scholarly journals On the Estimates in Various Spaces to the Control Function of the Extremum Problem for Parabolic Equation

2021 ◽  
Vol 16 ◽  
pp. 187-192
Author(s):  
Irina Astashova ◽  
Alexey Filinovskiy ◽  
Dmitriy Lashin
Keyword(s):  
Parabolic Equation ◽  
Free Convection ◽  
Existence And Uniqueness ◽  
Control Function ◽  
Extremum Problem ◽  
Convection Term ◽  
Temperature Deviation ◽  
One Dimensional ◽  
Control Functions ◽  
Quadratic Cost

For the minimization problem with pointwise observation governed by a one-dimensional parabolic equation with a free convection term and a depletion potential, we formulate a result on the existence and uniqueness of a minimizer from a prescribed set. We use a weight quadratic cost functional showing the temperature deviation. We obtain estimates for the norm of control functions in terms of the value of the quality functional in different functional spaces. It gives us a possibility to estimate the required internal energy of the system. To prove these results we establish the positivity principle.

scholarly journals On The Dense Controllability for The Parabolic Problem with Time-Distributed Functional

2018 ◽  
Vol 71 (1) ◽  
pp. 9-25
Author(s):  
Irina V. Astashova ◽  
Alexey V. Filinovskiy
Keyword(s):  
Heat Equation ◽  
Control Problem ◽  
Existence And Uniqueness ◽  
Parabolic Problem ◽  
Control Function ◽  
One Dimensional ◽  
Cost Functional ◽  
Control Functions ◽  
Quadratic Cost

Abstract We consider a control problem for one-dimensional heat equation with quadratic cost functional. We prove the existence and uniqueness of a control function from a prescribed set, and study the structure of the set of accessible temperature functions. We also prove the dense controllability of the problem for some set of control functions.

scholarly journals Well posedness for a class of ultra-parabolic equations with discontinuous flux

Filomat ◽  
2012 ◽  
Vol 26 (4) ◽  
pp. 793-800
Author(s):  
Jela Susic
Keyword(s):  
Weak Solution ◽  
Parabolic Equation ◽  
Classical Solution ◽  
Parabolic Equations ◽  
Existence And Uniqueness ◽  
Well Posedness ◽  
Convection Term ◽  
Discontinuous Flux ◽  
Parabolic Term

We prove existence and uniqueness of a weak solution to an ultra-parabolic equation with discontinuous convection term. Due to degeneracy in the parabolic term, the equation does not admit the classical solution. Equations of this type describe processes where transport is negligible in some directions.

scholarly journals On properties of minimizers of a control problem with time-distributed functional related to parabolic equations

2019 ◽  
Vol 39 (5) ◽  
pp. 595-609
Author(s):  
I. V. Astashova ◽  
A. V. Filinovskiy
Keyword(s):  
Mathematical Model ◽  
Optimal Control ◽  
Control Problem ◽  
Temperature Control ◽  
Parabolic Equations ◽  
Variable Coefficients ◽  
Qualitative Properties ◽  
One Dimensional ◽  
Control Functions ◽  
Quadratic Cost

We consider a control problem given by a mathematical model of the temperature control in industrial hothouses. The model is based on one-dimensional parabolic equations with variable coefficients. The optimal control is defined as a minimizer of a quadratic cost functional. We describe qualitative properties of this minimizer, study the structure of the set of accessible temperature functions, and prove the dense controllability for some set of control functions.

On the classical solvability of mixed problems for a second-order one-dimensional parabolic equation

Filomat ◽  
2017 ◽  
Vol 31 (16) ◽  
pp. 5241-5262
Author(s):  
Nebojsa Lazetic
Keyword(s):  
Parabolic Equation ◽  
Existence And Uniqueness ◽  
Initial Function ◽  
Fourier Method ◽  
Continuous Functions ◽  
Classical Solutions ◽  
One Dimensional ◽  
Mixed Problems ◽  
Upper Bound Estimate ◽  
Stability Of The Solution

We prove the existence and uniqueness of classical solutions to mixed problems for the equation ?u/?t(x,t)-?2u/?x2(x,t) + q(x) u(x,t) = f (x,t) on a rectangle ?? = [a,b]x[0,T], with arbitrary self-adjoint homogenous boundary conditions. We assume that q and f are continuous functions, that f (x,?) satisfies a H?lder condition uniformly with respect to x, and the initial function belongs to the class ?W(1)p (a,b) (1 < p ? 2 ). Also, an upper-bound estimate for the solution and, as a consequence, a kind of stability of the solution with respect to the initial function are established. Moreover, some convergence rate estimates for the series defining solutions (and their first derivatives) are given. A modification of the Fourier method is used. Based on the obtained results, we also study the mixed problems on an unbounded rectangle ??? = [a,b]x

Hypothesis on the Solvability of Parabolic Equations with nonlocal Condition

2002 ◽  
Vol 7 (1) ◽  
pp. 93-104 ◽  
Author(s):  
Mifodijus Sapagovas
Keyword(s):  
Parabolic Equation ◽  
Parabolic Equations ◽  
Existence And Uniqueness ◽  
Nonlocal Condition ◽  
Nonlocal Conditions ◽  
Numerical Algorithms ◽  
Uniqueness Of The Solution

Numerous and different nonlocal conditions for the solvability of parabolic equations were researched in many articles and reports. The article presented analyzes such conditions imposed, and observes that the existence and uniqueness of the solution of parabolic equation is related mainly to ”smallness” of functions, involved in nonlocal conditions. As a consequence the hypothesis has been made, stating the assumptions on functions in nonlocal conditions are related to numerical algorithms of solving parabolic equations, and not to the parabolic equation itself.

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 Existence and Uniqueness of Solutions for the p(x)-Laplacian Equation with Convection Term

Mathematics ◽  
2020 ◽  
Vol 8 (10) ◽  
pp. 1768
Author(s):  
Bin-Sheng Wang ◽  
Gang-Ling Hou ◽  
Bin Ge
Keyword(s):  
Existence And Uniqueness ◽  
Variable Exponent ◽  
Convection Term ◽  
Pseudomonotone Operators ◽  
Uniqueness Of Solutions ◽  
Laplacian Equation ◽  
Uniqueness Of The Solution ◽  
Gradient Based ◽  
Surjectivity Result ◽  
Quasilinear Elliptic

In this paper, we consider the existence and uniqueness of solutions for a quasilinear elliptic equation with a variable exponent and a reaction term depending on the gradient. Based on the surjectivity result for pseudomonotone operators, we prove the existence of at least one weak solution of such a problem. Furthermore, we obtain the uniqueness of the solution for the above problem under some considerations. Our results generalize and improve the existing results.

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