The central part in the process of solving the observer problem for nonlinear systems is to find a solution of a partial differential equation of first order. The original method proposed to solve this equation used expansions into Taylor polynomials, however, it suffers from rather restrictive assumptions while the approach proposed here allows to generalize these requirements. Its characteristic feature is that it is based on the application of the Finite Element Method. An illustrating example...

This paper outlines the manner in which Thom’s theory of catastrophes fits into the Hamilton-Jacobi theory of partial differential equations. The representation of solutions of a first order partial differential equation as lagrangian manifolds allows one to study the local structure of their singularities. The structure of generic singularities is closely related to Thom’s concept of the elementary catastrophe associated to a singularity. Three concepts of the stability of a singularity are discussed....

We study the growth rate of a cell population that follows an age-structured PDE with time-periodic coefficients. Our motivation comes from the comparison between experimental tumor growth curves in mice endowed with intact or disrupted circadian clocks, known to exert their influence on the cell division cycle. We compare the growth rate of the model controlled by a time-periodic control on its coefficients with the growth rate of stationary models of the same nature, but with averaged coefficients....

Let (x,y,z) ∈ ℂ³. In this paper we shall study the solvability of singular first order partial differential equations of nilpotent type by the following typical example: $Pu(x,y,z):=(y{\partial }_x-z{\partial }_y)u(x,y,z)=f(x,y,z){\in }_{x,y,z}$, where $P=y{\partial }_x-z{\partial }_y{:}_{x,y,z}{\to }_{x,y,z}$. For this equation, our aim is to characterize the solvability on ${}_{x,y,z}$ by using the Im P, Coker P and Ker P, and we give the exact forms of these sets.

Let there be given a differential operator on ${ℝ}^n$ of the form $D={\sum }_{i,j=1}^n{a}_{ij}·x_j\partial /\partial x_i+\mu$, where $A=\left(a_{ij}\right)$ is a real matrix and μ is a complex number. We study the following question: To what extent the mapping $D:S^{\text{'}}\left({ℝ}^n\right)\to S^{\text{'}}\left({ℝ}^n\right)$ is surjective? We shall give some conditions on A and μ which assure the surjectivity of D.

Nous démontrons l’unicité des solutions faibles pour une classe d’équations de transport dont les vitesses sont partiellement à variations bornées. Nous nous intéressons à des champs de vecteurs du type$$a_1\left(x_1\right)·{\partial }_{x_1}+a_2(x_1,x_2)·{\partial }_{x_2},a_1\in BV\left({ℝ}_{x_1}^{N_1}\right),a_2\in L_{x_1}^1\left(BV\left({ℝ}_{x_2}^{N_2}\right)\right),$$avec une borne sur la divergence de chacun des champs $a_1,a_2$. Ce modèle a été étudié récemment dans [LL] par C. Le Bris et P.-L. Lions avec une régularité $W^{1,1}$ ; nous montrons ici également que, dans le cas $W^{1,1}$, le contrôle $L^{\infty }$ de la divergence totale du champ est suffisant. Notre méthode consiste à démontrer...

This paper has two objectives. First, we prove the existence of solutions to the general advection-diffusion equation subject to a reasonably smooth initial condition. We investigate the behavior of the solution of these problems for large values of time. Secondly, a numerical scheme using the Sinc-Galerkin method is developed to approximate the solution of a simple model of turbulence, which is a special case of the advection-diffusion equation, known as Burgers' equation. The approximate solution...