Hilbert Space Methods and Linear Parabolic Differential Equations

1990 ◽  
pp. 402-451
Author(s):  
Eberhard Zeidler
Keyword(s):  
Hilbert Space ◽  
Differential Equations ◽  
Parabolic Differential Equations

scholarly journals On some stochastic parabolic differential equations in a Hilbert space

2005 ◽  
Vol 2005 (2) ◽  
pp. 167-173 ◽  
Author(s):  
Khairia El-Said El-Nadi
Keyword(s):  
Hilbert Space ◽  
Partial Differential Equations ◽  
Differential Operator ◽  
Differential Equations ◽  
Partial Differential Operator ◽  
Parabolic Differential Equations ◽  
Partial Differential ◽  
Mixed Problems ◽  
Uniqueness Of The Solution ◽  
Weiner Process

We consider some stochastic difference partial differential equations of the form du(x,t,c)=L(x,t,D)u(x,t,c)dt+M(x,t,D)u(x,t−a,c)dw(t), where L(x,t,D) is a linear uniformly elliptic partial differential operator of the second order, M(x,t,D) is a linear partial differential operator of the first order, and w(t) is a Weiner process. The existence and uniqueness of the solution of suitable mixed problems are studied for the considered equation. Some properties are also studied. A more general stochastic problem is considered in a Hilbert space and the results concerning stochastic partial differential equations are obtained as applications.

scholarly journals On Regional Boundary Gradient Strategic Sensors In Diffusion Systems

2021 ◽  
Vol 20 ◽  
pp. 66-78
Author(s):  
Raheam Al-Saphory ◽  
Ahlam Y Al-Shaya
Keyword(s):  
Hilbert Space ◽  
Differential Equations ◽  
Distributed Parameter Systems ◽  
Distributed Parameter ◽  
Parabolic Differential Equations ◽  
Continuous Type ◽  
Diffusion Systems ◽  
Regional Boundary ◽  
Boundary Gradient

This paper is aimed at investigating and introducing the main results regarding the concept of Regional Boundary Gradient Strategic Sensors (RBGS-sensors  the in Diffusion Distributed Parameter Systems (DDP-Systems  . Hence, such a method is characterized by Parabolic Differential Equations (PDEs  in which the behavior of the dynamic is created by a Semigroup ( of Strongly Continuous type (SCSG  in a Hilbert Space (HS) . Additionally , the grantee conditions which ensure the description for such sensors are given respectively to together with the Regional Boundary Gradient Observability (RBG-Observability  can be studied and achieved . Finally , the results gotten are applied to different situations with altered sensors positions are undertaken and examined.

Output stabilisation for a class of linear parabolic differential equations

1988 ◽  
Vol 110 (1-2) ◽  
pp. 125-133
Author(s):  
Takao Nambu
Keyword(s):  
Hilbert Space ◽  
Differential Equations ◽  
Finite Number ◽  
The State ◽  
Sufficient Condition ◽  
Linear Functionals ◽  
Algebraic Condition ◽  
Parabolic Differential Equations ◽  
Feedback Controls

SynopsisWe study the output stabilisation for a class of linear parabolic differential equations in a Hilbert space by means of feedback controls. The output is given as a finite number of linear functionals. Stabilisationof the state, of course, implies stabilisation of the output. In the present paper, however, we give a sufficient condition (an algebraic condition on the above functionals) for the output stabilisation, which is weakerin some sense than that for the state stabilisation.

Global attracting sets of neutral stochastic functional integro-differential equations driven by a fractional Brownian motion

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
A. Bakka ◽  
S. Hajji ◽  
D. Kiouach
Keyword(s):  
Hilbert Space ◽  
Fixed Point ◽  
Brownian Motion ◽  
Differential Equations ◽  
Fractional Brownian Motion ◽  
Sufficient Conditions ◽  
Integrodifferential Equations ◽  
Hurst Parameter ◽  
Fixed Point Principle ◽  
Stochastic Functional

Abstract By means of the Banach fixed point principle, we establish some sufficient conditions ensuring the existence of the global attracting sets of neutral stochastic functional integrodifferential equations with finite delay driven by a fractional Brownian motion (fBm) with Hurst parameter H ∈ ( 1 2 , 1 ) {H\in(\frac{1}{2},1)} in a Hilbert space.

scholarly journals Differential Games for an Infinite 2-Systems of Differential Equations

Mathematics ◽  
2021 ◽  
Vol 9 (13) ◽  
pp. 1467
Author(s):  
Muminjon Tukhtasinov ◽  
Gafurjan Ibragimov ◽  
Sarvinoz Kuchkarova ◽  
Risman Mat Hasim
Keyword(s):  
Hilbert Space ◽  
Differential Equations ◽  
Differential Games ◽  
Differential Game ◽  
Infinite System ◽  
The State ◽  
Geometric Constraints ◽  
Control Parameters ◽  
Systems Of Differential Equations ◽  
Pursuit And Evasion

A pursuit differential game described by an infinite system of 2-systems is studied in Hilbert space l2. Geometric constraints are imposed on control parameters of pursuer and evader. The purpose of pursuer is to bring the state of the system to the origin of the Hilbert space l2 and the evader tries to prevent this. Differential game is completed if the state of the system reaches the origin of l2. The problem is to find a guaranteed pursuit and evasion times. We give an equation for the guaranteed pursuit time and propose an explicit strategy for the pursuer. Additionally, a guaranteed evasion time is found.

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