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 \[ \begin{cases} z_{t}(t,x) = \Delta z(t,x) + 1_{\omega}u(t,x)+f(t,z,u(t,x)) & \text{in} \quad (0, \tau] \times \Omega,\\ z = 0, & \quad \text{on} \quad (0, \tau) \times \partial \Omega, \\ z(0,x) = z_{0}(x), & x \in\Omega, \end{cases} \] 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 \(f:[0, \tau] \times \mathbb R \times \mathbb R \rightarrow \mathbb R\) is smooth enough and there are \(a,b, c \in \mathbb R\), with \(c \neq -1\), such that \[ \sup_{(t,z,u) \in Q_{\tau}} |f(t,z,u) -az-cu-b | < \infty, \] where \(Q_{\tau}= [0, \tau] \times \mathbb R \times \mathbb R\). 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.


93B05 Controllability
93C25 Control/observation systems in abstract spaces
35K05 Heat equation
Full Text: Euclid


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