Necessary and Sufficient Conditions for Exponential Stability and Ultimate Boundedness of Systems Governed by Stochastic Partial Differential Equations

2000 ◽  
Vol 62 (1) ◽  
pp. 311-320 ◽  
Author(s):  
Kai Liu
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Exponential Stability ◽  
Stochastic Partial Differential Equations ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Ultimate Boundedness ◽  
Partial Differential ◽  
Necessary And Sufficient

Symmetry-based algorithms to relate partial differential equations: I. Local symmetries

1990 ◽  
Vol 1 (3) ◽  
pp. 189-216 ◽  
Author(s):  
G. W. Bluman ◽  
S. Kumei
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Sufficient Conditions ◽  
Linear Partial Differential Equation ◽  
Variable Coefficients ◽  
Necessary And Sufficient Conditions ◽  
Partial Differential ◽  
Flow Equations ◽  
Local Symmetries ◽  
Necessary And Sufficient

Simple and systematic algorithms for relating differential equations are given. They are based on comparing the local symmetries admitted by the equations. Comparisons of the infinitesimal generators and their Lie algebras of given and target equations lead to necessary conditions for the existence of mappings which relate them. Necessary and sufficient conditions are presented for the existence of invertible mappings from a given nonlinear system of partial differential equations to some linear system of equations with examples including the hodograph and Legendre transformations, and the linearizations of a nonlinear telegraph equation, a nonlinear diffusion equation, and nonlinear fluid flow equations. Necessary and sufficient conditions are also given for the existence of an invertible point transformation which maps a linear partial differential equation with variable coefficients to a linear equation with constant coefficients. Other types of mappings are also considered including the Miura transformation and the invertible mapping which relates the cylindrical KdV and the KdV equations.

scholarly journals On stabilization of partial differential equations by noise

2001 ◽  
Vol 161 ◽  
pp. 155-170 ◽  
Author(s):  
Tomás Caraballo ◽  
Kai Liu ◽  
Xuerong Mao
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Exponential Stability ◽  
Stochastic Partial Differential Equations ◽  
Finite Dimension ◽  
Infinite Dimension ◽  
Stability Criteria ◽  
Partial Differential ◽  
Almost Sure Exponential Stability

Some results on stabilization of (deterministic and stochastic) partial differential equations are established. In particular, some stability criteria from Chow [4] and Haussmann [6] are improved and subsequently applied to certain situations, on which the original criteria commonly do not work, to ensure almost sure exponential stability. This paper also extends to infinite dimension some results due to Mao [9] on stabilization of differential equations in finite dimension.

Sign in / Sign up

Export Citation Format

Share Document