scholarly journals Approximate Controllability of Semilinear Impulsive Evolution Equations

2015 ◽  
Vol 2015 ◽  
pp. 1-7 ◽  
Author(s):  
Hugo Leiva
Keyword(s):  
Linear Operator ◽  
Evolution Equation ◽  
Hilbert Spaces ◽  
Approximate Controllability ◽  
Evolution Equations ◽  
Smooth Functions ◽  
Bounded Linear ◽  
Unbounded Linear Operator ◽  
Approximately Controllable ◽  
Impulsive Evolution Equations

We prove the approximate controllability of the following semilinear impulsive evolution equation:z'=Az+Bu(t)+F(t,z,u), z∈Z, t∈(0,τ], z(0)=z0, z(tk+)=z(tk-)+Ik(tk,z(tk),u(tk)), k=1,2,3,…,p,where0<t1<t2<t3<⋯<tp<τ,ZandUare Hilbert spaces,u∈L2(0,τ;U),B:U→Zis a bounded linear operator,Ik,F:[0,τ]×Z×U→Zare smooth functions, andA:D(A)⊂Z→Zis an unbounded linear operator inZwhich generates a strongly continuous semigroup{T(t)}t≥0⊂Z. We suppose thatFis bounded and the linear system is approximately controllable on[0,δ]for allδ∈(0,τ). Under these conditions, we prove the following statement: the semilinear impulsive evolution equation is approximately controllable on[0,τ].

scholarly journals Approximate controllability of random impulsive quasilinear evolution equation

Filomat ◽  
2020 ◽  
Vol 34 (5) ◽  
pp. 1611-1620
Author(s):  
A. Vinodkumar ◽  
C. Loganathan ◽  
S. Vijay
Keyword(s):  
Evolution Equation ◽  
Approximate Controllability ◽  
Evolution Equations ◽  
Linear Operators ◽  
Sufficient Condition ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
C0 Semigroup

In this paper, we study the approximate controllability of quasilinear evolution equation with random impulsive moments with less restriction and sufficient condition. The results are obtained by the theory of C0 semigroup of bounded linear operators on evolution equations.

scholarly journals Existence and approximate controllability of Sobolev type fractional stochastic evolution equations

2014 ◽  
Vol 62 (2) ◽  
pp. 205-215 ◽  
Author(s):  
N.I. Mahmudov
Keyword(s):  
Linear System ◽  
Hilbert Spaces ◽  
Approximate Controllability ◽  
Evolution Equations ◽  
Mild Solutions ◽  
Sufficient Conditions ◽  
Sobolev Type ◽  
Stochastic Evolution ◽  
Stochastic Evolution Equations ◽  
Approximately Controllable

Abstract We study the existence of mild solutions and the approximate controllability concept for Sobolev type fractional semilinear stochastic evolution equations in Hilbert spaces. We prove existence of a mild solution and give sufficient conditions for the approximate controllability. In particular, we prove that the fractional linear stochastic system is approximately controllable in [0, b] if and only if the corresponding deterministic fractional linear system is approximately controllable in every [s, b], 0 ≤ s < b. An example is provided to illustrate the application of the obtained results.

scholarly journals A New Inequality for Frames in Hilbert Spaces

2018 ◽  
Vol 2018 ◽  
pp. 1-6
Author(s):  
Zhong-Qi Xiang
Keyword(s):  
Linear Operator ◽  
Hilbert Spaces ◽  
Bounded Linear Operator ◽  
Bounded Linear ◽  
Bessel Sequences

We obtain a new inequality for frames in Hilbert spaces associated with a scalar and a bounded linear operator induced by two Bessel sequences. It turns out that the corresponding results due to Balan et al. and Găvruţa can be deduced from our result.

scholarly journals Dua Formula Eksponensial (Operator Linear dari Semigrup)

2021 ◽  
Vol 18 (1) ◽  
pp. 41-46
Author(s):  
L Meisaroh
Keyword(s):  
Linear Operator ◽  
Bounded Linear Operator ◽  
Infinitesimal Generator ◽  
The Other ◽  
Bounded Linear ◽  
Unbounded Linear Operator ◽  
C0 Semigroup

Assumed A is infinitesimal generator of C0-semigroup T(t) on X. This could be defined as T(t)=etA, applies if A is a bounded linear operator. Not if A is unbounded linear operator, then it will result in one possibility that show T(t) could be represented as etA. This paper will discuss and detail the proof of the other two formulas that show T(t) could be represented as etA.

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.

(F,G)-operator frames for ℒ(ℋ,𝒦)

2020 ◽  
Vol 18 (05) ◽  
pp. 2050031
Author(s):  
J. Sedghi Moghaddam ◽  
A. Najati ◽  
F. Ghobadzadeh
Keyword(s):  
Linear Operator ◽  
Hilbert Spaces ◽  
Bounded Linear Operator ◽  
Bounded Operators ◽  
Closed Range ◽  
Bounded Linear ◽  
Atomic Systems ◽  
Pseudo Inverse

The concept of [Formula: see text]-frames was recently introduced by Găvruta7 in Hilbert spaces to study atomic systems with respect to a bounded linear operator. Let [Formula: see text] be a unital [Formula: see text]-algebra, [Formula: see text] be finitely or countably generated Hilbert [Formula: see text]-modules, and [Formula: see text] be adjointable operators from [Formula: see text] to [Formula: see text]. In this paper, we study a class of [Formula: see text]-bounded operators and [Formula: see text]-operator frames for [Formula: see text]. We also prove that the pseudo-inverse of [Formula: see text] exists if and only if [Formula: see text] has closed range. We extend some known results about the pseudo-inverses acting on Hilbert spaces in the context of Hilbert [Formula: see text]-modules. Further, we also present some perturbation results for [Formula: see text]-operator frames in [Formula: see text].

Approximate Controllability of Partial Fractional Neutral Integro-Differential Inclusions With Infinite Delay in Hilbert Spaces

2016 ◽  
Vol 11 (1) ◽  
Author(s):  
Zuomao Yan ◽  
Hongwu Zhang
Keyword(s):  
Hilbert Spaces ◽  
Differential Inclusions ◽  
Approximate Controllability ◽  
Infinite Delay ◽  
Sufficient Conditions ◽  
Multivalued Operators ◽  
Approximately Controllable ◽  
Fractional System ◽  
The Fixed Point Theorem ◽  
Necessary And Sufficient

We study the approximate controllability of a class of fractional partial neutral integro-differential inclusions with infinite delay in Hilbert spaces. By using the analytic α-resolvent operator and the fixed point theorem for discontinuous multivalued operators due to Dhage, a new set of necessary and sufficient conditions are formulated which guarantee the approximate controllability of the nonlinear fractional system. The results are obtained under the assumption that the associated linear system is approximately controllable. An example is provided to illustrate the main results.

scholarly journals Approximate Controllability of Fractional Sobolev-Type Evolution Equations in Banach Spaces

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
N. I. Mahmudov
Keyword(s):  
Fixed Point ◽  
Linear System ◽  
Fixed Point Theorem ◽  
Approximate Controllability ◽  
Evolution Equations ◽  
Sufficient Conditions ◽  
Sobolev Type ◽  
Complete Controllability ◽  
Approximately Controllable ◽  
Sobolev Type Differential Equations

We discuss the approximate controllability of semilinear fractional Sobolev-type differential system under the assumption that the corresponding linear system is approximately controllable. Using Schauder fixed point theorem, fractional calculus and methods of controllability theory, a new set of sufficient conditions for approximate controllability of fractional Sobolev-type differential equations, are formulated and proved. We show that our result has no analogue for the concept of complete controllability. The results of the paper are generalization and continuation of the recent results on this issue.

Periodic Solutions for Nonlinear Evolution Equations in RN

2013 ◽  
Vol 860-863 ◽  
pp. 2830-2833
Author(s):  
Li Hong Zhang ◽  
Wei Jie Li
Keyword(s):  
Linear Operator ◽  
Periodic Solutions ◽  
Bounded Linear Operator ◽  
Existence And Uniqueness ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Periodic Problem ◽  
Nonlinear Evolution Equations ◽  
Bounded Linear

The aim of this paper is to establish the existence and uniqueness of periodic solutions for a nonlinear periodic problem: in RN where A(t, x) is a nonlinear map and B is a bounded linear operator from RNto RN .

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