arXiv subjects

Kateryna Khalina

Publications and source records attributed to Kateryna Khalina.

4 recordsLinked to original sources

arXiv2025

Approximate Controllability Problems for the Heat Equation in a Half-Plane Controlled by the Dirichlet Boundary Condition with a Bounded Control

In the paper, the problems of approximate controllability are studied for the control system $w_t=Δw$, $w(0,x_2,t)=u(x_2,t)$, $x_1\in\mathbb R_+=(0,+\infty)$, $x_2\in\mathbb R$, $t\in(0,T)$, where $u$ is a control belonging to a special subset of $L^\infty(\mathbb R\times (0,T))\cap L^2(\mathbb R\times (0,T))$. It is proved that each initial state belonging to $L^2(\mathbb R_+\times\mathbb R)$ is approximately controllable to an arbitrary end state belonging to $L^2(\mathbb R_+\times\mathbb R)$ by applying these controls. A numerical algorithm of solving the approximate controllability problem for this system is given. The results are illustrated by an example.

math.OC↗

arXiv2025

Controllability Problems for the Heat Equation in a Half-Plane Controlled by the Neumann Boundary Condition with a Point-Wise Control

In the paper, the problems of controllability and approximate controllability are studied for the control system $w_t=Δw$, $w_{x_1}(0,x_2,t)=u(t)δ(x_2)$, $x_1>0$, $x_2\in\mathbb R$, $t\in(0,T)$, where $u\in L^\infty(0,T)$ is a control. To this aid, it is investigated the set $\mathcal{R}_T(0)\subset L^2((0,+\infty)\times\mathbb R)$ of its end states which are reachable from $0$. It is established that a function $f\in\mathcal{R}_T(0)$ can be represented in the form $f(x)=g\big(|x|^2\big)$ a.e. in $(0,+\infty)\times\mathbb R$ where $g\in L^2(0,+\infty)$. In fact, we reduce the problem dealing with functions from $L^2((0,+\infty)\times\mathbb R)$ to a problem dealing with functions from $L^2(0,+\infty)$. Both a necessary and sufficient condition for controllability and a sufficient condition for approximate controllability in a given time $T$ under a control $u$ bounded by a given constant are obtained in terms of solvability of a Markov power moment problem. Using the Laguerre functions (forming an orthonormal basis of $L^2(0,+\infty)$), necessary and sufficient conditions for approximate controllability and numerical solutions to the approximate controllability problem are obtained. It is also shown that there is no initial state that is null-controllable in a given time $T$. The results are illustrated by an example.

math.AP↗

arXiv2022

Controllability Problems for the Heat Equation with Variable Coefficients on a Half-Axis Controlled by the Neumann Boundary Condition

In the paper, the problems of controllability and approximate controllability are studied for the control system $w_t=\frac{1}ρ\left(kw_x\right)_x+γw$, $\left.\left(\sqrt{\frac{k}ρ}w_x\right)\right|_{x=0}=u$, $x>0$, $t\in(0,T)$, where $u$ is a control, $u\in L^\infty(0,T)$. It is proved that each initial state of the control system is approximately controllable to any target state in a given time $T>0$. To obtain this result, the transformation operator generated by the equation data $ρ$, $k$, $γ$ is applied. The results are illustrated by examples.

math.OC↗

arXiv2018

Reachability and Controllability Problems for the Heat Equation on a Half-Axis

In the paper, problems of controllability, approximate controllability, reachability and approximate reachability are studied for the control system $w_t=w_{xx}$, $w(0,\cdot)=u$, $x>0$, $t\in(0,T)$, where $u\in L^\infty(0,T)$ is a control. It is proved that each end state of this system is approximately reachable in a given time $T$, and each its initial state is approximately controllable in a given time $T$. A necessary and sufficient condition for reachability in a given time $T$ is obtained in terms of solvability a Markov power moment problem. It is also shown that there is no initial state that is null-controllable in a given time $T$. The results are illustrated by examples.

math.AP↗