arXiv subjects
Publications and source records attributed to Frédéric Jean.
In recent years, several notions of non-rigidity of horizontal vectors in Carnot groups have been proposed, motivated, in particular, by the characterization of monotone sets and Whitney extension properties. In this note we compare some of these notions.
Robust control of complex engineered and biological systems hinges on the integration of feedforward and feedback mechanisms. This is exemplified in neural motor control, where feedforward muscle co-contraction complements sensory-driven feedback corrections to ensure stable behaviors. However, deriving a general continuous-time framework to determine such optimal control policies for partially observable, stochastic, nonlinear, and high-dimensional systems remains a formidable computational challenge. Here, we introduce a framework that extends neighboring optimal control by enabling the feedforward plan to explicitly account for feedback uncertainties and latencies. Using statistical linearization, we transform the stochastic problem into an approximately equivalent deterministic optimization within a tractable, augmented state space that retains critical nonlinearities, offering both mechanistic interpretability and theoretical guarantees on approximation fidelity. We apply this framework to human neuromechanics, demonstrating that muscle co-contraction emerges as an optimal adaptation to task demands, given the characteristics of our sensorimotor system. Our results provide a computational foundation for neuromotor control and a generalizable tool for the control of nonlinear stochastic systems.
In this paper, we bring a complete solution to the Ovals problem, as formulated in [3] and [24].
A longstanding open question in sub-Riemannian geometry is the following: are sub-Riemannian length minimizers smooth? We give a negative answer to this question, exhibiting an example of a $C^2$ but not $C^3$ length-minimizer of a real-analytic (even polynomial) sub-Riemannian structure.
Inverse optimal control problem emerges in different practical applications, where the goal is to design a cost function in order to approximate given optimal strategies of an expert. Typical application is in robotics for generation of human motions. In this paper we analyze a general class of non autonomous inverse linear quadratic problems. This class of problems is of particular interest because it arises as a linearization of a nonlinear problem around an optimal trajectory. The addressed questions are the injectivity of the inverse problem and the reconstruction. We show that the nonlinear problem admits the same characterization of the injectivity as the autonomous one. In the autonomous case we show moreover that the injectivity property is generic in the considered class. We also provide a numerical test of the reconstruction algorithm in the autonomous setting.
Statistical linearization has recently seen a particular surge of interest as a numerically cheap method for robust control of stochastic differential equations. Although it has already been successfully applied to control complex stochastic systems, accessibility and controllability properties of statistical linearization, which are key to make the robust control problem well-posed, have not been investigated yet. In this paper, we bridge this gap by providing sufficient conditions for the accessibility and controllability of statistical linearization. Specifically, we establish simple sufficient algebraic conditions for the accessibility and controllability of statistical linearization, which involve the rank of the Lie algebra generated by the drift only. In addition, we show these latter algebraic conditions are essentially sharp, by means of a counterexample, and that they are generic with respect to the drift and the initial condition.
This paper presents a joint optimisation framework for optimal estimation and stochastic optimal control with imperfect information. It provides a estimation and control scheme that can be decomposed into a classical optimal estimation step and an optimal control step where a new term coming from optimal estimation is added to the cost. It is shown that a specific particle filter algorithm allows one to solve the first step approximately in the case of Mean Square Error minimisation and under suitable assumptions on the model. Then, it is shown that the estimation-based control step can justify formally the use of Explicit dual controllers which are most of the time derived from empirical matters. Finally, a relevant example from Aerospace engineering is presented.
This paper presents state estimation and stochastic optimal control gathered in one global optimization problem generating dual effect i.e. the control can improve the future estimation. As the optimal policy is impossible to compute, a sub-optimal policy that preserves this coupling is constructed thanks to the Fisher Information Matrix (FIM) and a Particle Filter. This method has been applied to the localization and guidance of a drone over a known terrain with height measurements only. The results show that the new method improves the estimation accuracy compared to nominal trajectories.
This paper presents a dual receding horizon output feedback controller for a general non linear stochastic system with imperfect information. The novelty of this controller is that stabilization is treated, inside the optimization problem, as a negative drift constraint on the control that is taken from the theory of stability of Markov chains. The dual effect is then created by maximizing information over the stabilizing controls which makes the global algorithm easier to tune than our previous algorithm. We use a particle filter for state estimation to handle nonlinearities and multimodality. The performance of this method is demonstrated on the challenging problem of terrain aided navigation.
The goal of robust motion planning consists of designing open-loop controls which optimally steer a system to a specific target region while mitigating uncertainties and disturbances which affect the dynamics. Recently, stochastic optimal control has enabled particularly accurate formulations of the problem. Nevertheless, despite interesting progresses, these problem formulations still require expensive numerical computations. In this paper, we start bridging this gap by leveraging statistical linearization. Specifically, through statistical linearization we reformulate the robust motion planning problem as a simpler deterministic optimal control problem subject to additional constraints. We rigorously justify our method by providing estimates of the approximation error, as well as some controllability results for the new constrained deterministic formulation. Finally, we apply our method to the powered descent of a space vehicle, showcasing the consistency and efficiency of our approach through numerical experiments.
This paper studies a vertical powered descent problem in the context of planetary landing, considering glide-slope and thrust pointing constraints and minimizing any final cost. In a first time, it proves the Max-Min-Max or Max-Singular-Max form of the optimal control using the Pontryagin Maximum Principle, and it extends this result to a problem formulation considering the effect of an atmosphere. It also shows that the singular structure does not appear in generic cases. In a second time, it theoretically analyzes the optimal trajectory for a more specific problem formulation to show that there can be at most one contact or boundary interval with the state constraint on each Max or Min arc.
In this paper, we study the $L^p$-asymptotic stability of the one-dimensional linear damped wave equation with Dirichlet boundary conditions in $[0,1]$, with $p\in (1,\infty)$. The damping term is assumed to be linear and localized to an arbitrary open sub-interval of $[0,1]$. We prove that the semi-group $(S_p(t))_{t\geq 0}$ associated with the previous equation is well-posed and exponentially stable. The proof relies on the multiplier method and depends on whether $p\geq 2$ or $1<p<2$.
H. Weyl in 1921 demonstrated that for a connected manifold of dimension greater than $1$, if two Riemannian metrics are conformal and have the same geodesics up to a reparametrization, then one metric is a constant scaling of the other one. In the present paper, we investigate the analogous property for sub-Riemannian metrics. In particular, we prove that the analogous statement, called the Weyl projective rigidity, holds either in real analytic category for all sub-Riemannian metrics on distributions with a specific property of their complex abnormal extremals, called minimal order, or in smooth category for all distributions such that all complex abnormal extremals of their nilpotent approximations are of minimal order. This also shows, in real analytic category, the genericity of distributions for which all sub-Riemannian metrics are Weyl projectively rigid and genericity of Weyl projectively rigid sub-Riemannian metrics on a given bracket generating distributions. Finally, this allows us to get analogous genericity results for projective rigidity of sub-Riemannian metrics, i.e.when the only sub-Riemannian metric having the same sub-Riemannian geodesics, up to a reparametrization, with a given one, is a constant scaling of this given one. This is the improvement of our results on the genericity of weaker rigidity properties proved in recent paper arXiv:1801.04257[math.DG].
We prove the $C^{1}$ regularity for a class of abnormal length-minimizers in rank $2$ sub-Riemannian structures. As a consequence of our result, all length-minimizers for rank $2$ sub-Riemannian structures of step up to $4$ are of class $C^{1}$.
In \cite{gjha} we give a detailed analysis of spherical Hausdorff measures on sub-Riemannian manifolds in a general framework, that is, without the assumption of equiregularity. The present paper is devised as a complement of this analysis, with both new results and open questions. The first aim is to extend the study to other kinds of intrinsic measures on sub-Riemannian manifolds, namely Popp's measure and general (i.e., non spherical) Hausdorff measures. The second is to explore some consequences of \cite{gjha} on metric measure spaces based on sub-Riemannian manifolds.
In this paper we study the Hausdorff volume in a non equiregular sub-Riemannian manifold and we compare it with a smooth volume. We first give the Lebesgue decomposition of the Hausdorff volume. Then we study the regular part, show that it is not commensurable with the smooth volume, and give conditions under which it is a Radon measure. We finally give a complete characterization of the singular part. We illustrate our results and techniques on numerous examples and cases (e.g. to generic sub-Riemannian structures).
In this paper we study the complexity of the motion planning problem for control-affine systems. Such complexities are already defined and rather well-understood in the particular case of nonholonomic (or sub-Riemannian) systems. Our aim is to generalize these notions and results to systems with a drift. Accordingly, we present various definitions of complexity, as functions of the curve that is approximated, and of the precision of the approximation. Due to the lack of time-rescaling invariance of these systems, we consider geometric and parametrized curves separately. Then, we give some asymptotic estimates for these quantities.
This paper is a starting point towards computing the Hausdorff dimension of submanifolds and the Hausdorff volume of small balls in a sub-Riemannian manifold with singular points. We first consider the case of a strongly equiregular submanifold, i.e., a smooth submanifold N for which the growth vector of the distribution D and the growth vector of the intersection of D with TN are constant on N. In this case, we generalize the result in [12], which relates the Hausdorff dimension to the growth vector of the distribution. We then consider analytic sub-Riemannian manifolds and, under the assumption that the singular point p is typical, we state a theorem which characterizes the Hausdorff dimension of the manifold and the finiteness of the Hausdorff volume of small balls B(p,ρ) in terms of the growth vector of both the distribution and the intersection of the distribution with the singular locus, and of the nonholonomic order at p of the volume form on M evaluated along some families of vector fields.