Convergence speed of unsteady distributed consensus: decay estimate along the settling spanning-trees

Results for estimating the convergence rate of non-stationary distributed consensus algorithms are provided, on the basis of qualitative (mainly topological) as well as basic quantitative information (lower-bounds on the matrix entries). The results appear to be tight in a number of instances and are illustrated through simple as well as more sophisticated examples. The main idea is to follow propagation of information along certain spanning trees which arise in the communication graph.Comment: 27 pages, 5 figure

Convergence results for continuous-time dynamics arising in ant colony optimization

This paper studies the asymptotic behavior of several continuous-time dynamical systems which are analogs of ant colony optimization algorithms that solve shortest path problems. Local asymptotic stability of the equilibrium corresponding to the shortest path is shown under mild assumptions. A complete study is given for a recently proposed model called EigenAnt: global asymptotic stability is shown, and the speed of convergence is calculated explicitly and shown to be proportional to the difference between the reciprocals of the second shortest and the shortest paths.Comment: A short version of this paper was published in the preprints of the 19th World Congress of the International Federation of Automatic Control, Cape Town, South Africa, 24-29 August 201

Tight estimates for non-stationary consensus with fixed underlying spanning tree

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A cooperative conjugate gradient method for linear systems permitting multithread implementation of low complexity

This paper proposes a generalization of the conjugate gradient (CG) method used to solve the equation $Ax=b$ for a symmetric positive definite matrix $A$ of large size $n$. The generalization consists of permitting the scalar control parameters (= stepsizes in gradient and conjugate gradient directions) to be replaced by matrices, so that multiple descent and conjugate directions are updated simultaneously. Implementation involves the use of multiple agents or threads and is referred to as cooperative CG (cCG), in which the cooperation between agents resides in the fact that the calculation of each entry of the control parameter matrix now involves information that comes from the other agents. For a sufficiently large dimension $n$, the use of an optimal number of cores gives the result that the multithread implementation has worst case complexity $O(n^{2+1/3})$ in exact arithmetic. Numerical experiments, that illustrate the interest of theoretical results, are carried out on a multicore computer.Comment: Expanded version of manuscript submitted to the IEEE-CDC 2012 (Conference on Decision and Control

On Positiveness of Matrix-Valued Polynomials and Robust Semidefinite Programming

Projet SOSSOThis report is devoted to the study of robust semidefinite programming. We show that to the issue of computing the worst-case optimal value of semidefinite programs depending polynomially upon a finite number of bounded scalar parameters, one may associate a countable family of standard semidefinite programs, whose optimal values converge monotonically towards the requested quantity. The results is linked to representation formula and positiveness criterion for matrix-valued polynomials

Stability of nonlinear delay systems : delay-independent small gain theorem and frequency domain interpretation of the Lyapunov-Krasovskii method

Projet SOSSOThe purpose of this note is to study the relationship between a certain stability criterion for nonlinear delay systems, obtained via Lyapunov-Krasovs- kii method, and a delay-independent version of the small gain theorem. We show that, contrary to the delay-free case (in which Kalman-Yakubovich-Popo- v lemma ensures the equivalence of the two approaches), the first method assumes stronger hypothesis than the second one. However, numerical verificati- on of the former is in general NP-hard, whereas the latter leads to linear matrix inequalities. The difference between the two approaches is precisely stated, and, among other benefits, this permits to exhibit classes of problems for which the Lyapunov-Krasovskii method is not conservative

A feedback control perspective on biological control of dengue vectors by Wolbachia infection

International audienceControlling diseases such as dengue fever, chikungunya and zika fever by introduction of the intracellular parasitic bacterium $Wolbachia$ in mosquito populations which are their vectors, is presently quite a promising tool to reduce their spread. While description of the conditions of such experiments has received ample attention from biologists, entomologists and applied mathematicians, the issue of effective scheduling of the releases remains an interesting problem. Having in mind the important uncertainties present in the dynamics of the two populations in interaction, we attempt here to identify general ideas for building feedback-based release strategies, enforceable to a variety of models and situations. These principles are exemplified by several feedback control laws whose stabilizing properties are demonstrated, illustrated numerically and compared, when applied to a model retrieved from [P.-A. Bliman et al., Ensuring successful introduction of $Wolbachia$ in natural populations of $Aedes aegypti$ by means of feedback control. $J. of Math. Bio.$ 76(5):1269-1300, 2018]. The contribution is believed to be also of potential interest to tackle other important issues related to the biological control of vectors and pests. A crucial use of the theory of monotone dynamical systems is made in the derivations

Extension of Popov Absolute Stability Criterion to Nonautonomous Systems with Delays

Projet SOSSOThis paper extends in a simple way the classical absolute stability Popov criterion to multivariable systems with delays and with time-varying memoryles- s nonlinearities subject to sector conditions. The proposed sufficient conditions are expressed in the frequency domain, a form well-suited for robustness issues, and lead to simple graphical interpretations for scalar systems. Apart from the usual conditions, the results assume basically a generalized sector condition on the derivative of the nonlinearities with respect to time. Results for local and global stability are given, the latter concerning in particular the linear time-varying ones. For rational transfers, the frequency conditions are equivalent to some easy-to-ch- eck Linear Matrix Inequalities: this leads to a tractable method of numerical resolution by approximation. As an illustration, a numerical example is provided