On the admissibility of unboundedness properties of forced deterministic and stochastic sublinear Volterra summation equations

In this paper we consider unbounded solutions of perturbed convolution Volterra summation equations. The equations studied are asymptotically sublinear, in the sense that the state--dependence in the summation is of smaller than linear order for large absolute values of the state. When the perturbation term is unbounded, it is elementary to show that solutions are also. The main results of the paper are mostly of the following form: the solution has an additional unboundedness property $U$ if and only if the perturbation has property $U$. Examples of property $U$ include monotone growth, monotone growth with fluctuation, fluctuation on $\mathbb{R}$ without growth, existence of time averages. We also study the connection between the times at which the perturbation and solution reach their running maximum, and the connection between the size of signed and unsigned running maxima of the solution and forcing term.Comment: 45 page

Mass concentration in a nonlocal model of clonal selection

Self-renewal is a constitutive property of stem cells. Testing the cancer stem cell hypothesis requires investigation of the impact of self-renewal on cancer expansion. To understand better this impact, we propose a mathematical model describing dynamics of a continuum of cell clones structured by the self-renewal potential. The model is an extension of the finite multi-compartment models of interactions between normal and cancer cells in acute leukemias. It takes a form of a system of integro-differential equations with a nonlinear and nonlocal coupling, which describes regulatory feedback loops in cell proliferation and differentiation process. We show that such coupling leads to mass concentration in points corresponding to maximum of the self-renewal potential and the model solutions tend asymptotically to a linear combination of Dirac measures. Furthermore, using a Lyapunov function constructed for a finite dimensional counterpart of the model, we prove that the total mass of the solution converges to a globally stable equilibrium. Additionally, we show stability of the model in space of positive Radon measures equipped with flat metric. The analytical results are illustrated by numerical simulations

Non-fragile H∞ control with randomly occurring gain variations, distributed delays and channel fadings

This study is concerned with the non-fragile H∞ control problem for a class of discrete-time systems subject to randomly occurring gain variations (ROGVs), channel fadings and infinite-distributed delays. A new stochastic phenomenon (ROGVs), which is governed by a sequence of random variables with a certain probabilistic distribution, is put forward to better reflect the reality of the randomly occurring fluctuation of controller gains implemented in networked environments. A modified stochastic Rice fading model is then exploited to account for both channel fadings and random time-delays in a unified representation. The channel coefficients are a set of mutually independent random variables which abide by any (not necessarily Gaussian) probability density function on [0, 1]. Attention is focused on the analysis and design of a non-fragile H∞ outputfeedback controller such that the closed-loop control system is stochastically stable with a prescribed H∞ performance. Through intensive stochastic analysis, sufficient conditions are established for the desired stochastic stability and H∞ disturbance attenuation, and the addressed non-fragile control problem is then recast as a convex optimisation problem solvable via the semidefinite programme method. An example is finally provided to demonstrate the effectiveness of the proposed design method

Min-Max Predictive Control of a Pilot Plant using a QP Approach

47th IEEE Conference on Decision and Control 9-11 Dec. 2008The practical implementation of min-max MPC (MMMPC) controllers is limited by the computational burden required to compute the control law. This problem can be circumvented by using approximate solutions or upper bounds of the worst possible case of the performance index. In a previous work, the authors presented a computationally efficient MMMPC control strategy in which a close approximation of the solution of the min-max problem is computed using a quadratic programming problem. In this paper, this approach is validated through its application to a pilot plant in which the temperature of a reactor is controlled. The behavior of the system and the controller are illustrated by means of experimental results

Problèmes de stabilisation au bord pour des systèmes d'équations aux dérivées partielles hyperboliques en dimension un d'espace

Dans cette thèse, nous étudions le problème de stabilisation au bord de systèmes généraux d'équations aux dérivées partielles hyperboliques. Plus précisément, l'étude se focalise sur des systèmes où le transport est uniquement scalaire et où le sens de propagation de l'information est fixé. En outre, le contrôle choisi sera la plupart du temps sous la forme d'une loi de retour d'état (ou feedback) linéaire que l'on perturbera éventuellement par l'effet d'une saturation. Le travail est séparé en deux parties bien distinctes ; l'une se concentre sur des méthodes de Lyapunov, tandis que l'autre va plutôt utiliser des techniques propres au linéaire. Pour la première partie, deux travaux principaux sont présentés. Dans un premier temps, nous ne considérons que des équations de transport linéaires à vitesses positives et cherchons à stabiliser exponentiellement le système dans L8 grâce à un feedback linéaire saturé. La méthode consiste à utiliser des techniques classiques de Lyapunov afin d'exhiber un bassin d'attraction et d'en donner une estimation fine. On généralise ensuite ce travail dans un cadre BV pour les systèmes de lois de conservation scalaires couplées au bord. Secondement, un système de lois de conservation scalaires à vitesses positives est discrétisé en utilisant un schéma à limiteur de pente. En s'inspirant des méthodes issues du cadre continu, une fonctionnelle de Lyapunov discrète est étudiée pour prouver la stabilisation exponentielle BV par feedback linéaire de la solution discrète. Pour la seconde partie, deux études sont également exposées mais cette fois-ci, dans un cadre totalement linéaire. D'une part, il s'agit d'établir la possibilité de construire un feedback issu d'un placement de pôles pour stabiliser exponentiellement des edps hyperboliques linéaires avec couplage au bord et dans le domaine. D'autre part, nous développons une théorie du backstepping discrétisé pour stabiliser en temps fini un schéma numérique modélisant un système 2 × 2 avec couplage au bord et au sein du domaine.In this thesis, we study the problem of boundary stabilization of general hyperbolic systems of partial differential equations. More precisely, the analysis focuses on systems where the transport term is scalar and for which the information propagates in a fixed direction. In addition, the chosen control is most of the time a state feedback law for which a saturation is possibly applied. The work is divided into two distinct parts, one focusing on Lyapunov techniques while the other one uses the linearity of the problem. In the first part of the thesis, two main works are presented. In the first one, only linear transport equations with positive velocities are considered. The main goal is to design a saturated linear feedback in order to stabilize exponentially the open-loop system in L 8 . The method consists of using classical Lyapunov techniques to exhibit a basin of attraction for which a fine estimate is given. We also extend this work to nonlinear scalar conservation laws in a BV framework. In the other work, thanks to a slope limiter scheme, a system of scalar conservation laws is discretized. Inspired by "continuous" Lyapunov methods, a discrete Lyapunov functional is studied to prove the exponential BV stabilization of the discrete solution using a linear feedback. In the second part of the thesis, two works are exposed as well, this time in a full linear framework. On the one hand, we study systems of linear transport equations of arbitrary dimension, coupled on the domain and at the boundary. Designing a controller from a pole placement algorithm, the exponential stabilization is proved in L 2 . On the other hand, we develop a numerical Backstepping theory in order to stabilize in finite time a numerical scheme modeling a 2 × 2 linear system with in domain and boundary couplings

Impulsive perturbations to differential equations: stable/unstable pseudo-manifolds, heteroclinic connections, and flux

State-dependent time-impulsive perturbations to a two-dimensional autonomous flow with stable and unstable manifolds are analysed by posing in terms of an integral equation which is valid in both forwards- and backwards-time. The impulses destroy the smooth invariant manifolds, necessitating new definitions for stable and unstable pseudo-manifolds. Their time-evolution is characterised by solving a Volterra integral equation of the second kind with discontinuous inhomogeniety. A criteria for heteroclinic trajectory persistence in this impulsive context is developed, as is a quantification of an instantaneous flux across broken heteroclinic manifolds. Several examples, including a kicked Duffing oscillator and an underwater explosion in the vicinity of an eddy, are used to illustrate the theory