Optimal Stabilization of the Solutions of a Parabolic Boundary-value Problem Using Bounded Lumped Control

1999 ◽  
Vol 31 (12) ◽  
pp. 45-52 ◽  
Author(s):  
Vladimir E. Kapustyan
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value ◽  
Parabolic Boundary ◽  
Optimal Stabilization

scholarly journals Uniqueness and stability of solutions for a type of parabolic boundary value problem

1989 ◽  
Vol 12 (4) ◽  
pp. 735-739
Author(s):  
Enrique A. Gonzalez-Velasco
Keyword(s):  
Parabolic Equation ◽  
Boundary Value Problem ◽  
Nonnegative Solution ◽  
Boundary Value ◽  
Stability Of Solutions ◽  
General Boundary Conditions ◽  
Boundary Values ◽  
Parabolic Boundary ◽  
One Dimensional ◽  
The One

We consider a boundary value problem consisting of the one-dimensional parabolic equationgut=(hux)x+q, where g, h and q are functions of x, subject to some general boundary conditions. By developing a maximum principle for the boundary value problem, rather than the equation, we prove the uniqueness of a nonnegative solution that depends continuously on boundary values.

scholarly journals On integral representation of the solutions of a model $\vec{2b}$-parabolic boundary value problem

2019 ◽  
Vol 11 (1) ◽  
pp. 193-203
Author(s):  
N.I. Turchyna ◽  
S.D. Ivasyshen
Keyword(s):  
Boundary Value Problem ◽  
Integral Representation ◽  
Boundary Conditions ◽  
Boundary Value ◽  
Integral Representations ◽  
Parabolic Boundary ◽  
Dirac Delta ◽  
Representation Of Solutions ◽  
Green’S Matrix ◽  
Complementing Condition

A general boundary value problem for Eidelman type $\overrightarrow{2b}$-parabolic system of equation without minor terms in the equations and boundary conditions, and with constant coefficients in the group of major terms is considered in the region $$\{(t,x_1,\dots,x_n)\in \mathbb{R}^{n+1}|t\in(0,T], x_j\in\mathbb{R}, j\in\{1,\dots,n-1\}, x_n>0\},$$ $T>0$, $n\ge 2$. It is assumed that the boundary conditions are connected with the system of equations by the complementing condition, which is analogous to the Lopatynsky complementing condition. Integral representations of the solutions for such a problem are derived. The kernels of the integrals from this representation form the Green's matrix of the problem. It is revealed that, in general, not all the elements of the Green's matrix are ordinary functions. Some of them contain terms that are linear combinations of Dirac delta functions and their derivatives. This occurs in cases when the boundary conditions include derivatives with respect to the variables $t$ and $x_n$ of orders that are equal or greater than the highest orders of derivatives with respect to these variables in the equations of the system. The obtained results are important, in particular, for the establishing of the correct solvability and integral representation of solutions for more general $\overrightarrow{2b}$-parabolic boundary value problems.

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