An improved result of exponential stability of stochastic semilinear evolution equations

Sufficient conditions for the exponential stability of a class ofnonlinear, non-autonomous stochastic differential equations in infinitedimensions are studied. The analysis consists of introducing a suitableapproximating solution systems and using a limiting argument to pass onstability of strong solutions to mild ones. As a consequence, the classicalcriteriaof stability in A. Ichikawa [8] are improved and extended to cover a class ofnon-autonomous stochastic evolution equations.Two examples are investigated to illustrate our theory.

Moment decay rates of solutions of stochastic differential equations

The objective of this paper is to investigate the p-ίh moment asymptotic stability decay rates for certain finite-dimensional Itό stochastic differential equations. Motivated by some practical examples, the point of our analysis is a special consideration of general decay speeds, which contain as a special case the usual exponential or polynomial type one, to meet various situations. Sufficient conditions for stochastic differential equations (with variable delays or not) are obtained to ensure their asymptotic properties. Several examples are studied to illustrate our theory.

A class of generalized Lévy Laplacians in infinite dimensional calculus

A class of generalized Lévy Laplacians which contain as a special case the ordinary Lévy Laplacian are considered. Topics such as limit average of the second order functional derivative with respect to a certain equally dense (uniformly bounded) orthonormal base, the relations with Kuo’s Fourier transform and other infinite dimensional Laplacians are studied.

Birth-death processes with an instantaneous reflection barrier

A new structure with the special property that an instantaneous reflection barrier is imposed on the ordinary birth-death processes is considered. An easy-checking criterion for the existence of such Markov processes is first obtained. The uniqueness criterion is then established. In the nonunique case, all the honest processes are explicitly constructed. Ergodicity properties for these processes are investigated. It is proved that honest processes are always ergodic without necessarily imposing any extra conditions. Equilibrium distributions for all these ergodic processes are established. Several examples are provided to illustrate our results.

Birth-death processes with disaster and instantaneous resurrection

A new structure with the special property that instantaneous resurrection and mass disaster are imposed on an ordinary birth-death process is considered. Under the condition that the underlying birth-death process is exit or bilateral, we are able to give easily checked existence criteria for such Markov processes. A very simple uniqueness criterion is also established. All honest processes are explicitly constructed. Ergodicity properties for these processes are investigated. Surprisingly, it can be proved that all the honest processes are not only recurrent but also ergodic without imposing any extra conditions. Equilibrium distributions are then established. Symmetry and reversibility of such processes are also investigated. Several examples are provided to illustrate our results.

An industrial implementation of a methodology for the optimisation of product design tolerances

Today, the key to commercial success in manufacturing is the timely development of new products that are not only functionally fit for purpose but offer high performance and quality throughout their entire lifecycle. In principle, this demands the introduction of a fully developed and optimised product from the outset. To accomplish this, manufacturing companies must leverage existing knowledge in their current technical, manufacturing and service capabilities. This is especially true in the field of tolerance selection and application, the subject area of this research. Tolerance knowledge must be readily available and deployed as an integral part of the product development process. This paper describes a methodology and framework,currently under development in a UK manufacturer, to achieve this objective.