Singularly perturbed differential inclusions: An averaging approach

We consider nonlinear, singularly perturbed differential inclusions and apply the averaging method in order to construct a limit differential inclusion for slow motion. The main approximation result states that the existence and regularity of the limit differential inclusion suffice to describe the limit behavior of the slow motion. We give explicit approximation rates for the uniform convergence on compact time intervals. The approach works under controllability or stability properties of fast motion.     

Sweeping process with regular and nonregular sets

Differential inclusions involving the normal cone to a moving set are investigated. A special attention is paid to the sweeping process associated with sets for which no regularity assumption is required.     

Discontinuous control problems for non-convex dynamics and near viability for singularly perturbed control systems

The aim of this paper is to study two classes of discontinuous control problems without any convexity assumption on the dynamics. In the first part we characterize the value function for the Mayer problem and the supremum cost problem using viscosity tools and the notion of ε-viability (near viability). These value functions are given with respect to discontinuous cost functionals. In the second part we obtain results describing the ε-viability (near viability) of singularly perturbed control systems.     

Characterization of the optimal trajectories for the averaged dynamics associated to singularly perturbed control systems

The aim of this paper is to study singularly perturbed control systems. Firstly, we provide linearized formulation version for the calculus of the value function associated with the averaged dynamics. Secondly, we obtain necessary and sufficient conditions in order to identify the optimal trajectory of the averaged system.     

Singularly perturbed finite element methods

Summary This paper introduces a new piecewise linear finite element, which is designed to handle singularly perturbed ordinary differential equations. Both pointwise and global estimates (which are independent of the perturbation parameter) are obtained.     

Sufficient conditions for optimality for differential inclusions of parabolic type and duality

Sufficient conditions for optimality are derived for partial differential inclusions of parabolic type on the basis of the apparatus of locally conjugate mapping, and duality theorems are proved. The duality theorems proved allow one to conclude that a sufficient condition for an extremum is an extremal relation for the direct and dual problems.      

Filippov and invariance theorems for mutational inclusions of tubes

The framework of transitions and mutational calculus inspired by shape optimization allows the notions of derivative, tangent cone, and differential equation to be extended to a metric space and especially to the family of all nonempty compact subsets of a given domainE. It gives tools to study the evolution of tubes and fundamental theorems such as those of Cauchy-Lipschitz, Nagumo, or Lyapunov, well known in vector spaces, can be adapted to mutational equations. The present paper deals with mutational inclusions of tubes which include many tube control problems and an adaptation of the Filippov theorem is proved. As a consequence, an invariance theorem is stated.     

The study of Jacobi and cyclic Jacobi matrix eigenvalue problems using Sturm-Liouville theory

We study the eigenvalues of matrix problems involving Jacobi and cyclic Jacobi matrices as functions of certain entries. Of particular interest are the limits of the eigenvalues as these entries approach infinity. Our approach is to use the recently discovered equivalence between these problems and a class of Sturm-Liouville problems and then to apply the Sturm-Liouville theory.     

Three anti-periodic solutions for second-order impulsive differential inclusions via nonsmooth critical point theory

A nonsmooth version of a three critical point theorem of Ricceri (due to Iannizzotto) is used to obtain three anti-periodic solutions for a second-order impulsive differential inclusions with a perturbed nonlinearity and two parameters.     

BV solutions of nonconvex sweeping process differential inclusion with perturbation

This paper is devoted to the study of differential inclusions, particularly discontinuous perturbed sweeping processes in the infinite-dimensional setting. On the one hand, the sets involved are assumed to be prox-regular and to have a variation given by a function which is of bounded variation and right continuous. On the other hand, the perturbation satisfies a linear growth condition with respect to a fixed compact subset. Finally, the case where the sets move in an absolutely continuous way is recovered as a consequence.     

Symmetry and-commutativity of topological*-algebras

An advertibly complete locallym-convex (lmc)^*-algebraE is symmetric if and only if each normed (inverse limit) factorE/N , A, ofE is symmetric in the respective Banach factorE , A, ofE. Every locally C^*-algebra is symmetric. If denotes the continuous positive functionals on an lmc^*-algebraE and withL _f ={x E: f(x ^* x) =0}, thenE is, by definition,-commutative if for anyx, y E.-commutativity and commutativity coincide in lmcC ^*-algebras, so that an lmc^*-algebra with a bounded approximate identity is-commutative if and only if its enveloping algebra is commutative. Several standard results for commutative lmc^*-algebras are also obtained in the-commutative case, as for instance, the nonemptiness of the Gel'fand space of a suitable-commutative lmc^*-algebra, the automatic continuity of positive functionals when the algebras involved factor, as well as that the spectral radius is a continuous submultiplicative semi-norm, when the algebras considered are moreover symmetric. An application of the latter result yields a spectral characterization of-commutativity.     

Concentration of measure and isoperimetric inequalities in product spaces

The concentration of measure phenomenon in product spaces roughly states that, if a set A in a product Ω^N of probability spaces has measure at least one half, “most” of the points of Ωn are “close” to A. We proceed to a systematic exploration of this phenomenon. The meaning of the word “most” is made rigorous by isoperimetrictype inequalities that bound the measure of the exceptional sets. The meaning of the work “close” is defined in three main ways, each of them giving rise to related, but different inequalities. The inequalities are all proved through a common scheme of proof. Remarkably, this simple approach not only yields qualitatively optimal results, but, in many cases, captures near optimal numerical constants. A large number of applications are given, in particular to Percolation, Geometric Probability, Probability in Banach Spaces, to demonstrate in concrete situations the extremely wide range of application of the abstract tools. Dedicated to Vitali Milman     

Evolution equations governed by the sweeping process

This paper is concerned with variants of the sweeping process introduced by J.J. Moreau in 1971. In Section 4, perturbations of the sweeping process are studied. The equation has the formX(t) -N _C(t) (X(t)) +F(t, X(t)). The dimension is finite andF is a bounded closed convex valued multifunction. WhenC(t) is the complementary of a convex set,F is globally measurable andF(t, ·) is upper semicontinuous, existence is proved (Th. 4.1). The Lipschitz constants of the solutions receive particular attention. This point is also examined for the perturbed version of the classical convex sweeping process in Th. 4.1. In Sections 5 and 6, a second-order sweeping process is considered:X (t) -N _C(X(t)) (X(t)). HereC is a bounded Lipschitzean closed convex valued multifunction defined on an open subset of a Hilbert space. Existence is proved whenC is dissipative (Th. 5.1) or when allC(x) are contained in a compact setK (Th. 5.2). In Section 6, the second-order sweeping process is solved in finite dimension whenC is continuous.     

Singularly perturbed elliptic systems

We study the existence and concentration behavior of positive solutions for a class of Hamiltonian systems (two coupled nonlinear stationary Schrödinger equations). Combining the Legendre–Fenchel transformation with mountain pass theorem, we prove the existence of a family of positive solutions concentrating at a point in the limit, where related functionals realize their minimum energy. In some cases, the location of the concentration point is given explicitly in terms of the potential functions of the stationary Schrödinger equations.     

An equational approach to products of relatively regular operators

A bounded linear operatorT on Banach spaceX is called relatively regular if its nullspaceN(T) and rangeR(T) are closed complemented subspaces ofX. It is known that the product of two relatively regular operators is not necessarily relatively regular. This paper shows how to find conditions, more general than those previously known, to ensure that two relatively regular operators have relatively regular product.This work was supported in part by NRC Operating Grant A3985 and Canada Council Leave Fellowship.     

Inequalities for sums of independent geometrical random variables

Summary We give a survey of known results regarding Schur-convexity of probability distribution functions. Then we prove that the functionF(p ₁,...,p_n;t)=P(X₁+...+X_n≤t) is Schur-concave with respect to (p ₁,...,p_n) for every realt, whereX _i are independent geometric random variables with parametersp _i. A generalization to negative binomial random variables is also presented.