Moussa Balde

A note on stability conditions for planar switched systems

This article is concerned with the stability problem for the planar linear switched system , where the real matrices A 1, A 2 ∈ ℝ2×2 are Hurwitz and u(·) : [0, ∞ [→ {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(·). The new conditions unify those given in previous papers and are simpler to be verified since we reduced to study 4 cases instead of 20. Most of the cases are analysed in terms of the function .

Geometry of the Limit Sets of Linear Switched Systems

This paper is concerned with asymptotic stability properties of linear switched systems. Under the hypothesis that all the subsystems share a nonstrict 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 nonchaotic inputs, which generalize the different notions of inputs with dwell time. Next we turn our attention to the behavior for possibly chaotic inputs. Finally, we give a sufficient condition for a system composed of a pair of Hurwitz matrices to be asymptotically stable for all inputs.

A Mathematical Model of a Fishery with Variable Market Price: Sustainable Fishery/Over-exploitation

We present a mathematical bioeconomic model of a fishery with a variable price. The model describes the time evolution of the resource, the fishing effort and the price which is assumed to vary with respect to supply and demand. The supply is the instantaneous catch while the demand function is assumed to be a monotone decreasing function of price. We show that a generic market price equation (MPE) can be derived and has to be solved to calculate non trivial equilibria of the model. This MPE can have 1, 2 or 3 equilibria. We perform the analysis of local and global stability of equilibria. The MPE is extended to two cases: an age-structured fish population and a fishery with storage of the resource.

Relaxed conditions for the stability of switched nonlinear triangular systems under arbitrary switching

This paper addresses the stability of a class of nonlinear switching systems under arbitrary switching. It focuses on switching systems whose modes are made of a cascade of nonlinear scalar systems. It relaxes the stability conditions proposed by Angeli and Liberzon, by relying on the Strong iISS property which is known to be preserved under cascade interconnection. Its applicability is illustrated by the study of switched bilinear systems in triangular form.

A Bioeconomic Model of a Multi-site Fishery with Nonlinear Demand Function: Number of Sites Optimizing the Total Catch

Abstract We present a mathematical model of a fishery on several sites with a variable price. The model takes into account the evolution during the time of the resource, fish and boat movement between the different sites, fishing effort and price that varies with respect to supply and demand. We suppose that the movements of the boats and resource as well as the variation of the price go on at a fast time scale. We use methods of aggregation of variables in order to reduce the number of variables and we derive a reduced model governing two global variables, respectively the biomass of the resource and the fishing effort of the whole fishery. We look for the existence of equilibria of the aggregated model and perform local stability analysis. Two main cases can occur. The first one corresponds to over-exploitation leading to fish extinction. At extinction, the fishing effort tends to a positive value. The second case corresponds to a durable fishery equilibrium which is globally asymptotically stable. In the later case, we show that there exists a number of fishing sites that optimizes the total catch of the fishery.

Observability and observer design for kinetic networks

In this paper, we deal with the observability and observer design of a certain class of reactors (batch reactors) which are globally asymptotically feedback stabilisable. We show how to design two types of high-gain observers: a Luenberger-like high-gain observer and a high-gain extended Kalman filter. Results of convergence of these estimators follow from our observability analysis. We prove that the non-linear feedback controller using these high-gain observers stabilise such reactors. In fact, we develop an observer-based control structure for a non-linear model of batch reactors. Two examples of such reactors that correspond to strongly connected networks and that have zero deficiency are presented in this paper. Simulations show the good performance of such an observer-based control structure.