Alain Pietrus

On the discretization of switched linear systems

Abstract A bilinear single-control system which can be viewed as a control formulation of a linear switched system is considered. The control is restricted to take values either (i) in { 0 , 1 } (switched system), or (ii) in [ 0 , 1 ] (relaxed system). In more practical considerations the control is often allowed to change only at the points of a given time-net with a step length h . The paper investigates what is the approximation error in terms of the reachable set in the two cases (i) and (ii). The error estimates that follow directly from known results are of order h and h , respectively. In the present paper estimations of order h and h 1.5 are proved in a constructive way. The second one makes use of the effect of non-accumulation of errors established earlier by the second author.

The Euler Method for Linear Control Systems Revisited

Although optimal control problems for linear systems have been profoundly investigated in the past more than 50 years, the issue of numerical approximations and precise error analyses remains challenging due the bang-bang structure of the optimal controls. Based on a recent paper by M. Quincampoix and V.M. Veliov on metric regularity of the optimality conditions for control problems of linear systems the paper presents new error estimates for the Euler discretization scheme applied to such problems. It turns out that the accuracy of the Euler method depends on the “controllability index” associated with the optimal solution, and a sharp error estimate is given in terms of this index. The result extends and strengthens in several directions some recently published ones.

A General Iterative Procedure for Solving Nonsmooth Generalized Equations

We present a general iterative procedure for solving generalized equations in the nonsmooth framework. To this end, we consider a class of functions admitting a certain type of approximation and establish a local convergence theorem that one can apply to a wide range of particular problems.

The Cahn-Hilliard equation for an isotropic deformable continuum

Abstract In this note, we study a Cahn-Hilliard equation in a deformable elastic isotropic continuum. We define boundary conditions associated with the problem and obtain the existence and uniqueness of solutions. We also study the long time behavior of the system.

High order discrete approximations to Mayer's problems for linear systems

This paper presents a discretization scheme for Mayers type optimal control problems of linear systems. The scheme is based on second order Volterra--Fliess approximations, and on an augmentation of the control variable in a control set of higher dimension. Compared with the existing results, it has the advantage of providing a higher order accuracy, which may make it more efficient when aiming for a certain precision. Error estimations (depending on the controllability index of the system at the solution) are proved by using a recent result about stability of the optimal solution with respect to disturbances. Numerical results are provided which show the sharpness of the error estimations.

A Hummel–Seebeck type method for variational inclusions

We study the stability of a Hummel–Seebeck like method for solving variational inclusions of the form 0 ∈ f(x) + G(x), where f is a single-valued function while G stands for a set-valued mapping, both of them acting in Banach spaces. Then, we investigate a measure of conditioning of these inclusions under canonical perturbations.

On the convergence of some methods for variational inclusions

In this paper, we study variational inclusions of the following form 0 ∈f(x) + g(x) + F(x) (*) wheref is differentiable in a neighborhood of a solutionx* of (*) andg is differentiable atx* and F is a set-valued mapping with closed graph acting in Banach spaces. The method introduced to solve (*) is superlinear and quadratic when ∇f is Lipschitz continuous.ResumenEn este artículo se estudian inclusiones variacionales de la forma 0 ∈f(x) + g(x) + F(x) (*) dondef es diferenciable en un entorno de la soluciónx* de (*),g es diferenciable enx* yF es una aplicación con gráfica cerrada entre espacios de Banach. El método introducido para resolver (*) es superlineal y cuadrático cuando ∇f es continuo y verifica la condición de Lipschtz.

A differential inclusion approach for modeling and analysis of dynamical systems under uncertainty

In this paper we deal with the application of differential inclusions to modeling nonlinear dynamical systems under uncertainty in parameters. In this case, differential inclusions seem to be better suited to modeling practical situations under uncertainty and imprecision than formulations by means of fuzzy differential equations. We develop a practical algorithm to approximate the reachable sets of a class of nonlinear differential inclusion, which eludes the computational problems of a previous set-valued version of the Heun’s method. Our algorithm is based on a complete discretization (time and state space) of the differential inclusion and it suits hardware features, handling the memory used by the method in a controlled fashion during all iterations. As a case of study, we formulate a differential inclusion to model an epidemic outbreak of dengue fever under Cuban conditions. The model takes into account interaction of human and mosquito populations as well as vertical transmission in the mosquito population. It is studied from the theoretical point of view to apply the Practical Algorithm. Also, we estimate the temporal evolution of the different human and mosquito populations given by the model in the Dengue 3 epidemic in Havana 2001, through the computation of the reachable sets using the Practical Algorithm.

Dengue model described by differential inclusions

In this paper we formulate a differential inclusion to model an epidemic outbreak of Dengue fever in the Cuban conditions. The model takes into account interaction of human and mosquito populations as well as vertical transmission in the mosquito population. Finally, we propose a mathematical framework allowing us to make suitable predictions about the populations of humans, mosquitoes and eggs infected during the epidemic time.

A globally convergent method for solving nonlinear equations without the differentiability condition

We give a general iterative method which computes the maximal real rootxmax of a one variable Lipschitzian function in a given interval. The method generates a monotonically decreasing sequence which converges towardsxmax or demonstrates the non-existence of a real root in the considered interval. We show that the method is globally convergent and locally linearly convergent. We also compute the number of iterations needed to reach the given accuracy.