Speaker
Description
Consider the following control system:
$w_t=w_{xx}$, $x\in(0,+\infty)$, $ t\in(0,T)$,
$w(0,\cdot)=u$, $t\in(0,T)$,
$w(\cdot,0)=w^0$, $x\in(0,+\infty)$,
where $T>0$ is a constant, $w^0$ is a given function, $u\in L^\infty(0,T)$ is a control. The control system is considered in Sovolev spaces.
An initial state $w^0$ of control system (1)-(3) is said to be null-controllable in a given time $T > 0$ if we can find a control $u\in L^\infty(0,T)$ such that the state of the solution to the control system at $t = T$ satisfies the condition $w(\cdot, T) = 0$. An initial state $w^0$ of control system (1)-(3) is said to be approximately controllable to a target state $w^T$ in a given time $T > 0$ if for each neighbourhood of a target state $w^T$ there exists a control $u\in L^\infty(0,T)$ such that the end state of the solution to the control system (at $t = T$) belongs to this neighbourhood of $w^T$.
We prove that any initial state of the control system (except the zero one) is not null-controllable in a given time $T>0$.
We also prove that each initial state of the control system is approximately controllable to any target state in a given time $T>0$.
The results are illustrated by examples.
The results on controllability of the heat equation controlled by the Dirichlet boundary condition was published in [1-3].
References
[1] L. Fardigola and K. Khalina, Reachability and controllability problems for the heat equation on a half-axis, J. Math. Phys. Anal. Geom. 15 (2019), 57-78.
[2] L. Fardigola and K. Khalina, Controllability problems for the heat equation with variable coefficients on a half-axis, ESAIM Control Optim. Calc. Var. 28 (2022), Art. No. 41.
[3] L. Fardigola and K. Khalina, Controllability problems for the heat equation in a half-plane controlled by the Dirichlet boundary condition with a point-wise control, J. Math. Phys. Anal. Geom. 18 (2022), 75-104.
