Operator-theoretic solution of stochastic systems

A (stochastic) operator-theoretic approach leads to expresssions for inverses of linear and nonlinear stochastic operators—useful for the solution of linear or nonlinear stochastic differential equations. Operator equations are developed for inverses of linear or nonlinear stochastic operators. Series expressions are obtained which allow writing the solution y=$\text{F}$^?1x of the operator equation $\text{F}$y=x. Special cases are studied in which $\text{F}$ may be linear or nonlinear, deterministic or stochastic in various combinations.     

Solutions entropiques des problèmes non locauxEntropy solutions of nonlocal problems

We introduce a notion of entropy solution for the nonlocal problem $\text{C}\text{f+f=ψ}$ on $\text{?Ω}$, where $\text{ψ∈}\text{L}{}^1\text{(?Ω)}$ and $\text{C}$ is a nonlinear capacity operator. We prove its existence and uniqueness. This notion of solution allows also to solve a general elliptic problem with nonlinear boundary conditions. To cite this article: K. Ammar, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 751–756.     

Multi-dimensional G-Brownian motion and related stochastic calculus under G-expectation

We develop a notion of nonlinear expectation–G$G$-expectation–generated by a nonlinear heat equation with infinitesimal generator G$G$. We first study multi-dimensional G$G$-normal distributions. With this nonlinear distribution we can introduce our G$G$-expectation under which the canonical process is a multi-dimensional G$G$-Brownian motion. We then establish the related stochastic calculus, especially stochastic integrals of Itô’s type with respect to our G$G$-Brownian motion, and derive the related Itô’s formula. We have also obtained the existence and uniqueness of stochastic differential equations under our G$G$-expectation.     

Regularity and decay of solutions of nonlinear harmonic oscillators

We prove sharp analytic regularity and decay at infinity of solutions of variable coefficients nonlinear harmonic oscillators. Namely, we show holomorphic extension to a sector in the complex domain, with a corresponding Gaussian decay, according to the basic properties of the Hermite functions in ${\mathbb{R}}^d$. Our results apply, in particular, to nonlinear eigenvalue problems for the harmonic oscillator associated to a real-analytic scattering, or asymptotically conic, metric in ${\mathbb{R}}^d$, as well as to certain perturbations of the classical harmonic oscillator.     

On w-compatible mappings and common coupled coincidence point in cone metric spaces

Recently, Abbas et al. [M. Abbas, M.A. Khan, S. Radenovi?, Common coupled fixed point theorems in cone metric spaces for $w$-compatible mapping, Applied Mathematics and Computation (2010) doi:10.1016/j.amc.2010.05.042] introduced the concept of $w$-compatible mappings and obtained results on coupled coincidence point for nonlinear contractive mappings in a cone metric space. In the present paper, we introduce the concept of a common coupled coincidence point of the mappings $F,G:X\times X\to X$ and $f:X\to X$ and we prove some theorems for nonlinear contractive mappings in a cone metric space with a cone having nonempty interior. Our results generalize several well known comparable results in the literature.     

Stabilization and tracking control for a class of nonlinear systems

This paper discusses stabilization and tracking control using linear matrix inequalities for a class of systems with Lipschitz nonlinearities. A nonlinear state feedback stabilization control is proposed for systems containing a more general case of Lipschitz nonlinearity. The main objective of the present study is to provide, for multi-input multi-output nonlinear systems, a tracking control approach based on nonlinear state feedback, which guarantees global asymptotic output and state tracking with zero tracking error in the steady state. Further, the tracking control is formulated for optimal disturbance rejection, using $L_2$ gain reduction based performance criteria. The proposed methodologies are illustrated herein using two simulation examples of chaotic and unstable dynamical systems.     

Integration theory for Hammerstein operators

We develop in this article a strong nonlinear integral and obtain a Riesz-type theorem (utilizing this integral) for the class of (nonlinear) Hammerstein operators. The integral is extended to the class $\text{M}$_E($\text{B}$) of E-valued totally $\text{B}$-measurable functions and convergence theorems are studied. Then an exchange of information is carried out between the operators and the corresponding set functions; for example, the implication of the operator being compact or unconditionally summing is drawn. In the latter case it is shown that the representing set function is analogous to strongly bounded set functions. A vast body of literature exists for both of these concepts.     

Controllability of a 4 × 4 quadratic reaction–diffusion system

We consider a $4\times 4$ nonlinear reaction–diffusion system posed on a smooth domain Ω of ${\mathbb{R}}^N$ ($N\ge 1$) with controls localized in some arbitrary nonempty open subset ω of the domain Ω. This system is a model for the evolution of concentrations in reversible chemical reactions. We prove the local exact controllability to stationary constant solutions of the underlying reaction–diffusion system for every $N\ge 1$ in any time $T>0$. A specificity of this control system is the existence of some invariant quantities in the nonlinear dynamics. The proof is based on a linearization which uses return method and an adequate change of variables that creates cross diffusion which will be used as coupling terms of second order. The controllability properties of the linearized system are deduced from Carleman estimates. A Kakutani's fixed-point argument enables to go back to the nonlinear parabolic system. Then, we prove a global controllability result in large time for $1\le N\le 2$ thanks to our local controllability result together with a known theorem on the asymptotics of the free nonlinear reaction–diffusion system.     

Existence of periodic solutions for nonlinear differential equations with a p-Laplacian-like operator

The existence of periodic solutions for nonlinear differential equations with a $p$-Laplacian-like operator is studied by applying a new generalization of polar coordinates.     

Nonlinear boundary value problems and operators TT1

We consider nonlinear boundary value problems of the type $\text{L? + N? = 0}$ for the existence of solutions. It is assumed that L is a 2nth-order linear differential operator in the real Hilbert space S = L²[a, b] which admits a decomposition of the form $\text{L =}{}^{\mathrm{TT1}}$ where T is an nth-order linear differential operator and N is a nonlinear operator defined on a subspace of S. The decomposition of L induces a natural decomposition of the generalized inverse of L. Using the method of “alternative problems,” we split the boundary value problem into an equivalent system of two equations. The theory of monotone operators and the theory of nonlinear Hammerstein equations are then utilized to consider the solvability of the equivalent system.