arXiv subjects

Philippe Jouan

Publications and source records attributed to Philippe Jouan.

13 recordsLinked to original sources

arXiv2023

The Allen-Cahn equation with nonlinear truncated Laplacians: description of radial solutions

We consider the Allen-Cahn equation with the so-called truncated Laplacians, which are fully nonlinear differential operators that depend on some eigenvalues of the Hessian matrix. By monitoring the sign of a quantity that is responsible for switches from a first order ODE regime to a second order ODE regime (and vice versa), we give a nearly complete description of radial solutions. In particular, we reveal the existence of surprising unbounded radial solutions. Also radial solutions cannot oscillate, which is in sharp contrast with the case of the Laplacian operator, or that of Pucci's operators.

math.AP↗

arXiv2022

General Nilpotent and Solvable Approximations of Almost-Riemannian Structures

It is first shown that the nilpotent or the solvable approximation of an almost-Riemannian structure at a singular point is always a linear almost-Riemannian structure on a Lie group or a homogeneous space. The generic properties of almost-Riemannian structures are then investigated in all dimensions and the generic nilpotent and solvable approximations are identified.

math.OC↗

arXiv2020

Jordan decomposition and the recurrent set of flows of automorphisms

In this paper we show that any linear vector field $\mathcal{X}$ on a connected Lie group $G$ admits a Jordan decomposition and the recurrent set of the associated ow of automorphisms is given as the intersection of the fixed points of the hyperbolic and nilpotent components of its Jordan decomposition.

math.DS↗

arXiv2018

Control sets of linear systems on semi-simple Lie groups

In this paper we study the main properties of control sets with nonempty interior of linear control systems on semisimple Lie groups. We show that, unlike the solvable case, linear control systems on semisimple Lie groups may have more than one control set with nonempty interior and that they are contained in right translations of the one around the identity.

math.OC↗

arXiv2017

Isometries of almost-Riemannian structures on Lie groups

A simple Almost-Riemannian Structure on a Lie group G is defined by a linear vector field (that is an infinitesimal automorphism) and dim(G) -- 1 left-invariant ones. It is first proven that two different ARSs are isometric if and only if there exists an isometry between them that fixes the identity. Such an isometry preserves the left-invariant distribution and the linear field. If the Lie group is nilpotent it is an automorphism. These results are used to state a complete classification of the ARSs on the 2D affine and the Heisenberg groups.

math.OC↗

arXiv2015

Convergence Rate of Nonlinear Switched Systems

This paper is concerned with the convergence rate of the solutions of nonlinear switched systems. We first consider a switched system which is asymptotically stable for a class of inputs but not for all inputs. We show that solutions corresponding to that class of inputs converge arbitrarily slowly to the origin. Then we consider analytic switched systems for which a common weak quadratic Lyapunov function exists. Under two different sets of assumptions we provide explicit exponential convergence rates for inputs with a fixed dwell-time.

math.OC↗

arXiv2015

Almost-Riemannian Geometry on Lie Groups

A simple Almost-Riemmanian Structure on a Lie group G is defined by a linear vector field and dim(G)-1 left-invariant ones. We state results about the singular locus, the abnormal extremals and the desingularization of such ARS's, and these results are illustrated by examples on the 2D affine and the Heisenberg groups.These ARS's are extended in two ways to homogeneous spaces, and a necessary and sufficient condition for an ARS on a manifold to be equivalent to a general ARS on a homogeneous space is stated.

math.DG↗

arXiv2015

Controllability of Linear Systems on Generalized Heisenberg Groups

This paper is devoted to the study of controllability of linear systems on generalized Heisenberg groups. Some general necessary controllability conditions and some sufficient ones are provided. We introduce the notion of decoupled systems, and more precise controllability criteria are stated for them.

math.OC↗

arXiv2015

Stability of uniformly bounded switched systems and Observability

This paper mainly deals with switched linear systems defined by a pair of Hurwitz matrices that share a common but not strict quadratic Lyapunov function. Its aim is to give sufficient conditions for such a system to be GUAS.We show that this property of being GUAS is equivalent to the uniform observability on $[0,+\infty)$ of a bilinear system defined on a subspace whose dimension is in most cases much smaller than the dimension of the switched system.Some sufficient conditions of uniform asymptotic stability are then deduced from the equivalence theorem, and illustrated by examples.The results are partially extended to nonlinear analytic systems.

math.OC↗

arXiv2012

Asymptotic Stability of Uniformly Bounded Nonlinear Switched Systems

We study the asymptotic stability properties of nonlinear switched systems under the assumption of the existence of a common weak Lyapunov function. We consider the class of nonchaotic inputs, which generalize the different notions of inputs with dwell-time, and the class of general ones. For each of them we provide some sufficient conditions for asymptotic stability in terms of the geometry of certain sets. The results, which extend those of Balde, Jouan about linear systems, are illustrated by many examples.

math.OC↗

arXiv2010

Finite Time and Exact Time Controllability on Compact Manifolds

It is first shown that a smooth controllable system on a compact manifold is finite time controllable. The technique of proof is close to the one of Sussmann's orbit theorem, and no rank condition is required. This technique is also used to give a new and elementary proof of the equivalence between controllability for essentially bounded inputs and for piecewise constant ones. Two sufficient conditions for controllability at exact time on a compact manifold are then stated. Some applications, in particular to linear systems on Lie groups, are provided.

math.OC↗

arXiv2010

Geometry of the Limit Sets of Linear Switched Systems

The paper is concerned with asymptotic stability properties of linear switched systems. Under the hypothesis that all the subsystems share a non strict quadratic Lyapunov function, we provide a large class of switching signals for which a large class of switched systems are asymptotically stable. For this purpose we define what we call non chaotic inputs, which generalize the different notions of inputs with dwell time. Next we turn our attention to the behaviour for possibly chaotic inputs. To finish we give a sufficient condition for a system composed of a pair of Hurwitz matrices to be asymptotically stable for all inputs.

math.OC↗

arXiv2008

Equivalence of Control Systems with Linear Systems on Lie Groups and Homogeneous Spaces

The aim of this paper is to prove that a control affine system on a manifold is equivalent by diffeomorphism to a linear system on a Lie group or a homogeneous space if and only the vector fields of the system are complete and generate a finite dimensional Lie algebra. A vector field on a connected Lie group is linear if its flow is a one parameter group of automorphisms. An affine vector field is obtained by adding a left invariant one. Its projection on a homogeneous space, whenever it exists, is still called affine. Affine vector fields on homogeneous spaces can be characterized by their Lie brackets with the projections of right invariant vector fields. A linear system on a homogeneous space is a system whose drift part is affine and whose controlled part is invariant. The main result is based on a general theorem on finite dimensional algebras generated by complete vector fields, closely related to a theorem of Palais, and which have its own interest. The present proof makes use of geometric control theory arguments.

math.OC↗