Although several assertions were presented as the maximum principles for delay differential equations, they can be only interpreted in a corresponding sense as analogs of classical ones for ordinary differential equations and do not imply important corollaries, reached on the basis of the finite-dimensional fundamental systems.
Along this line, by a duality between linear stochastic differential delay equations (SDDEs) and anticipated backward stochastic differential equations (ABSDEs) established in [9], the maximum principle for stochastic delay optimal control problems was studied by [10 13].
In this paper, we investigate the averaging principle for stochastic delay differential equations (SDDEs) and SDDEs with pure jumps.
A componentwise estimate of exponential convergence is obtained for a class of delayed Hopfield type neural networks by using a method based on a comparison principle of delay differential systems.
As a similar process to the stochastic differential equation case, we can derive the averaging principle for SDE with delay.
LaSalle-type invariance principle for stochastic differential delay equations is employed to investigate the globally almost surely asymptotical stability of the error dynamical system.
The inclusion principle for distributed-time-delay systems is defined.
It can be seen that ω 1 ( e ( t ) ) > ω 2 ( e ( t ) ) for any e ( t ) ≠ 0. Therefore, applying a LaSalle-type invariance principle for the stochastic differential delay equation, we can conclude that the controlled network (2) can be synchronized with the trajectory s ( t ) for almost every initial data.
Suddenly, Dr. Hopfield realized that the brain -- or at least the sand mouse brain -- uses a "novel, simple, powerful, plausible" computational principle for dealing with time delays.
Furthermore, given the uncertainty of parameter estimation, the timing results of this work should not be interpreted quantitatively, but rather as a general principle for antiviral strategies that delaying the onset of wide-scale treatment can potentially reduce the overall disease burden while preventing large resistant outbreaks.
