Well-posedness and exponential stability of an abstract evolution equation with infinite memory and time delay

2013 ◽  
Vol 30 (4) ◽  
pp. 507-526 ◽  
Author(s):  
A. Guesmia
Keyword(s):  
Time Delay ◽  
Evolution Equation ◽  
Exponential Stability ◽  
Well Posedness ◽  
Infinite Memory ◽  
Abstract Evolution Equation

Well-posedness and exponential stability of a flexible structure with second sound and time delay

2018 ◽  
Vol 98 (16) ◽  
pp. 2903-2915 ◽  
Author(s):  
Gang Li ◽  
Yue Luan ◽  
Jiangyong Yu ◽  
Feida Jiang
Keyword(s):  
Time Delay ◽  
Exponential Stability ◽  
Flexible Structure ◽  
Well Posedness ◽  
Second Sound

scholarly journals Well-posedness for a fourth-order equation of Moore–Gibson–Thompson type

2021 ◽  
pp. 1-18
Author(s):  
Carlos Lizama ◽  
Marina Murillo-Arcila
Keyword(s):  
Evolution Equation ◽  
Fourth Order ◽  
Fractional Laplacian ◽  
Order Equation ◽  
Well Posedness ◽  
Fourth Order Equation ◽  
Abstract Evolution Equation ◽  
Laplacian Operators ◽  
Vector Valued ◽  
With Memory

In this paper, we completely characterize, only in terms of the data, the well-posedness of a fourth order abstract evolution equation arising from the Moore–Gibson–Thomson equation with memory. This characterization is obtained in the scales of vector-valued Lebesgue, Besov and Triebel–Lizorkin function spaces. Our characterization is flexible enough to admite as examples the Laplacian and the fractional Laplacian operators, among others. We also provide a practical and general criteria that allows Lp–Lq-well posedness.

scholarly journals Exponential Stability of Stochastic Differential Equation with Mixed Delay

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Wenli Zhu ◽  
Jiexiang Huang ◽  
Xinfeng Ruan ◽  
Zhao Zhao
Keyword(s):  
Differential Equation ◽  
Time Delay ◽  
Stochastic Differential Equation ◽  
Exponential Stability ◽  
Sufficient Conditions ◽  
Lyapunov Stability Theory ◽  
Martingale Inequality ◽  
Mixed Delay ◽  
Exponentially Stable ◽  
Theoretical Results

This paper focuses on a class of stochastic differential equations with mixed delay based on Lyapunov stability theory, Itô formula, stochastic analysis, and inequality technique. A sufficient condition for existence and uniqueness of the adapted solution to such systems is established by employing fixed point theorem. Some sufficient conditions of exponential stability and corollaries for such systems are obtained by using Lyapunov function. By utilizing Doob’s martingale inequality and Borel-Cantelli lemma, it is shown that the exponentially stable in the mean square of such systems implies the almost surely exponentially stable. In particular, our theoretical results show that if stochastic differential equation is exponentially stable and the time delay is sufficiently small, then the corresponding stochastic differential equation with mixed delay will remain exponentially stable. Moreover, time delay upper limit is solved by using our theoretical results when the system is exponentially stable, and they are more easily verified and applied in practice.

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