Thus with every function L or ℓ in L is associated the optimal synthesis of P 2 (L) or P 1 (ℓ), that is the set of solutions of this problem for all q 1 ∈ R 2 × S 1 . The inverse optimal control problem of the goal-oriented locomotion consists in finding an inverse to this mapping.

The main purpose of this paper, as a first step toward the analysis of this inverse problem, is to show for that mapping a continuity property that we call robustness (Section 4). We will also prove in Section 3 the existence of the solutions P 2 (L) and P 1 (ℓ), and their continuity with respect to q 1 . Both parts make use of the equations of the Pontryagin maximum principle that we state now.

Application of the Pontryagin Maximum

Principle. The Pontryagin maximum principle (PMP) establishes some first-order necessary conditions to be satisfied by every optimal trajectory. However, the PMP in its classical form [START_REF] Agrachev | Control theory from the geometric viewpoint[END_REF][START_REF] Pontryagin | The mathematical theory of optimal processes[END_REF] requires that the optimal control is bounded in the L ∞ topology, which is an information that we do not possess at this stage. Nevertheless, refined versions of the PMP deal also with unbounded controls. In particular it is easy to check that our problems meet all the hypotheses required in [START_REF] Arutyunov | A simple 'finite approximations' proofs of the Pontryagin maximum principle under reduced differentiability hypotheses[END_REF]Theorem 2.3], and thus we can state the following.

Proposition 1. Every solution of P 2 (L) (resp. P 1 (ℓ)) satisfies the PMP.

Remark 3. The PMP does not actually guarantee the existence of optimal trajectories. In Section 3 we prove the existence of solutions for every final point q 1 , both for problem P 1 (ℓ) and P 2 (L).

In order to apply the PMP to the problem P 2 (L), we define the following Hamiltonian function:

H(Q, P, ν, u) = p x cos θ + p y sin θ + p θ κ + p κ u + ν(1 + L(κ, u)),
where P = (p x , p y , p θ , p κ ) ∈ R 4 and ν ∈ R.

We recall that the PMP states that if Q(•) is an optimal trajectory corresponding to a control u(•) defined on an interval [0, T ], then there exist an absolutely continuous function P (•) : [0, T ] → R 4 and ν ≤ 0 such that the pair (P (•), ν) is non-trivial and (Q(•), P (•)) satisfies the Hamiltonian system

Q(t) = ∂H ∂P (Q(t), P (t), ν, u(t)) Ṗ (t) = -∂H ∂Q (Q(t), P (t), ν, u(t)) . (5)
Moreover, the PMP also states that the n-tuple (Q(•), P (•), ν, u(t)) satisfies the maximality condition

H(Q(t), P (t), ν, u(t)) = max v∈R H(Q(t), P (t), ν, v) for t ∈ [0, T ].
This condition establishes a relation between the optimal control u(•), κ(•) and p κ (•):

p κ (t) = -ν ∂L ∂u (κ(t), u(t)) (6) for t ∈ [0, T ].
As the final time is free, the Hamiltonian is identically zero along the optimal trajectory, i.e. H(Q(t), P (t), ν, u(t)) ≡ 0. Since the values κ(0) and κ(T ) are free, we get also the following transversality conditions:

p κ (0) = p κ (T ) = 0.
The equation for Q in ( 5) is nothing but (3), while the equation for the adjoint vector P , also called the adjoint equation, can be explicitly rewritten as

           ṗx = 0 ṗy = 0 ṗθ = p x sin θ -p y cos θ ṗκ = -p θ -ν ∂L ∂κ (7)
If ν = 0, we can always normalize it by putting ν = -1. In this case, we say that the pair (Q(•), P (•)) is a normal extremal, while if ν = 0 the pair (Q(•), P (•)) is said to be an abnormal extremal. Thanks to Remark 4 below, in the following we always set ν = -1.

Remark 4. The problem P 2 (L) does not admit abnormal extremals. Indeed, from the maximality condition applied to the Hamiltonian H(Q, P, 0, u) = p x cos θ + p y sin θ + p θ κ + p κ u we obtain that p κ ≡ 0, and then, from ṗκ = -p θ , that p θ ≡ 0. Since ṗθ = p x sin θ -p y cos θ ≡ 0 and H(Q, P, 0, u) = p x cos θ + p y sin θ ≡ 0, we get that the whole adjoint vector vanishes, contradicting the PMP. Moreover, we deduce that with any solution Q(•) of P 2 (L) it is associated one and only one normal extremal trajectory

(Q(•), P (•)). Indeed, if (Q(•), P 1 (•)) and (Q(•), P 2 (•)) were two normal extremals (with ν = -1), then it is easy to check that (Q(•), P 1 (•) -P 2 (•)) would be an abnormal extremal.
Notice that the absence of abnormal extremals, together with [START_REF] Bayen | Asymptotic analysis of an optimal control problem connected to the human locomotion[END_REF], implies that all the pairs (Q(•), P (•)) satisfying the PMP are C 1 .

Concerning the problem P 1 (ℓ), its associated Hamiltonian is

H(q, p, ν, κ) = p x cos θ + p y sin θ + p θ κ + ν(1 + ℓ(κ)),
where p = (p x , p y , p θ ) ∈ R 3 and ν ≤ 0, and the Hamiltonian equation for the pair (q(•), p(

•)) is q(t) = ∂H ∂p (q(t), p(t), ν, κ(t)) ṗ(t) = -∂H ∂q (q(t), p(t), ν, κ(t)) . (8)
The n-tuple (q(•), p(•), ν, κ(•)) satisfies the maximality condition

H(q(t), p(t), ν, κ(t)) = max v∈R H(q(t), p(t), ν, v)
for t ∈ [0, T ], and, as the final time is free, the Hamiltonian is zero along the optimal trajectory: H(q(t), p(t), ν, κ(t)) ≡ 0. The maximality condition links p θ with κ:

p θ (t) = -νℓ ′ (κ(t)) (9)
for t ∈ [0, T ]. We notice that in this case there are no transversality conditions. The adjoint equation is

     ṗx = 0 ṗy = 0 ṗθ = p x sin θ -p y cos θ
We remark that also the problem P 1 (ℓ) does not admit abnormal extremals, so that we set, without loss of generality, ν = -1.

Existence and stability of the optimal solutions

Given a function L or ℓ in L, with every terminal point q 1 ∈ R 2 × S 1 one associates the corresponding solutions of P 2 (L) or P 1 (ℓ). The aim of this section is to study this mapping; we first prove that it is well-defined (existence of solutions of P 2 (L) or P 1 (ℓ)), and then that it is continuous with respect to q 1 (stability of the solutions). Namely, we will show that when a sequence of final points converges to a point q 1 , any sequence of corresponding optimal trajectories converges uniformly to the set of optimal trajectories having q 1 as final point.

3.1. Existence of optimal solutions. First of all, we notice that the set of trajectories of (3) (resp. ( 4)) satisfying the boundary conditions of problem P 2 (L) (resp. P 1 (ℓ)) is non-empty. This is a straightforward consequence of classical controllability results (see for instance [START_REF] Agrachev | Control theory from the geometric viewpoint[END_REF]). Concerning problem P 2 (L), the main result of this section, stated below, guarantees that, among these trajectories, there always exists a minimizer.

Theorem 1. For every choice of q 1 = (x 1 , y 1 , θ 1 ) ∈ R 2 × S 1 such that (x 1 , y 1 ) = (0, 0) there exist T > 0, a control function ū ∈ L p ([0, T ]
) and an initial value κ0 such that the corresponding trajectory Q(•) solves problem P 2 (L).

In order to prove the previous theorem we will make use of a standard result in calculus of variation, and a technical lemma, stated below.

Theorem 2 ([11, Theorem 1.3]). Let n, N ∈ N, p ≥ 1, Ω ⊂ R n be a bounded open set with Lipschitz boundary, f : Ω × R N × R N ×n →
R be a non-negative continuous function and

J(ϕ(•)) = Ω f (x, ϕ(x), ∇ϕ(x))dx. If the function s → f (x, r, s) is convex, then J is (sequentially) weakly lower semicontinuous in W 1,p (Ω). Lemma 1. Let u n : [0, T n ] → R, κ n 0 ∈ R satisfy J L (T n , κ n 0 , u n ) ≤ M for every n, for some M > 0. Consider the solution Q n (•) = (x n (•), y n (•), θ n (•), κ n (•)
) of the control system (3) associated with the control u n (•) and with initial condition Q n (0) = (0, 0, 0, κ n 0 ), and assume that lim n q n (T

n ) = lim n (x n (T n ), y n (T n ), θ n (T n )) = q 1 , for some q 1 = (x 1 , y 1 , θ 1 ) ∈ R 2 × S 1 , with (x 1 , y 1 ) = (0, 0).
Then there exist a time T > 0 and a function ū : [0, T ] → R such that, up to subsequences, T = lim n T n and, possibly putting

u n = 0 on [T n , T ] if T n ≤ T , (u n (•)) n weakly converges to ū(•) in L p ([0, T ]). Moreover, the functions Q n (•) converge uniformly to a solution Q(•) = (x(•), ȳ(•), θ(•), κ(•)) of (3) associated with ū(•). In particular, (x( T ), ȳ( T ), θ( T ))) = q 1 .
Finally, the following relation holds

J L ( T , κ(0), ū) ≤ lim inf n J L ( T , κ n 0 , u n ) ≤ lim inf n J L (T n , κ n 0 , u n ). ( 10
)
Proof. By (H3) and the special structure of the cost, it is easy to see that T n and u n L p are uniformly bounded and therefore, up to subsequences, we have that (T n ) n converges to T . Whenever T n < T , we extend u n to the whole [0, T ] taking u n (t) = 0 for t ∈ [T n , T ]. Without loss of generality, we can assume that (u n (•)) n weakly converges to some ū(•) in L p ([0, T ]).

Since

|κ n (t) -κ n (t ′ )| ≤ t t ′ |u n (s)| ds ≤ |t -t ′ | 1/q u n L p , (11)
where as usual 1/q + 1/p = 1, we see that, for every n, κ n (•) is Hölder continuous and the family (κ n (•)) n is uniformly equicontinuous. We claim that the sequence (κ n (•)) n is uniformly bounded. By contradiction, assume that κ n (•) ∞ goes to infinity. Since T n and u n L p are bounded and by [START_REF] Dacorogna | Direct methods in the calculus of variations[END_REF], for n large enough κ n (•) has constant sign on [0, T n ]. We fix n sufficiently large and, without loss of generality, we assume that κ n (•) is positive. Let α be defined by the relation (

x n (T n ), y n (T n )) = |(x n (T n ), y n (T n ))|(cos α, sin α), where | • | denotes the Euclidean norm on R 2 . Let I + h , h = 1, . . .

, N + and I -

h , h = 1, . . . , N -be the subintervals of [0, T n ] where cos(θ n (t) -α) is respectively non-negative or non-positive. Then |N --N + | ≤ 1 and, since for every h (except pos-

sibly h = 1, N -, N + ) it holds π = I ± h κ n (s)ds ≤ µ(I ± h ) κ n ∞ , we get N -, N + ≤ T n κ n ∞ 2π + 3
2 (here and in the following we denote by µ(J) the measure of a measurable set J ⊂ R).

Let us observe that from [START_REF] Dacorogna | Direct methods in the calculus of variations[END_REF] we get that κ n (t) ≥ κ n ∞ 1+ε , where ε =

(T n ) 1/q u n p κ n ∞-(T n ) 1/q u n p > 0 if n is large enough. Then for every h = 1, . . . , N + we have that I + h cos(θ n (t) -α)dt ≤ 1+ε κ n ∞ I + h cos(θ n (t) -α) κ n (t)dt ≤ 2 1+ε κ n ∞ . Moreover for every h = 2, . . . , N --1 we have that I - h cos(θ n (t)-α)dt ≤ 1 κ n ∞ I - h cos(θ n (t)- α) κ n (t)dt = -2 κ n ∞ . Then |(x n (T n ), y n (T n ))| = (x n (T n ), y n (T n )), (cos α, sin α) = T n 0 (cos θ n (t), sin θ n (t)), (cos α, sin α) dt = T n 0 cos(θ n (t) -α)dt ≤ 2N + 1 + ε κ n ∞ - 2N --4 κ n ∞ ≤ εT n π + 6 + 3ε κ n ∞ .
This leads to a contradiction, since ε tends to zero as n goes to infinity, while |(x n (T n ), y n (T n ))| is bounded from below and T n is bounded. Thus (κ n (•)) n is uniformly bounded and therefore, by Ascoli-Arzelà Theorem, the functions κ n (•) converge uniformly, up to extracting a subsequence, to a function κ : [0, T ] → R. This implies the uniform convergence of

Q n (•) to Q(•) on [0, T ].
Finally, in view of (H2), equation ( 10) is a direct consequence of Theorem 2.

Proof of Theorem 1. Fix q 1 = (x 1 , y 1 , θ 1 ) and consider a minimizing sequence

κ n ∈ R, u n : [0, T n ] → R for problem P 2 (L), that is lim n J L (T n , κ n 0 , u n ) = inf J L (T, κ 0 , u),
where κ n 0 , u n and κ 0 , u are such that the trajectories 3) with initial condition Q n (0) = (0, 0, 0, κ n 0 ) and Q(0) = (0, 0, 0, κ 0 ), respectively, satisfy (x n (T n ), y n (T n ), θ n (T n )) = (x(T ), y(T ), θ(T )) = q 1 . It is easy to prove that the sequence (u n (•)) n satisfies the hypotheses of Lemma 1, and therefore it weakly converges to some ū(•) in L p ([0, T ]) whose corresponding trajectory Q(•) solves problem P 2 (L).

Q n (•) = (x n (•), y n (•), θ n (•), κ n (•)), Q(•) = (x(•), y(•), θ(•), κ(•)) of (
The counterpart of Theorem 1 for problem P 1 (ℓ) is the following. Theorem 3. For every choice of q 1 ∈ R 2 × S 1 there exist T > 0 and a control function κ(•) ∈ L p ′ ([0, T ]) such that the corresponding trajectory q(•) solves problem P 1 (ℓ).

The previous result follows directly from the following adaptation of Lemma 1, which, in turn, is an easy consequence of Ascoli-Arzelà Theorem and of Theorem 2 applied with ϕ = θ.

Lemma 2. Let κ n : [0, T n ] → R satisfy J ℓ (T n , κ n ) ≤ M
for every n and some M > 0. Let q n (•) be a solution of the control system (4) associated with the control κ n (•) with q n (0) = q 0 = (0, 0, 0), and assume that lim n q n (T n ) = q 1 , for some q 1 ∈ R 2 × S 1 . Then there exist a time T > 0 and a function κ : [0, T ] → R such that up to subsequences, T = lim n T n and, possibly putting

κ n = 0 on [T n , T ] if T n ≤ T , κ n (•) weakly converges to κ(•) in L p ([0, T ]).
Moreover, the functions q n (•) converge uniformly to the corresponding solution q(•) of (4) associated with κ(•), and

J ℓ ( T , κ) ≤ lim inf n J ℓ ( T , κ n ) ≤ lim inf n J ℓ (T n , κ n ).
3.2. Stability of the optimal solutions. We start this section by stating an accessory result about the regularity of the value function associated with problems P 2 (L) and P 1 (ℓ). For every q 1 ∈ R 2 × S 1 we define the value function J * L (q 1 ) (resp. J * ℓ (q 1 )) as the minimum of the functional J L (resp. J ℓ ) among all trajectories of (3) (resp. ( 4)) steering q 0 = (0, 0, 0) to q 1 . Proposition 2. For ♮ = L, ℓ, the value function J * ♮ (•) is continuous at every q 1 = (x 1 , y 1 , θ 1 ) such that (x 1 , y 1 ) = (0, 0).

Proof. Take ♮ = L; the proof for ♮ = ℓ is analogous. Let us consider a time-reparametrization of the control system (3)

           ẋ = v cos θ ẏ = v sin θ θ = vκ κ = vu , (12)
where v(•) > 0 is essentially bounded. Equation ( 12) is considered as a control system with control function (u(•), v(•)). Let Q 0 = (0, 0, 0, κ 0 ) for some κ 0 ∈ R and, for T > 0, let

E Q 0 ,T (•) : L ∞ ([0, T ], R 2 ) → R 2 × S 1 × R denote the end-point mapping, that is E Q 0 ,T (u, v) = Q(T ), where Q(•)
is the solution of ( 12), associated with the control (u, v) and satisfying

Q(0) = Q 0 .
It is easy to see that the control system [START_REF] Darlot | The inactivation principle: Mathematical solutions minimizing the absolute work and biological implications for the planning of arm movements[END_REF] does not admit singular controls, that is controls (u, v) such that the (Fréchet) differential at (u, v) of the end-point mapping is not surjective (see [START_REF] Bonnard | Singular trajectories and their role in control theory[END_REF]). Indeed, any singular control satisfies the Weak maximum principle [8, Theorem 6] with associated Hamiltonian H = v(p x cos θ + p y sin θ + p θ κ + p κ u), and the maximality condition implies that p x cos θ + p y sin θ + p θ κ + p κ u = 0; there are no extremals (Q(•), P (•)) satisfying the latter condition, since this would entail the existence of abnormal extremals for problem P 2 (L), which is excluded by Remark 4.

Let ū : [0, T ] → R be an optimal control for P 2 (L) with final condition q 1 , and let Q(•) denote the associated trajectory. Then the absence of singular controls for [START_REF] Darlot | The inactivation principle: Mathematical solutions minimizing the absolute work and biological implications for the planning of arm movements[END_REF] implies that the mapping E Q(0), T (•) is open at (ū, 1).

It is easy to see that the functional J(κ 0 , u, v) = T 0 v(t)(1 + L(κ(t), u(t))) dt, evaluated on the solution Q(•) of ( 12) with Q(0) = Q(0) = (0, 0, 0, κ0 ) associated with (u, v), is continuous with respect to the L ∞ ([0, T ], R 2 ) topology for (u, v), and moreover we have that J(κ 0 , u, v) coincides with the value of the functional J L evaluated on a time reparametrization of Q(•) depending on v. Then for every ε > 0 there exists an open neighborhood U of (ū, 1) such that J(κ 0 , u, v) ≤ J * L (q 1 ) + ε for every (u, v) ∈ U, while by definition of J * L we have

J * L (E Q(0), T (u, v)) ≤ J(κ 0 , u, v).
Since the end-point mapping is open and ε is arbitrary, we get that lim sup q→q 1 J * L (q) ≤ J * L (q 1 ), that is J * L is upper semicontinuous. It remains to prove that J * L is lower semicontinuous. Let (q n 1 ) n be a sequence of points converging to q 1 , and, for every n, u n : [0, T n ] → R be an optimal control steering the system from q 0 to q n 1 , with initial curvature κ n 0 . By the upper semicontinuity of J * L we get the uniform boundedness of J L (T n , κ n 0 , u n ), thus we can apply Lemma 1 and obtain lim inf n J * L (q n 1 ) = lim inf n J L (T n , κ n 0 , u n ) ≥ J L ( T , κ0 , ū), where ū : [0, T ] → R steers the system from q 0 to q 1 , with initial curvature κ0 . By definition of J * L we have that J L ( T , κ0 , ū) ≥ J * L (q 1 ), and this completes the proof.

The previous result shows that the cost (value function) is continuous with respect to the final point. We will prove now a continuity property of the optimal trajectories and the associate adjoint vectors with respect to the final point.

Fix a point q 1 = (x 1 , y 1 , θ 1 ) ∈ R 2 × S 1 . Given a cost L ∈ L 2 , we define E 2 (L, q 1 ) as the set of curves (Q(•), P (•)) such that Q(•) is a solution of P 2 (L) with final condition q 1 and P (•) the associated adjoint vector. We endow the set of such continuous curves (Q(•), P (•)) with the distance d ∞ associated with the norm • ∞ of the uniform convergence.

Similarly, given a cost ℓ ∈ L 1 we define E 1 (ℓ, q 1 ) as the set of curves (q(•), p(•)) such that q(•) is a solution of P 1 (ℓ) with final condition q 1 and p(•) the associated adjoint vector. As above, we endow the set of such continuous curves (q(•), p(•)) with the distance associated with the norm of the uniform convergence that we denote again d ∞ .

Theorem 4. Let q 1 = (x 1 , y 1 , θ 1 ) be a point in R 2 × S 1 and (q n 1 ) n a sequence of points converging to q 1 .

• Fix L ∈ L 2 and assume (x 1 , y 1 ) = (0, 0). Then, for any sequence

(Q n (•), P n (•)) n such that, for every n, (Q n (•), P n (•)) ∈ E 2 (L, q n 1 ), we have lim n→∞ d ∞ ((Q n (•), P n (•)) , E 2 (L, q 1 )) = 0.
• Fix ℓ ∈ L 1 and assume (x 1 , y 1 ) = (0, 0). Then, for any sequence (q n (•), p n (•)) n such that, for every n, (q n (•),

p n (•)) ∈ E 1 (ℓ, q n 1 ), we have lim n→∞ d ∞ ((q n (•), p n (•)) , E 1 (ℓ, q 1 )) = 0.
The first item of this theorem is a direct consequence of the lemma below and the second one results from a straighforward adaptation of this lemma. Lemma 3. Fix L ∈ L 2 , q 1 = (x 1 , y 1 , θ 1 ) ∈ R 2 × S 1 such that (x 1 , y 1 ) = 0, and let (q n 1 ) n be a sequence of points converging to q 1 . Then, for any sequence

(Q n (•), P n (•)) n with (Q n (•), P n (•)) ∈ E 2 (L, q n 1 )
, there exists a subsequence (Q n k (•), P n k (•)) k which uniformly converges to an extremal ( Q(•), P (•)) in E 2 (L, q 1 ).

Proof. For every integer n, let u n be the optimal control associated with (Q n (•), P n (•)). By the continuity of the value function, we have that J * L (q n 1 ) is uniformly bounded. Thus the sequence (u n ) n satisfies the hypotheses of Lemma 1, which implies the uniform convergence of Q n (•), up to subsequences, to a trajectory Q(•) = (x(•), ȳ(•), θ(•), κ(•)) associated with ū : [0, T ] → R. By [START_REF] Chitour | Optimal control models of the goal-oriented human locomotion[END_REF] and the continuity of the value function, we have that Q(•) is a solution of P 2 (L) with final point q 1 .

Let us now prove the uniform convergence of the adjoint vector. For every integer n we set

Q n (•) = (x n (•), y n (•), θ n (•), κ n (•)) and P n (•) = (p n x (•), p n y (•), p n θ (•), p n κ (•)).
We then proceed in two step: first of all, we prove that the sequence of the initial conditions P n (0) is bounded (and therefore it converges, up to subsequences); then, since for every n the extremal (Q n (•), P n (•)) satisfies the Hamiltonian system (5), we get the convergence of the whole adjoint vector.

By the adjoint equation we get that p n x , p n y are constant and we write (p n x , p n y ) = ρ n (cos α n , sin α n ), ρ n > 0. Let us first show that ρ n is uniformly bounded. Assume by contradiction that this is not true. Up to restricting to an appropriate subsequence we can assume that ( For every n, consider p n κ (t) for t ∈ I n . Since ṗn κ = -p n θ + ∂L ∂κ , from (2) we have that there exists a family of subintervals I n′ ⊂ I n with µ(I n′ ) ≥ c > 0 such that lim n inf I n′ |p n κ | = ∞. Since κ n (•) is uniformly bounded and by [START_REF] Bayen | Asymptotic analysis of an optimal control problem connected to the human locomotion[END_REF] we get that |u n | is unbounded in I n′ , which contradicts the uniform boundedness of u n in the L p norm. Then ρ n is uniformly bounded.

Q n (•)) n uniformly converges to Q(•) = (x(•), ȳ(•), θ(•), κ(•)), (α n ) n converges
Let us now consider the case in which θ(•) is constant, that is θ(•) ≡ κ(•) ≡ ū(•) ≡ 0. First of all, we notice that α * * = 0; indeed, if not, then we can proceed as above and get a contradiction. The Hamiltonian at t = 0 writes H = ρ n cos α n +p n θ (0)κ n (0)-1-L(κ n (0), 0) = 0. If (ρ n ) n goes to infinity, then also |p n θ (0)| does, as well as the ratio |p n θ (0)|/ρ n . This comes from the fact that the Hamiltonian is null and κ n (0) tends to zero. Since | ṗn θ |/ρ n ≤ 1, we get as above that |p n θ (•)| goes to infinity on [0, T n ], which leads to a contradiction. We can then conclude that ρ n is uniformly bounded.

In particular, we obtain that | ṗn θ | is uniformly bounded with respect to n, and therefore that |p n θ (0)| is uniformly bounded: if not, in fact, |p n θ (•)| would go to infinity on the whole interval [0, T n ], and then also |p n κ (•)| would go to infinity (on some appropriate subinterval), and we would get a contradiction.

Then |P n (0)| is uniformly bounded and therefore, up to subsequences, we get that (P n (0)) n converges to some P * .

Let us now consider the system

Qn = ∂H ∂P (Q n , P n , -1, Υ(κ n , p n κ )) Ṗ n = -∂H ∂Q (Q n , P n , -1, Υ(κ n , p n κ )) (13)
where Υ(κ, p κ ) is the control expressed in terms of κ and p κ by means of the implicit function theorem applied to [START_REF] Bayen | Asymptotic analysis of an optimal control problem connected to the human locomotion[END_REF], namely Υ(κ,

p κ ) = ∂L ∂u (κ, •) -1 (p κ ).
Since the initial conditions (Q n (0), P n (0)) of ( 13) converge to ( Q(0), P * ), we have the convergence of the solutions (Q n (•), P n (•)) to some solution ( Q(•), P * (•)) of [START_REF] Kalman | When is a linear control system optimal?[END_REF]. By uniqueness of the adjoint vector (see Remark 4) we get that P * (•) = P (•).

Robustness

In this section we study the continuity of the mapping which associates with a cost in L the set of solutions of the corresponding optimal problem P 2 (L) or P 1 (ℓ) with fixed terminal point. We will show that when a sequence of costs converge to a cost L or ℓ in L, any sequence of optimal trajectories associated with these costs converges uniformly to the set of optimal trajectories of respectively P 2 (L) or P 1 (ℓ). This can be understood as a property of robustness of these optimal control problems. The class of costs L being defined as the disjoint union of two sub-classes L 1 and L 2 , we have to distinguish several cases: convergence of a sequence in L 2 to a cost in L 2 as well, of a sequence in L 2 to a cost in L 1 , and of a sequence in L 1 to a cost in L 1 . In each case, we have to specify the notion of convergence we use.

Robustness of P 2 (L)

. Consider functions L 0 and L ε , ε ∈ (0, ε], belonging to L 2 , and assume that L 0 satisfies (H3) with p = p. Let L ε converge to L 0 in the following sense:

(H4) there exists a constant C > 0 such that for every (r, s) one has

|L ε (r, s) -L 0 (r, s)| + ∂L ε ∂r (r, s) - ∂L 0 ∂r (r, s) ≤ Cε(|s| p + 1).
(H5) ∂ 2 Lε ∂s 2 (r, s) converges uniformly to ∂ 2 L 0 ∂s 2 (r, s) on compact subsets of R 2 . We notice that hypothesis (H4) in particular implies that for every ε > 0 small enough L ε satisfies condition (H3) with p = p.

As in Section 3.2, we use E 2 (L, q 1 ) to denote the set of curves (Q(•), P (•)) such that Q(•) is a solution of P 2 (L) with final condition q 1 and P (•) the associated adjoint vector and we endow the set of continuous curves (Q(•), P (•)) with the distance d ∞ associated with the norm of the uniform convergence.

Theorem 5. Fix q 1 ∈ R 2 × S 1 and let L ε , ε ≥ 0, be a family of costs in L 2 satisfying hypotheses (H4)-(H5), with L 0 satisfying (H3) for p = p. Then, for any family

(Q ε (•), P ε (•)) ∈ E 2 (L ε , q 1 ), ε ≥ 0, we have lim ε→0 d ∞ ((Q ε (•), P ε (•)) , E 2 (L 0 , q 1 )) = 0.
This theorem is a direct consequence of the following lemma. Lemma 4. Under the hypotheses of Theorem 5, if (ε n ) n is a sequence converging to 0, then, for any sequence (Q εn (•), P εn (•)) n such that, for every n, (Q εn (•), P εn (•)) ∈ E 2 (L εn , q 1 ), there exists a subsequence which uniformly converges to an extremal (Q * (•), P * (•)) in E 2 (L 0 , q 1 ).

Proof. We choose first an extremal (Q 0 (•), P 0 (•)) ∈ E 2 (L 0 , q 1 ). For ε = 0 and ε ∈ (ε n ) n , we call u ε : [0, T ε ] → R the optimal control of the problem P 2 (L ε ) associated with the extremal

(Q ε (•), P ε (•)) and we denote Q ε (•) = (x ε (•), y ε (•), θ ε (•), κ ε (•)) and P ε (•) = (p ε x , p ε y , p ε θ (•), p ε κ (•)). Let us introduce the functional J ε 2 (T, κ 0 , u) = T 0 (1 + L ε (κ(s), u(s))
) ds, for every ε ∈ [0, ε], which is evaluated along trajectories of (3) starting from (0, 0, 0, κ 0 ). By definition, for every integer n we have J εn 2 (T εn , κ εn 0 , u εn ) ≤ J εn 2 (T 0 , κ 0 0 , u 0 ), and moreover

|J εn 2 (T 0 , κ 0 0 , u 0 ) -J 0 2 (T 0 , κ 0 0 , u 0 )| ≤ T 0 0 |L εn (κ 0 (s), u 0 (s)) -L 0 (κ 0 (s), u 0 (s))| ds ≤ Cε n ( u 0 p L p + T 0 ), ( 14
)
so that J εn 2 (T εn , κ εn 0 , u εn ) ≤ J 0 2 (T 0 , κ 0 0 , u 0 ) + Cε n ( u 0 p L p + T 0 )
. By (H3), (H4) and the special structure of the cost we thus have that both T εn and u εn L p are uniformly bounded. In particular there exists a subsequence, that we denote again (ε k ) k , converging to zero such that (T ε k ) k converges to some T * > 0. Whenever T ε k < T * , we extend the control u ε k (•) on the interval [0, T * ], putting it equal to zero on (T

ε k , T * ]. Since (u ε k ) k is a bounded sequence in L p([0, T * ]), up to subsequences it weakly converges to some u * (•) ∈ L p([0, T * ]).
We can then repeat the argument of Lemma 1 to prove that κ ε k 0 is bounded, so that it converges to some κ * 0 , up to subsequences. We call

Q * (•) = (x * (•), y * (•), θ * (•), κ * (•)) the solution of (3) associated with u * (•) with initial condition Q * (0) = (0, 0, 0, κ * 0 ). Then, the weak convergence of (u ε k (•)) k to u * (•) implies that (κ ε k (•)) k converges to κ * (•) pointwise on [0, T * ]
and that κ ε k (•) is (uniformly) Hölder continuous for every ε k > 0. Then by Ascoli-Arzelà Theorem, we obtain that (κ ε k (•)) k converges uniformly to κ * (•).

Therefore, we obtain that the trajectories Q ε k (•) uniformly converge to Q * (•). In particular, (x * (T * ), y * (T * ), θ * (T * )) = q 1 .

Since by definition u 0 is a minimum for P 2 (L 0 ), it holds J 0 2 (T 0 , κ 0 0 , u 0 ) ≤ J 0 2 (T * , κ * 0 , u * ). From Theorem 2 we get that J 0 2 is weakly lower semicontinuous, and therefore lim inf

k→∞ J 0 2 (T ε k , κ ε k , u ε k ) ≥ lim inf k→∞ J 0 2 (T * , κ ε k 0 , u ε k ) ≥ J 0 2 (T * , κ * 0 , u * ),
where we used the fact that

u ε k (•) is set equal to zero on [T ε k , T * ] if T ε k < T * .
Moreover, by hypothesis (H4), we have that for every ε

∈ (ε n ) n , J 0 2 (T ε , κ ε 0 , u ε ) ≤ J ε 2 (T ε , κ ε 0 , u ε ) + Cε(T ε + u ε p L p ) ≤ J ε 2 (T 0 , κ 0 0 , u 0 ) + Cε(T ε + u ε p L p )
. Passing to the lim inf and applying ( 14) we obtain

J 0 2 (T * , κ * 0 , u * ) ≤ lim inf k→∞ J ε k 2 (T 0 , κ 0 0 , u 0 ) + Cε k (T ε k + u ε k p L p ) = J 0 2 (T 0 , κ 0 0 , u 0 ), that is, u * is a minimum.
Let us now prove the convergence of the adjoint vectors. As above, we first show that P ε k (0) is bounded, and then we prove that the right-hand side of the adjoint equation ( 5) converges uniformly on compact sets: then also its solution converges uniformly.

As usual, we write (p ε x , p ε y ) = ρ ε (cos α ε , sin α ε ), ρ ε > 0. Let us prove that ρ ε k is uniformly bounded. Assume by contradiction that this is not true. Up to restricting to an appropriate subsequence we can assume that (α ε k ) k converges to α * while ρ ε k goes to infinity. By previous results θ ε k (•) converges to θ * (•); as in Lemma 3, we can assume that θ * (•) may be constant or not.

In the first case, if θ * is constant then κ * ≡ u * ≡ 0, and therefore Q * (•) is the straight line connecting q 0 = (0, 0, 0) to q 1 = (T * , 0, 0). Moreover, since the Hamiltonian is identically zero along extremal trajectories and from the transversality conditions we get that P * (•) ≡ (1, 0, 0, 0); since we know that the only solution for P 2 (L ε ) with final point q 1 = (T * , 0, 0) is the straight line for every ε > 0, we obtain that T ε k = T * and P ε k (t) ≡ (1, 0, 0, 0), for every k.

Consider the second case, that is θ * (•) is not constant. Applying the same argument of Lemma 3, we can assume that there exist η > 0 and a family of intervals

I ε k with µ(I ε k ) ≥ η such that lim k inf I ε k |p ε k θ (t)| = ∞. Since ṗε k κ = -p ε k θ + ∂Lε k
∂κ and by [START_REF] Arechavaleta | Optimizing principles underlying the shape of trajectories in goal oriented locomotion for humans[END_REF] and hypothesis (H4), we obtain that there exist a c > 0 and a family of subintervals

I ε k ′ ⊂ I ε k with µ(I ε k ′ ) ≥ c > 0 such that lim k inf I ε k ′ |p ε k κ | = ∞.
This implies that also |u ε k | is unbounded in I ε k ′ , which contradicts the uniform boundedness of u ε k L p . Then we obtain that ρ ε k is uniformly bounded.

In particular, this implies that ṗε k θ is uniformly bounded with respect to ε k . As in Lemma 3, we can conclude that also p ε k θ (0) is uniformly bounded and then we get the convergence of the initial conditions P ε k (0) to some (p * *

x , p * * y , p * * θ (0), 0), up to subsequences. Let us now consider the system, for ε

∈ (ε n ) n ,            ṗε θ = p ε x cos θ ε -p ε y sin θ ε ṗε κ = -p ε θ + ∂Lε ∂κ (κ ε , Υ ε (κ ε , p ε κ )) θε = κ ε κε = Υ ε (κ ε , p ε κ ) , (15)
where Υ ε (κ, p κ ) is the control expressed in function of κ and p κ by means of the implicit function theorem and the maximality condition [START_REF] Bayen | Asymptotic analysis of an optimal control problem connected to the human locomotion[END_REF], namely Υ ε (κ,

p κ ) = ∂Lε ∂u (κ, •) -1 (p κ )
Let us prove that Υ ε k (•, •) converges to Υ 0 (•, •) uniformly on the compact subsets of R 2 . Fix r and consider υ r ε k (s) = Υ ε k (r, s). The function υ r ε k is the solution of the Cauchy problem

   υ r′ ε k = 1 ∂ 2 Lε k ∂u 2 (r, υ r ε k ) υ r ε k (0) = 0 .
Thanks to hypothesis (H5), the right-hand side of the equation converges uniformly on compact sets, so that also the solution of the Cauchy problem converges. Then Υ ε k (•, •) converges to Υ 0 (•, •) uniformly on the compact subsets of R 2 . The latter, together with hypothesis (H4) and the uniform convergence of the pair (p ε k x , p ε k y ), implies that the right-hand side of (15) converges uniformly on compact subsets as ε k goes to zero. Since also the initial conditions of (15) converge, we obtain that the quadruple (p

ε k θ (•), p ε k κ (•), θ ε k (•), κ ε k (•)) converges uniformly on [0, T ] to the solution (p * * θ (•), p * * κ (•), θ * (•), κ * (•)) of the system            ṗ * * θ = p * * x cos θ * -p * * y sin θ * ṗ * * κ = -p * * θ + ∂L 0 ∂κ (κ * , Υ 0 (κ * , p * * κ )) θ * = κ * κ * = Υ 0 (κ * , p * * κ ) (16)
with initial condition (p * * θ , 0, 0, κ * ). Call P * * (•) = (p * * x , p * * y , p * * θ (•), p * * κ (•)); since the pair (Q * (•), P * * (•)) satisfies the PMP, by the uniqueness of the adjoint vector we have that P * * (•) = P * (•).

Robustness of P 1 (ℓ) w.r.t. P 2 (L). Consider a cost ℓ 0 in L 1 , and a family of costs L ε in L 2 , ε ∈ (0, ε] converging to ℓ 0 in the following sense: (H4 ′ ) there exist three constants C, C 1 , C 2 > 0 such that for every (r, s) one has

C 1 ε(|s| p -1) ≤ L ε (r, s) -ℓ 0 (r) ≤ C 2 ε(|s| p + 1) (17)
∂L ε ∂r (r, s) -ℓ ′ 0 (r) ≤ Cε(|s| p + 1). ( 18
)
Remark 5. It is not our purpose here to find the minimal assumptions under which robustness results can be obtained. If needed, the hypotheses above could certainly be weakened. We notice moreover that (H4 ′ ) implies that, for ε ∈ (0, ε], L ε satisfies (H3) with p = p.

In the following we consider again the sets E 2 (L, q 1 ) and E 1 (ℓ, q 1 ) defined in Section 3.2, and we also use E 1 (ℓ, q 1 ) to denote the set of curves (Q(•), P (•)) such that Q(•) = (q(•), κ(•)) and P (•) = (p(•), 0), where (q(•), p(•)) ∈ E 1 (ℓ, q 1 ) and κ(•) is the associated optimal control. We endow these sets with the distance d ∞ associated with the norm of the uniform convergence. Theorem 6. Fix q 1 ∈ R 2 × S 1 . Let ℓ 0 be a cost in L 1 , and let L ε , ε ∈ (0, ε] be a family of functions in L 2 . Then, for any family

(Q ε (•), P ε (•)) ∈ E 2 (L ε , q 1 ), ε ≥ 0, we have lim ε→0 d ∞ (Q ε (•), P ε (•)) , E 1 (ℓ 0 , q 1 ) = 0.
This theorem is a direct consequence of the following lemma. Lemma 5. Under the hypotheses of Theorem 6, if (ε n ) n is a sequence converging to 0, then, for any sequence (Q εn (•), P εn (•)) n such that, for every n, (Q εn (•), P εn (•)) ∈ E 2 (L εn , q 1 ), there exists a subsequence which uniformly converges to a curve ((q * (•), κ * (•)), (p * (•), 0)) in E 1 (ℓ 0 , q 1 ).

Proof. For every ε ∈ (0, ε] we denote by J ε 2 (T, κ 0 , u) the functional T 0 1 + L ε (κ(s), u(s)) ds associated with a solution of (3) starting from (0, 0, 0, κ 0 ) and with control function u(•), and we set J 0 1 (T, κ)

= T 0 1 + ℓ 0 (κ(s)) ds. For every ε ∈ (ε n ) n , we call u ε : [0, T ε ] → R the optimal control of the problem P 2 (L ε ) associated with the extremal (Q ε (•), P ε (•)) and we denote Q ε (•) = (x ε (•), y ε (•), θ ε (•), κ ε (•)) and P ε (•) = (p ε x , p ε y , p ε θ (•), p ε κ (•)). Let κ 0 (•) : [0, T 0 ] → R
be an optimal control of the problem P 1 (ℓ 0 ) with final point q 1 . From the PMP (in particular, see Remark 4) we know that κ 0 (•) ∈ C 1 ([0, T 0 ]), so that we can define u 0 (•) = κ0 (•), which is a continuous function on [0, T 0 ]. Hypothesis (H4 ′ ) implies that, for every integer n,

C 1 ε n ( u εn p L p -T εn ) ≤ J εn 2 (T εn , κ εn (0), u εn ) -J 0 1 (T εn , κ εn ) ≤ J εn 2 (T εn , κ εn 0 , u εn ) -J 0 1 (T 0 , κ 0 ) ≤ ≤ J εn 2 (T 0 , κ 0 (0), u 0 ) -J 0 1 (T 0 , κ 0 ) ≤ T 0 0 L εn (κ 0 (s), u 0 (s)) -ℓ 0 (κ 0 (s)) ds ≤ C 2 ε n ( u 0 p L p + T 0
). Since moreover J εn 2 (T εn , κ εn (0), u εn ) ≥ T εn , we therefore obtain that both T εn and u εn L p are uniformly bounded.

As in the proof of Lemma 4, up to taking a subsequence, T ε k converges to some T * > 0. As above, we extend all the controls u ε k such that T ε k < T * putting them equal to zero on (T ε k , T * ]. Then we have that (u ε k (•)) k weakly converges to some u * (•) ∈ L p([0, T * ]), up to subsequences.

Again reasoning as in Lemma 1, we obtain that the trajectories

Q ε k (•) uniformly converge to Q * (•), where Q * (•) = (x * (•), y * (•), θ * (•), κ * (•))
is the solution of (3) associated with u * (•) with initial condition Q * (0) = (0, 0, 0, κ * 0 ), and κ * 0 is the limit of κ ε (0), up to subsequences. In particular, q * (•) = (x * (•), y * (•), θ * (•)) is the solution of (4) associated with the control κ * (•) with initial condition q 0 and final condition q 1 . By definition, J 0 1 (T 0 , κ 0 ) ≤ J 0 1 (T * , κ * ). By hypothesis (H2 ′ ) and Theorem 2 we obtain that J 0 1 is weakly lower semicontinuous in L p , and since κ ε k (•) converges uniformly to κ * (•) we have

J 0 1 (T * , κ * ) ≤ lim inf k→∞ J 0 1 (T * , κ ε k ) = lim inf k→∞ J 0 1 (T ε k , κ ε k ),
Moreover, by hypothesis (H4 ′ ), we have that for every ε ∈

(ε n ) n , J 0 1 (T ε , κ ε ) ≤ J ε 2 (T ε , κ ε (0), u ε ) ≤ J ε 2 (T 0 , κ 0 (0), u 0 ).
We thus obtain

J 0 1 (T * , κ * ) ≤ lim inf k→∞ J ε k 2 (T 0 , κ 0 (0), u 0 ) = J 0 1 (T 0 , κ 0 ),
that is q * (•) is a solution of the problem P 1 (ℓ 0 ) with final condition q 1 . Moreover, as proved above, the trajectories (x ε k (•), y ε k (•), θ ε k (•)) uniformly converge to (x * (•), y * (•), θ * (•)).

Let us now investigate the convergence of the adjoint vectors. Since the first two components of the adjoint vector are constant with respect to time, we write (p

ε k x , p ε k y ) = ρ ε k (cos α ε k , sin α ε k ), ρ ε k > 0.
Repeating the same argument of Lemma 3 and 4, we have that (ρ ε k ) k converges to some ρ * * and (α ε k ) k converges to α * * , up to subsequences. Indeed, as above, if ρ ε k was not uniformly bounded, then it is easy to see that u ε k L p would not be uniformly bounded.

The same argument proves that also p ε k θ (0) is uniformly bounded and then we get the convergence of the initial conditions (P ε k (0)) k to some (p * *

x , p * * y , p * * θ (0), 0), up to subsequences. Let us now consider the system

ṗε k θ = p ε k x cos θ ε k -p ε k y sin θ ε k θε k = κ ε k . ( 19
)
The right-hand side of (19) converges uniformly on [0, T * ], then we obtain that the pair (p ε k θ ( Robustness of P 1 (ℓ) w.r.t. P 1 (ℓ). Consider a cost ℓ 0 in L 1 satisfying (H3 ′ ) with p = p, and a family of costs ℓ ε in L 1 , ε ∈ (0, ε], converging to ℓ 0 in the following sense: (H4 ′′ ) there exists a constant C > 0 such that for every r one has |ℓ ε (r) -ℓ 0 (r)| ≤ Cε(|r| p + 1).

(H5 ′′ ) ℓ ′′ ε (r) converges uniformly to ℓ ′′ 0 (r) on compact subsets of R. Here again we consider the set of curves E 1 (ℓ, q 1 ) defined in Section 3.2, endowed with the distance d ∞ associated with the norm of the uniform convergence.

Theorem 7. Fix q 1 ∈ R 2 × S 1 and let ℓ ε be a family of costs in L 1 satisfying hypotheses (H4 ′′ )-(H5 ′′ ). Then, for any family (q ε (•), p ε (•)) ∈ E 1 (ℓ ε , q 1 ), ε ≥ 0, we have lim ε→0 d ∞ ((q ε (•), p ε (•)) , E 1 (ℓ 0 , q 1 )) = 0. The proof of this result is an easy adaptation of the one of Theorem 5.

Conclusion

In this paper we analyze continuity properties of a class of optimal control problems that is suitable to describe the human locomotion. These properties reflect the physiological nature of the problem, in the sense that human behavior is homogeneous and stable with respect to different kinds of perturbations. Since costs not satisfying such continuity properties are not suitable to describe the locomotion problem, this analysis is a necessary step to possibly reduce the class of candidate cost functions that can modelize the system.

The first property analyzed is what we call stability, namely, for an optimal control problem associated with a fixed cost, the continuity of the solutions with respect to small perturbations of the final point. This very natural property has been highlighted in the experiments performed by Laumond et al. [START_REF] Arechavaleta | Optimizing principles underlying the shape of trajectories in goal oriented locomotion for humans[END_REF][START_REF] Arechavaleta | On the nonholonomic nature of human locomotion[END_REF][START_REF] Arechavaleta | An optimality principle governing human walking[END_REF].

For an optimal control problem associated with a final point, we call robustness the continuity of the solutions with respect to small perturbations of the cost. This property is also natural and is important to specify a candidate cost, in order to perform numerical simulations of the model. Indeed, if the class of cost functions is robust, then the solution of the problem does not depend critically on the choice of a particular cost function, but it is possible to reduce the problem to the study of a simpler cost function. This is in particular very effective if we can reduce the number of parameters of the problem by choosing a cost function that does not depend on the angular acceleration and its derivatives. In this paper

Fig. 2 .

 2 Fig.2. Examples of trajectories for several final positions, drawn by using the data recorded by Arechavaleta et al.[START_REF] Arechavaleta | On the nonholonomic nature of human locomotion[END_REF] (with the kind permission of the authors).

  to some α * * and (ρ n ) n goes to infinity. Consider the case in which θ(•) is not constant. Then | sin( θ(t) -α * * )| > C for t belonging to some (closed) interval I = [a, a + τ ] ⊂ [0, T ] and for some C > 0, which implies that | sin(θ n (t) -α n )| ≥ C on I, for n suitably large. Then by equation (7) we get | ṗn θ | ≥ Cρ n on I and therefore there exist η > 0 and a family of intervals I n ⊂ I with µ(I n ) ≥ η such that lim n inf I n |p n θ (t)| = ∞. Indeed, for every t ∈ [a, a + τ /3], every t ′ ∈ [a + 2τ /3, a + τ ] and for every n we have that |p n θ (t) -p n θ (t ′ )| ≥ Cρ n τ /3, therefore it must be |p n θ (t)| ≥ Cρ n τ /6 on [a, a + τ /3] or [a + 2τ /3, a + τ ].

  •), θ ε k (•)) converges uniformly on [0, T * ] to the solution (p * * θ (•), θ * (•)) of the system ṗ * * θ = p * * x cos θ * -p * * y sin θ * θ * = κ * (20) with initial condition (p * * θ (0), 0). Moreover we have that p ε k κ (•) converges to zero uniformly. This comes immediately from the equation ṗε k κ(•) = -p ε k θ (•) + ∂Lε k∂κ , the convergence of the trajectories, and equations (18) and[START_REF] Boyd | Linear matrix inequalities in system and control theory[END_REF].Call p * * (•) = (p * * x , p * * y , p * * θ (•)); since the pair (q * (•), p * * (•)) satisfies the PMP, by the uniqueness of the adjoint vector we have that p * * (•) = p * (•).

we performed a first step in this direction, showing a continuity property between the problems associated with the class of costs that do depend on the angular acceleration and the problems associated with the class of cost that do only depend on the angular velocity.

This work was supported by the Digiteo grant Congeo, by the ANR project GCM, program "Blanche", project number NT09 504490, by the CNRS-INSIS through the PEPS grant Co-MoMo, and by the Commission of the European Communities under the 7th Framework Programme Marie Curie Initial Training Network (FP7-PEOPLE-2010-ITN), project SADCO, contract number 264735. 1072-3374/**/****-0001 c 20** Springer Science+Business Media, Inc.

0 ¡0 (x , y , ) 1 1 ¡ 1 (x , y ,