Exponential stability, exponential expansiveness, and exponential dichotomy of evolution equations on the half-line

1998 ◽  
Vol 32 (3) ◽  
pp. 332-353 ◽  
Author(s):  
Nguyen Van Minh ◽  
Frank R�biger ◽  
Roland Schnaubelt
Keyword(s):  
Exponential Stability ◽  
Exponential Dichotomy ◽  
Evolution Equations ◽  
Half Line

scholarly journals Approaching the Discrete Dynamical Systems by means of Skew-Evolution Semiflows

2016 ◽  
Vol 2016 ◽  
pp. 1-10 ◽  
Author(s):  
Codruţa Stoica
Keyword(s):  
Dynamical Systems ◽  
Exponential Stability ◽  
Stability Theory ◽  
Exponential Dichotomy ◽  
Evolution Equations ◽  
Discrete Dynamical Systems ◽  
Asymptotic Approach ◽  
Stability And Instability ◽  
Unified Study ◽  
The Stability

The aim of this paper is to highlight current developments and new trends in the stability theory. Due to the outstanding role played in the study of stable, instable, and, respectively, central manifolds, the properties of exponential dichotomy and trichotomy for evolution equations represent two domains of the stability theory with an impressive development. Hence, we intend to construct a framework for an asymptotic approach of these properties for discrete dynamical systems using the associated skew-evolution semiflows. To this aim, we give definitions and characterizations for the properties of exponential stability and instability, and we extend these techniques to obtain a unified study of the properties of exponential dichotomy and trichotomy. The results are underlined by several examples.

scholarly journals Linear evolution equations on the half-line with dynamic boundary conditions

2021 ◽  
pp. 1-33
Author(s):  
D. A. SMITH ◽  
W. Y. TOH
Keyword(s):  
Boundary Conditions ◽  
Heat Equation ◽  
Evolution Equations ◽  
Half Line ◽  
Robin Problem ◽  
Dynamic Boundary Conditions ◽  
Transform Method ◽  
Linear Evolution ◽  
Robin Condition ◽  
Linear Evolution Equations

The classical half-line Robin problem for the heat equation may be solved via a spatial Fourier transform method. In this work, we study the problem in which the static Robin condition $$bq(0,t) + {q_x}(0,t) = 0$$ is replaced with a dynamic Robin condition; $$b = b(t)$$ is allowed to vary in time. Applications include convective heating by a corrosive liquid. We present a solution representation and justify its validity, via an extension of the Fokas transform method. We show how to reduce the problem to a variable coefficient fractional linear ordinary differential equation for the Dirichlet boundary value. We implement the fractional Frobenius method to solve this equation and justify that the error in the approximate solution of the original problem converges appropriately. We also demonstrate an argument for existence and unicity of solutions to the original dynamic Robin problem for the heat equation. Finally, we extend these results to linear evolution equations of arbitrary spatial order on the half-line, with arbitrary linear dynamic boundary conditions.

scholarly journals Nonuniform Exponential Trichotomy for Linear Discrete-Time Systems in Banach Spaces

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Xiao-qiu Song ◽  
Tian Yue ◽  
Dong-qing Li
Keyword(s):  
Exponential Stability ◽  
Banach Spaces ◽  
Discrete Time ◽  
Difference Equations ◽  
Exponential Dichotomy ◽  
Linear Difference Equations ◽  
Discrete Time Systems ◽  
Exponential Trichotomy ◽  
Time Systems

The aim of this paper is to give several characterizations for nonuniform exponential trichotomy properties of linear difference equations in Banach spaces. Well-known results for exponential stability and exponential dichotomy are extended to the case of nonuniform exponential trichotomy.

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