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Interior controllability of the $nD$ semilinear heat equation. (English) Zbl 1243.93020
Summary: In this paper, we prove the interior approximate controllability of the following semilinear heat equation $$ \cases z_{t}(t,x) = \Delta z(t,x) + 1_{\omega}u(t,x)+f(t,z,u(t,x)) & \text{in} \quad (0, \tau] \times \Omega,\\ z = 0, & \quad \text{on} \quad (0, \tau) \times \partial \Omega, \\ z(0,x) = z_{0}(x), & x \in\Omega, \endcases $$ where $\Omega$ is a bounded domain in $\mathbb{R}^{N}(N\geq1), z_0 \in L^{2}(\Omega)$, $\omega$ is an open nonempty subset of $\Omega$, and $1_{\omega}$ denotes the characteristic function of the set $\omega$. The distributed control $u$ belong to $\in L^{2}([0,\tau]; L^{2}(\Omega;))$ and the nonlinear function $f:[0, \tau] \times \Bbb R \times \Bbb R \rightarrow \Bbb R$ is smooth enough and there are $a,b, c \in \Bbb R$, with $c \neq -1$, such that $$ \sup_{(t,z,u) \in Q_{\tau}} |f(t,z,u) -az-cu-b | < \infty, $$ where $Q_{\tau}= [0, \tau] \times \Bbb R \times \Bbb R$. Under this condition, we prove the following statement: For all open nonempty subset $\omega$ of $\Omega$ the system is approximately controllable on $[0, \tau]$. Moreover, we could exhibit a sequence of controls steering the nonlinear system (1) from an initial state $z_0$ to an $\epsilon$ neighborhood of the final state $z_1$ at time $\tau >0$, which is very important from a practical and numerical point of view.

93C25Control systems in abstract spaces
35K05Heat equation
Full Text: Euclid