arXiv subjects

Moussa Balde

Publications and source records attributed to Moussa Balde.

5 recordsLinked to original sources

arXiv2012

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↗

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

A note on stability conditions for planar switched systems

This paper is concerned with the stability problem for the planar linear switched system $\dot x(t)=u(t)A_1x(t)+(1-u(t))A_2x(t)$, where the real matrices $A_1,A_2\in \R^{2\times 2}$ are Hurwitz and $u(\cdot) [0,\infty[\to\{0,1\}$ is a measurable function. We give coordinate-invariant necessary and sufficient conditions on $A_1$ and $A_2$ under which the system is asymptotically stable for arbitrary switching functions $u(\cdot)$. The new conditions unify those given in previous papers and are simpler to be verified since we are reduced to study 4 cases instead of 20. Most of the cases are analyzed in terms of the function $Γ(A_1,A_2)={1/2}(\tr(A_1) \tr(A_2)- \tr(A_1A_2))$.

math.OC↗

arXiv2006

Stability of Planar Switched Systems: the Nondiagonalizable Case

Consider the planar linear switched system $\dot x(t)=u(t)Ax(t)+(1-u(t))Bx(t),$ where $A$ and $B$ are two $2\times2$ real matrices, $x \in \R^2$, and $u(.):[0,\infty[\to\{0,1\}$ is a measurable function. In this paper we consider the problem of finding a (coordinate-invariant) necessary and sufficient condition on $A$ and $B$ under which the system is asymptotically stable for arbitrary switching functions $u(.)$. This problem was solved in previous works under the assumption that both $A$ and $B$ are diagonalizable. In this paper we conclude this study, by providing a necessary and sufficient condition for asymptotic stability in the case in which $A$ and/or $B$ are not diagonalizable. To this purpose we build suitable normal forms for $A$ and $B$ containing coordinate invariant parameters. A necessary and sufficient condition is then found without looking for a common Lyapunov function but using "worst-trajectory'' type arguments.

math.OC↗

arXiv1999

On One-Parameter Families of Dido Riemannian Problems

Locally, isoperimetric problems on Riemannian surfaces are sub-Riemannian problems in dimension 3. The particular case of Dido problems corresponds to a class of singular contact sub-Riemannian metrics : metrics which have the charateristic vertor field as symmetry. We give a classification of the generic conjugate loci (i.e. classification of generic singularities of the exponential mapping) of a 1-parameter family of 3-d contact sub-Riemannian metrics associated to a 1-parameter family of Dido Riemannian problems.

math.DG↗