In this chapter, we survey recent results on the numerical solutions of the Hammerstein equations. Hammerstein equations arise naturally in connection with the Laplace equation with a certain class of nonlinear boundary conditions. The Hammerstein equations with smooth as well as weakly singular kernels will be treated.

Using the connection between closed solution curves of the Vortex Filament Equation and the periodic problem for the nonlinear Schrr odinger equation (NLS), we investigate the possibility of relating the knot types of nite-gap solutions to the Floquet spectra of the corresponding NLS potentials, in the special case of small amplitude curves close to multiply-covered circles.

This paper is intended to give a probabilistic representation for stochastic viscosity solution of semi-linear reflected stochastic partial differential equations with nonlinear Neumann boundary condition. We use its connection with reflected generalized backward doubly stochastic differential equations. AMS Subject Classification: 60H15; 60H20

A linear version of backward stochastic differential equations (BSDEs) was first studied by Bismut [4] as the adjoint processes in the maximum principal of stochastic control. Pardoux and Peng in [20] introduced the notion of nonlinear BSDE. Since then, the interest in BSDEs has increased. Indeed, BSDEs provide connection with mathematical finance [10], stochastic control [11], and stochastic g...

The paper provides a powerful basic concept for object manipulation with finger relocation and extends the previous approaches in the area of stratified manipulation. The algorithm gives also some improvements of the developed software for smooth and stratified motion planning (MP). The key element of the approach is based strongly on the theory of stratified control for nonlinear systems. The ...

We study existence and uniqueness of a variational solution in terms of stochastic variational inequalities (SVI) to stochastic nonlinear diffusion equations with a highly singular diffusivity term and multiplicative Stratonovich gradient-type noise. We derive a commutator relation for the unbounded noise coefficients in terms of a geometric Killing vector condition. The drift term is given by ...

Model predictive control (MPC) is a very popular controller design method in the process industry. A key advantage of MPC is that it can accommodate constraints on the inputs and outputs. Usually MPC uses linear or nonlinear discrete-time models. In this paper we give an overview of some results in connection with model predictive control (MPC) approaches for some tractable classes of discrete-...

We construct a model of innovation diffusion that incorporates a spatial component into a classical imitation-innovation dynamics first introduced by F. Bass. Relevant for situations where the imitation process explicitly depends on the spatial proximity between agents, the resulting nonlinear field dynamics is exactly solvable. As expected for nonlinear collective dynamics, the imitation mecha...

This paper shows the circuitry implementation and practical verification of the autonomous nonlinear oscillator. Since it is described by a single third-order differential equation, its state variables can be considered as the position, velocity and acceleration and thus have direct connection to a real physical system. Moreover, for some specific configurations of internal system parameters, i...

Quantum control of ultra-cold atoms: uncovering a novel connection between two paradigms of quantum nonlinear dynamics Jiao Wang ab; Anders S. Mouritzen cd; Jiangbin Gong ce a Temasek Laboratories, National University of Singapore, Singapore b Beijing-Hong Kong-Singapore Joint Center for Nonlinear and Complex Systems (Singapore), National University of Singapore, Singapore c Department of Physi...