## 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.

### MSC:

 93B05 Controllability 93C25 Control/observation systems in abstract spaces 35K05 Heat equation
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### References:

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