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The early stopping strategy consists in stopping the training process of a neural network (NN) on a set $S$ of input data before training error is minimal. The advantage is that the NN then retains good generalization properties, i.e. it gives good predictions on data outside $S$, and a good estimate of the statistical error (``population loss'') is obtained. We give here an analytical estimation of the optimal stopping time involving basically the initial training error vector and the eigenvalues of the ``neural tangent kernel''. This yields an upper bound on the population loss which is well-suited to the underparameterized context (where the number of parameters is moderate compared with the number of data). Our method is illustrated on the example of an NN simulating the MPC control of a Van der Pol oscillator., Comment: 7 pages, 2 figures

Methods based on "(Jacobian) matrix measure" to show the convergence of a dynamical system to a limit cycle (LC), generally assume that the measure is negative everywhere on the LC. We relax this assumption by assuming that the matrix measure is negative "on average" over one period of LC. Using an approximate Euler trajectory, we thus present a method that guarantees the LC existence, and allows us to construct a basin of attraction. This is illustrated on the example of the Van der Pol system., Comment: 6 pages, 3 figures

Electrical conductors are usually made of pure copper because this material shows outstanding electrical conductivity (IACS > 100%). Additive manufacturing can be used to further improve the performances of such electrical conductors because it enables to consider sophisticated geometries impossible to be fabricated by conventional processing routes such as casting, machining or hot forming. In the present work, Electron Beam Powder Bed Fusion is optimized to achieve dense pure copper specimens (relative density >99.9%). Only few residual spherical pores inherited from the as-received powder particles are detected in the samples as nicely revealed by X-ray microtomography scans. Three different build orientations, namely 0{\textdegree}, 45{\textdegree} and 90{\textdegree} with respect to the build direction are produced. The microstructures are characterized using optical microscopy and EBSD measurements. The electrical conductivity is measured using the Eddy current method as well as the four-probes method while the mechanical properties are assessed using Vickers microhardness and tensile mechanical testing. The electrical and mechanical performances are discussed in the light of the microstructural characterizations, and further compared with a cold-worked Cu-ETP sheet subjected to an annealing treatment. It demonstrates that E-PBF is able to reliably produce components made of pure Copper whose properties equal those of pure copper fabricated by more conventional processing routes, regardless of the build orientation.

The next generation of space systems will have to achieve more and more complex missions. In order to master the development cost and duration of such systems, an alternative to a manual design is to automatically synthesize the main parameters of the system. In this paper, we present an approach for the specific case of the scheduling of the flight control of a space launcher. The approach requires two successive steps: (1) the formalization of the problem to be solved in a parametric formal model and (2) the synthesis of the model parameters with a tool. We first describe the problem of the scheduling of a launcher flight control, then we show how this problem can be formalized with parametric stopwatch automata; we then present the results computed by the parametric timed model checker IMITATOR. We enhance our model by taking into consideration the time for switching context, and we compare the results to those obtained by other tools classically used in scheduling., Comment: This manuscript is the author version of the manuscript of the same name published in Fundamenta Informatica 182(1). This is an extended version of the manuscript published in the proceedings of the 19th International Conference on Application of Concurrency to System Design (ACSD 2019). arXiv admin note: substantial text overlap with arXiv:1903.07217

We show here how, using Euler's integration method and an associated function bounding the error in function of time, one can generate structures closely surrounding the invariant tori of dynamical systems. Such structures are constructed from a finite number of balls of $\mathbb{R}^n$ and encompass the deformations of the tori when small perturbations of the flow of the system occur., Comment: 15 pages, 7 figures. arXiv admin note: text overlap with arXiv:2012.09310

In this paper, we consider the application of optimal periodic control sequences to switched dynamical systems. The control sequence is obtained using a finite-horizon optimal method based on dynamic programming. We then consider Euler approximate solutions for the system extended with bounded perturbations. The main result gives a simple condition on the perturbed system for guaranteeing the existence of a stable limit cycle of the unperturbed system. An illustrative numerical example is provided which demonstrates the applicability of the method., Comment: This is the author version of the manuscript of the same name published in the proceedings of the 2021 American Control Conference (ACC 2021)

A method is given for generating a bounded invariant of a differential system with a given set of initial conditions around a point $x_0$. This invariant has the form of a tube centered on the Euler approximate solution starting at $x_0$, which has for radius an upper bound on the distance between the approximate solution and the exact ones. The method consists in finding a real $T>0$ such that the "snapshot" of the tube at time $t=(i+1)T$ is included in the snapshot at $t=iT$, for some integer $i$. In the phase space, the invariant is therefore in the shape of a torus. A simple additional condition is also given to ensure that the solutions of the system can never converge to a point of equilibrium. In dimension 2, this ensures that all solutions converge towards a limit cycle. The method is extended in case the dynamic system contains a parameter $p$, thus allowing the stability analysis of the system for a range of values of $p$. This is illustrated on classical Van der Pol's system.

The proceedings consist of a keynote paper by Alberto followed by 6 invited papers written by Lorenzo Clemente (U. Warsaw), Alain Finkel (U. Paris-Saclay), John Gallagher (Roskilde U. and IMDEA Software Institute) et al., Neil Jones (U. Copenhagen) et al., Michael Leuschel (Heinrich-Heine U.) and Maurizio Proietti (IASI-CNR) et al.. These invited papers are followed by 4 regular papers accepted at VPT 2020 and the papers of HCVS 2020 which consist of three contributed papers and an invited paper on the third competition of solvers for Constrained Horn Clauses. In addition, the abstracts (in HTML format) of 3 invited talks at VPT 2020 by Andrzej Skowron (U. Warsaw), Sophie Renault (EPO) and Moa Johansson (Chalmers U.), are included.

Set-based integration methods allow to prove properties of differential systems, which take into account bounded disturbances. The systems (either time-discrete, time-continuous or hybrid) satisfying such properties are said to be "robust". In the context of optimal control synthesis, the set-based methods are generally extensions of numerical optimal methods of two classes: first, methods based on convex optimization; second, methods based on the dynamic programming principle. Heymann et al. have recently shown that, for certain systems of low dimension, the second numerical method can give better solutions than the first one. They have built a solver (Bocop) that implements both numerical methods. We show in this paper that a set-based extension of a method of the second class which uses a guaranteed Euler integration method, allows us to find such good solutions. Besides, these solutions enjoy the property of robustness against uncertainties on initial conditions and bounded disturbances. We demonstrate the practical interest of our method on an example taken from the numerical Bocop solver. We also give a variant of our method, inspired by the method of Model Predictive Control, that allows us to find more efficiently an optimal control at the price of losing robustness.

The phenomenon of phase synchronization was evidenced in the 17th century by Huygens while observing two pendulums of clocks leaning against the same wall. This phenomenon has more recently appeared as a widespread phenomenon in nature, and turns out to have multiple industrial applications. The exact parameter values of the system for which the phenomenon manifests itself are however delicate to obtain in general, and it is interesting to find formal sufficient conditions to guarantee phase synchronization. Using the notion of reachability, we give here such a formal method. More precisely, our method selects a portion $S$ of the state space, and shows that any solution starting at $S$ returns to $S$ within a fixed number of periods $k$. Besides, our method shows that the components of the solution are then (almost) in phase. We explain how the method applies on the Brusselator reaction-diffusion and the biped walker examples.

In this paper, we introduce a concept called ''virtual co-embodiment'', which enables a user to share their virtual avatar with another entity (e.g., another user, robot, or autonomous agent). We describe a proof-of-concept in which two users can be immersed from a first-person perspective in a virtual environment and can have complementary levels of control (total, partial, or none) over a shared avatar. In addition, we conducted an experiment to investigate the influence of users' level of control over the shared avatar and prior knowledge of their actions on the users' sense of agency and motor actions. The results showed that participants are good at estimating their real level of control but significantly overestimate their sense of agency when they can anticipate the motion of the avatar. Moreover, participants performed similar body motions regardless of their real control over the avatar. The results also revealed that the internal dimension of the locus of control, which is a personality trait, is negatively correlated with the user's perceived level of control. The combined results unfold a new range of applications in the fields of virtual-reality-based training and collaborative teleoperation, where users would be able to share their virtual body., Comment: Accepted for publication in IEEE Transactions on Visualization and Computer Graphics

We show that, for any spatially discretized system of reaction-diffusion, the approximate solution given by the explicit Euler time-discretization scheme converges to the exact time-continuous solution, provided that diffusion coefficient be sufficiently large. By "sufficiently large", we mean that the diffusion coefficient value makes the one-sided Lipschitz constant of the reaction-diffusion system negative. We apply this result to solve a finite horizon control problem for a 1D reaction-diffusion example. We also explain how to perform model reduction in order to improve the efficiency of the method.

We have recently used a symbolic reachability method for controlling the stability of special hybrid systems called 'sampled switched systems'. We show here how the method can be extended in order to control the stability of more general hybrid systems with guard conditions and state resets. We illustrate the method through the example of a biped robot with 6 state variables, using a proportional-derivative (PD) controller. More specifically, we isolate a state region R such that, starting from a state located in R just after a footstep, the PD-control makes the robot state return to R at the end of the following footstep.

The next generation of space systems will have to achieve more and more complex missions. In order to master the development cost and duration of such systems, an alternative to a manual design is to automatically synthesize the main parameters of the system. In this paper, we present an approach on the specific case of the scheduling of the flight control of a space launcher. The approach requires two successive steps: (1) the formalization of the problem to be solved in a parametric formal model and (2) the synthesis of the model parameters with a tool. We first describe the problematic of the scheduling of a launcher flight control, then we show how this problematic can be formalized with parametric stopwatch automata; we then present the results computed by IMITATOR. We compare the results to the ones obtained by other tools classically used in scheduling., Comment: This work was supported by the Paris \^Ile-de-France Region (project DIM RFSI ASTREI). \'Etienne Andr\'e is partially supported by the ANR national research program PACS (ANR-14-CE28-0002), and the ERATO HASUO Metamathematics for Systems Design Project (No. JPMJER1603), JST

In this paper, we propose a new method for ensuring formally that a controlled trajectory stay inside a given safety set S for a given duration T. Using a finite gridding X of S, we first synthesize, for a subset of initial nodes x of X , an admissible control for which the Euler-based approximate trajectories lie in S at t $\in$ [0,T]. We then give sufficient conditions which ensure that the exact trajectories, under the same control, also lie in S for t $\in$ [0,T], when starting at initial points 'close' to nodes x. The statement of such conditions relies on results giving estimates of the deviation of Euler-based approximate trajectories, using one-sided Lipschitz constants. We illustrate the interest of the method on several examples, including a stochastic one.

The election of a leader in a network is a challenging task, especially when the processes are asynchronous, i.e., execute an algorithm with time-varying periods. Thales developed an industrial election algorithm with an arbitrary number of processes, that can possibly fail. In this work, we prove the correctness of a variant of this industrial algorithm. We use a method combining abstraction, the SafeProver solver, and a parametric timed model-checker. This allows us to prove the correctness of the algorithm for a large number p of processes (p=5000)., Comment: This is the author version of the manuscript of the same name published in the proceedings of the 20th International Conference on Verification, Model Checking, and Abstract Interpretation (VMCAI 2019)