On a fractional impulsive partial stochastic integro-differential equation with state-dependent delay and optimal controls

Stochastics ◽  
2016 ◽  
Vol 88 (8) ◽  
pp. 1115-1146 ◽  
Author(s):  
Zuomao Yan ◽  
Xiumei Jia
Keyword(s):  
Differential Equation ◽  
Optimal Controls ◽  
Integro Differential Equation ◽  
State Dependent ◽  
State Dependent Delay

Approximate controllability of impulsive fractional integro-differential equation with state-dependent delay in Hilbert spaces

2018 ◽  
Vol 36 (2) ◽  
pp. 603-622 ◽  
Author(s):  
Yong Zhou ◽  
S Suganya ◽  
M Mallika Arjunan ◽  
B Ahmad
Keyword(s):  
Differential Equation ◽  
Hilbert Spaces ◽  
Approximate Controllability ◽  
Semigroup Theory ◽  
Sufficient Conditions ◽  
Integro Differential Equation ◽  
State Dependent ◽  
State Dependent Delay ◽  
Non Linear ◽  
Approximately Controllable

Abstract In this paper, the problem of approximate controllability for non-linear impulsive fractional integro-differential equation with state-dependent delay in Hilbert spaces is investigated. We study the approximate controllability for non-linear impulsive integro-differential systems under the assumption that the corresponding linear control system is approximately controllable. By utilizing the methods of fractional calculus, semigroup theory, fixed-point theorem coupled with solution operator, sufficient conditions are formulated and proved. Finally, an example is provided to illustrate the proposed theory.

Periodic Solutions for a Class of Functional Differential Equations with State-Dependent Delay Close to Zero

2003 ◽  
Vol 13 (06) ◽  
pp. 807-841 ◽  
Author(s):  
R. Ouifki ◽  
M. L. Hbid
Keyword(s):  
Differential Equation ◽  
Hopf Bifurcation ◽  
Periodic Solutions ◽  
Functional Differential Equations ◽  
Delay Equation ◽  
Constant Delay ◽  
State Dependent ◽  
State Dependent Delay ◽  
Existence Of Periodic Solutions ◽  
Functional Differential

The purpose of the paper is to prove the existence of periodic solutions for a functional differential equation with state-dependent delay, of the type [Formula: see text] Transforming this equation into a perturbed constant delay equation and using the Hopf bifurcation result and the Poincaré procedure for this last equation, we prove the existence of a branch of periodic solutions for the state-dependent delay equation, bifurcating from r ≡ 0.

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