to be done in this area. Face recognition is a problem that scarcely clamoured for attention before the computer age but, having surfaced, has involved a wide range of techniques and has attracted the attention of some fine minds (David Mumford was a Fields Medallist in 1974). This singular application of mathematical modelling to a messy applied problem of obvious utility and importance but with no unique solution is a pretty one to share with students: perhaps, returning to the source of our opening quotation, we may invert Duncan's earlier observation, 'There is an art to find the mind's construction in the face!'.

Linear and nonlinear waves, by G. B. Whitham. Pp.636. £50. 1999. ISBN 0 471 35942 4 (Wiley).

关于Semigroups of Linear Operators and Applications to Partial Differential Equations的两个注解

In a pedagogical but exhaustive manner, this survey reviews the main results on input-to-state stability (ISS) for infinite-dimensional systems. This property allows for the estimation of the impac...

https://hal.archives-ouvertes.fr/hal-02921344/document

Input-to-State Stability of Infinite-Dimensional Systems: Recent Results and Open Questions

We consider the problem of global minimization of rational functions on ** (unconstrained case), and on an open, connected, semi-algebraic subset of **, or the (partial) closure of such a set (constrained case). We show that in the univariate case (n = 1), these problems have exact reformulations as semidefinite programming (SDP) problems, by using reformulations introduced in the PhD thesis of Jibetean [16]. This extends the analogous results by Nesterov [13] for global minimization of univariate polynomials.For the bivariate case (n = 2), we obtain a fully polynomial time approximation scheme (FPTAS) for the unconstrained problem, if an a priori lower bound on the infimum is known, by using results by De Klerk and Pasechnik [1].For the NP-hard multivariate case, we discuss semidefinite programming-based relaxations for obtaining lower bounds on the infimum, by using results by Parrilo [15], and Lasserre [12].

Global optimization of rational functions: A semidefinite programming approach

Boundary control of delayed ODE–heat cascade under actuator saturation

This article deals with the design of saturated controls in the context of partial
differential equations. It focuses on a Korteweg–de Vries equation, which is a nonlinear mathematical
model of waves on shallow water surfaces. Two different types of saturated controls are considered.
The well-posedness is proven applying a Banach fixed-point theorem, using some estimates of this
equation and some properties of the saturation function. The proof of the asymptotic stability of the
closed-loop system is separated in two cases: (i) when the control acts on all the domain, a Lyapunov
function together with a sector condition describing the saturating input is used to conclude on the
stability; (ii) when the control is localized, we argue by contradiction. Some numerical simulations
illustrate the stability of the closed-loop nonlinear partial differential equation.

/pdf/global-stabilization-of-a-korteweg-de-vries-equation-with-548j84tz3o.pdf

Global stabilization of a Korteweg-de Vries equation with saturating distributed control

This work studies the influence of some constraints on a stabilizing feedback law. It is considered an abstract nonlinear control system for which we assume that there exists a linear feedback law that makes the origin of the closed-loop system globally asymptotically stable. This controller is then modified via a cone-bounded nonlinearity. A well-posedness and a stability theorems are stated. The first theorem is proved thanks to the Schauder fixed-point theorem, the second one with an infinite-dimensional version of LaSalle's Invariance Principle. These results are illustrated on a linear Korteweg-de Vries equation by some simulations and on a nonlinear heat equation.

/pdf/cone-bounded-feedback-laws-for-m-dissipative-operators-on-4hv1o9qyan.pdf

Cone-bounded feedback laws for m-dissipative operators on Hilbert spaces

Stabilization of the linear Kuramoto-Sivashinsky equation with a delayed boundary control

This article deals with the design of saturated controls in the context of partial differential equations. It focuses on a Korteweg-de Vries equation, which is a nonlinear mathematical model of waves on shallow water surfaces. Two different types of saturated controls are considered. The well-posedness is proven applying a Banach fixed point theorem, using some estimates of this equation and some properties of the saturation function. The proof of the asymptotic stability of the closed-loop system is separated in two cases: i) when the control acts on all the domain, a Lyapunov function together with a sector condition describing the saturating input is used to conclude on the stability, ii) when the control is localized, we argue by contradiction. Some numerical simulations illustrate the stability of the closed-loop nonlinear partial differential equation. 1. Introduction. In recent decades, a great effort has been made to take into account input saturations in control designs (see e.g [39], [15] or more recently [17]). In most applications, actuators are limited due to some physical constraints and the control input has to be bounded. Neglecting the amplitude actuator limitation can be source of undesirable and catastrophic behaviors for the closed-loop system. The standard method to analyze the stability with such nonlinear controls follows a two steps design. First the design is carried out without taking into account the saturation. In a second step, a nonlinear analysis of the closed-loop system is made when adding the saturation. In this way, we often get local stabilization results. Tackling this particular nonlinearity in the case of finite dimensional systems is already a difficult problem. However, nowadays, numerous techniques are available (see e.g. [39, 41, 37]) and such systems can be analyzed with an appropriate Lyapunov function and a sector condition of the saturation map, as introduced in [39]. In the literature, there are few papers studying this topic in the infinite dimensional case. Among them, we can cite [18], [29], where a wave equation equipped with a saturated distributed actuator is studied, and [12], where a coupled PDE/ODE system modeling a switched power converter with a transmission line is considered. Due to some restrictions on the system, a saturated feedback has to be designed in the latter paper. There exist also some papers using the nonlinear semigroup theory and focusing on abstract systems ([20],[34],[36]). Let us note that in [36], [34] and [20], the study of a priori bounded controller is tackled using abstract nonlinear theory. To be more specific, for bounded ([36],[34]) and unbounded ([34]) control operators, some conditions are derived to deduce, from the asymptotic stability of an infinite-dimensional linear system in abstract form, the asymptotic stability when closing the loop with saturating controller. These articles use the nonlinear semigroup theory (see e.g. [24] or [1]). The Korteweg-de Vries equation (KdV for short)

We propose to solve polynomial hyperbolic partial differential equations (PDEs) with convex optimization This approach is based on a very weak notion of solution of the nonlinear equation, namely the measure-valued (mv) solution, satisfying a linear equation in the space of Borel measures The aim of this paper is, first, to provide the conditions that ensure the equivalence between the two formulations and, second, to introduce a method which approximates the infinite-dimensional linear problem by a hierarchy of convex, finite-dimensional, semidefinite programming problems This result is then illustrated on the celebrated Burgers equation We also compare our results with an existing numerical scheme, namely the Godunov scheme

/pdf/a-moment-approach-for-entropy-solutions-to-nonlinear-2kxad6lobd.pdf

Swann Marx

Papers

Global stabilization of a Korteweg-de Vries equation with saturating distributed control

Cone-bounded feedback laws for m-dissipative operators on Hilbert spaces

Stabilization of the linear Kuramoto-Sivashinsky equation with a delayed boundary control

Global stabilization of a Korteweg-de Vries equation with saturating distributed control

A moment approach for entropy solutions to nonlinear hyperbolic PDEs