scholarly journals A class of singularly perturbed evolution systems

1994 ◽  
Vol 7 (2) ◽  
pp. 179-190
Author(s):  
N. U. Ahmed
Keyword(s):  
Continuous Dependence ◽  
Evolution Equations ◽  
Sufficient Conditions ◽  
Coupled System ◽  
Singularly Perturbed ◽  
Evolution Operators ◽  
Semilinear Systems ◽  
Existence Of Periodic Solutions ◽  
Semigroup Generators ◽  
Evolution Systems

In this paper we study a class of evolution equations where the semigroup generators are singularly perturbed by a nonnegative real valued function of time. Sufficient conditions for existence of evolution operators and their compactness are given including continuous dependence on the perturbation. Further, for a coupled system of singularly perturbed semilinear systems in two Banach spaces, existence of periodic solutions and their stability are studied.

scholarly journals Optimal control of nonlocal fractional evolution equations in the α-norm of order $(1,2)$

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Azmat Ullah Khan Niazi ◽  
Naveed Iqbal ◽  
Wael W. Mohammed
Keyword(s):  
Optimal Control ◽  
Continuous Dependence ◽  
Evolution Equations ◽  
Mild Solutions ◽  
Fractional Evolution Equations ◽  
Lagrange Problem ◽  
Adequate Definition ◽  
Fractional Evolution Systems ◽  
Definition Of ◽  
Evolution Systems

AbstractThis paper investigates the optimal control for a class of nonlocal fractional evolution equations of order $\gamma \in (1,2)$ γ ∈ ( 1 , 2 ) in Banach spaces. An adequate definition of α-mild solutions is obtained and the existence, uniqueness and continuous dependence of α-mild solutions for the presented control system are also established. The existence of optimal pairs of nonlocal fractional evolution systems is also demonstrated with a view on the construction of the Lagrange problem. Finally, an example is propounded for the presentation of optimal control.

scholarly journals Blow-Up Phenomena for Certain Nonlocal Evolution Equations and Systems

2015 ◽  
Vol 2015 ◽  
pp. 1-7
Author(s):  
Mohamed Jleli ◽  
Bessem Samet
Keyword(s):  
Evolution Equation ◽  
Positive Solutions ◽  
Evolution Equations ◽  
Blow Up ◽  
Sufficient Conditions ◽  
Duality Argument ◽  
Nonlocal Evolution Equation ◽  
Global Nonexistence ◽  
Nonlocal Evolution Equations ◽  
Evolution Systems

We provide sufficient conditions for the nonexistence of global positive solutions to the nonlocal evolution equationutt(x,t)=(J ∗ u-u)(x,t)+up(x,t), (x,t)∈RN×(0,∞),(u(x,0),ut(x,0))=(u0(x),u1(x)),x∈RN,whereJ:RN→R+,p>1, and(u0,u1)∈Lloc1(RN;R+)×Lloc1(RN;R+). Next, we deal with global nonexistence for certain nonlocal evolution systems. Our method of proof is based on a duality argument.

scholarly journals Complete Controllability Conditions for Linear Singularly Perturbed Time-Invariant Systems with Multiple Delays via Chang-Type Transformation

Axioms ◽  
2019 ◽  
Vol 8 (2) ◽  
pp. 71 ◽  
Author(s):  
Olga Tsekhan
Keyword(s):  
Sufficient Conditions ◽  
Singularly Perturbed ◽  
Degenerate System ◽  
Multiple Delays ◽  
Complete Controllability ◽  
Singularly Perturbed System ◽  
System With Delay ◽  
Perturbed System ◽  
Time Invariant ◽  
Controllability Conditions

The problem of complete controllability of a linear time-invariant singularly-perturbed system with multiple commensurate non-small delays in the slow state variables is considered. An approach to the time-scale separation of the original singularly-perturbed system by means of Chang-type non-degenerate transformation, generalized for the system with delay, is used. Sufficient conditions for complete controllability of the singularly-perturbed system with delay are obtained. The conditions do not depend on a singularity parameter and are valid for all its sufficiently small values. The conditions have a parametric rank form and are expressed in terms of the controllability conditions of two systems of a lower dimension than the original one: the degenerate system and the boundary layer system.

scholarly journals On the finite approximate controllability for Hilfer fractional evolution systems with nonlocal conditions

2020 ◽  
Vol 18 (1) ◽  
pp. 529-539
Author(s):  
Xianghu Liu
Keyword(s):  
Approximate Controllability ◽  
Nonlocal Conditions ◽  
Sufficient Conditions ◽  
Fixed Point Problem ◽  
Point Problem ◽  
Evolution System ◽  
Controlled Systems ◽  
Uniqueness Of The Solution ◽  
Fractional Evolution Systems ◽  
Evolution Systems

Abstract The aim of this study is to investigate the finite approximate controllability of certain Hilfer fractional evolution systems with nonlocal conditions. To achieve this, we first transform the mild solution of the Hilfer fractional evolution system into a fixed point problem for a condensing map. Then, by using the topological degree approach, we present sufficient conditions for the existence and uniqueness of the solution of the Hilfer fractional evolution systems. Using the variational approach, we obtain sufficient conditions for the finite approximate controllability of semilinear controlled systems. Finally, an example is provided to illustrate main results.

scholarly journals Periodic solutions for periodic second-order differential equations with variable potentials

2018 ◽  
Vol 24 (2) ◽  
pp. 127-137
Author(s):  
Jaume Llibre ◽  
Ammar Makhlouf
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Periodic Solutions ◽  
Order Differential Equation ◽  
Sufficient Conditions ◽  
Second Order ◽  
Second Order Differential Equations ◽  
Second Order Differential Equation ◽  
Existence Of Periodic Solutions

Abstract We provide sufficient conditions for the existence of periodic solutions of the second-order differential equation with variable potentials {-(px^{\prime})^{\prime}(t)-r(t)p(t)x^{\prime}(t)+q(t)x(t)=f(t,x(t))} , where the functions {p(t)>0} , {q(t)} , {r(t)} and {f(t,x)} are {\mathcal{C}^{2}} and T-periodic in the variable t.

scholarly journals Periodic Solutions of a Periodic FitzHugh–Nagumo System

2015 ◽  
Vol 25 (13) ◽  
pp. 1550180 ◽  
Author(s):  
Jaume Llibre ◽  
Claudio Vidal
Keyword(s):  
Periodic Solutions ◽  
Sufficient Conditions ◽  
Differential Systems ◽  
Existence Of Periodic Solutions

Recently some interest has appeared for the periodic FitzHugh–Nagumo differential systems. Here, we provide sufficient conditions for the existence of periodic solutions in such differential systems.

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