Controllability, Observability, and Stability of Mathematical Models

International audienceThis article presents an overview of three fundamental concepts in Mathematical System Theory: controllability, stability and observability. These properties play a prominent role in the study of mathematical models and in the understanding of their behavior. They constitute the main research subject in Control Theory. Historically the tools and techniques of Automatic Control have been developed for artificial engineering systems but nowadays they are more and more applied to "natural systems". The main objective of this article is to show how these tools can be helpful to model and to control a wide variety of natural systems

Feedback stabilization of stochastic nonlinear composite systems

International audienceIn this paper, we study the global stabilization, by means of smooth state feedback, of partially linear composite stochastic systems

Observer design for a schistosomiasis model

This paper deals with the state estimation for a schistosomiasis infection dynamical model described by a continuous non linear system when only the infected human population is measured. The central idea will be studied following two major angles. On the one hand, when all the parameters of the model are supposed to be well known, we will construct a simple observer and a high-gain Luenberger observer based on a canonical controller form and conceived for the nonlinear dynamics where it is implemented. On the other hand, when the nonlinear uncertain continuous-time system is in a bounded-error context, we will introduce a method for designing a guaranteed interval observer. Numerical simulations are included in order to test the behavior and the performance of the given observers.Un observateur 'grand gain' non-linéaire est mis en œuvre pour évaluer l'évolution de dynamique d'une infection de la Bilharziose décrite par un modèle continu non linéaire [1]. On propose un modèle réduit du modèle [1] de la Bilharziose pour construire l'observateur. Des simulations numériques ont été faites pour tester le comportement et la performance de l'observateur proposé

A remark on the stabilization of partially linear composite systems

International audienceIn this paper, we study the global stabilization, by means of smooth state feedback, of partially linear composite systems. We show how to compute the stabilizing feedback thanks to a weak Lyapunov function for a nonlin- ear subsystem instead of a stricte one

Multi-patch and multi-group epidemic models: A new framework

International audienceWe develop a multi-patch and multi-group model that captures the dynamics of an infectious disease when the host is structured into an arbitrary number of groups and interacts into an arbitrary number of patches where the infection takes place. In this framework, we model host mobility that depends on its epidemiological status, by a Lagrangian approach. This framework is applied to a general SEIRS model and the basic reproduction number R0 is derived. The effects of heterogeneity in groups, patches and mobility patterns on R0 and disease prevalence are explored. Our results show that for a fixed number of groups, the basic reproduction number increases with respect to the number of patches and the host mobility patterns. Moreover, when the mobility matrix of susceptible individuals is of rank one, the basic reproduction number is explicitly determined and was found to be independent of the latter if the matrix is also stochastic. The cases where mobility matrices are of rank one capture important modeling scenarios. Additionally, we study the global analysis of equilibria for some special cases. Numerical simulations are carried out to showcase the ramifications of mobility pattern matrices on disease prevalence and basic reproduction number

Backstepping with bounded feedbacks

International audienceAn extension of the backstepping approach is proposed. It allows to globally asymptotically stabilize by bounded feedbacks families of nonlinear control systems. Explicit expressions of control laws and Lyapunov functions are given

Simultaneous stabilization of single-input nonlinear systems with bounded controls

International audienceIn this paper, the simultaneous stabilization of single-input nonlinear systems with bounded controls is considered. Using the Lyapunov approach and based on Lin–Sontag’s formula for bounded and continuous stabilizers for affine nonlinear systems, a constructive universal formula for the bounded simultaneous stabilization of single-input nonlinear systems is presented explicitly. An illustrative example is given to demonstrate the validity of the method

Nonlinear stabilization by adding integrators

International audienceThe global stabilization problem of a nonlinear control system with an integrator is considered when the initial system (subsystem without integrator) is stabilizable. It is supposed that the stabilizing feedback and the Lyapunov function for the initial system satisfy the Barbashin-Krasovski\u ı asymptotic stability theorem. Explicit formulae for stabilizing feedback of a nonlinear system with an integrator are derived. The use of Lyapunov functions with derivatives of constant but not fixed signs significantly simplifies the computing of stabilizing feedback. This is confirmed by examples. The global stabilization of a nonlinear control system of the form \dot x= f(x, y), \qquad \qquad \dot y=u \tag 1 is studied, where x∈ \bbfRⁿ, y∈ \bbfR^p, u∈ \bbfR^p and f is a smooth vector field such that f (0, 0) =0. It is proved that to find a feedback stabilizer for this system we do not need to have a strict Lyapunov function for the subsystem (2) \dot x= f(x, v), where v is the input. Moreover, it is proved how to asymptotically stabilize system (1) without stabilizing system (2)

On the stability of nonautonomous systems

International audienceIn (Kalitine, 1982), the use of semi definite Lyapunov functions for exploring the local stability of autonomous dynamical systems has been introduced. In this paper we give an extension of the results of (Kalitine, 1982) that allows to study the local stability of nonautonomous differential systems. We give an application to the Algebraic Riccati Equation