Approximate controllability of a semilinear heat equation (Q457883)

Summary: We apply Rothe's type fixed-point theorem to prove the interior approximate controllability of the following semilinear heat equation: \(z_t(t,x) = \Delta z(t,x) + f(t,z(t,x),u(t,x))\) in \((0,\tau] \times \Omega\), \(z=0\) on \((0,\tau) \times \partial \Omega\), \(z(0,x) = z_0(x)\), \(x \in \Omega\) where \(\Omega\) is a bounded domain in \(\mathbb R^N\), \(N \geq 1\), \(z_0 \in L^2(\Omega)\), \(\omega\) is an open nonempty subset of \(\Omega\). \(1_\omega\) denotes the characteristic function of the set \(\omega\), the distributed control \(u\) belongs to \(L^2(0,\tau;L^2(\Omega))\), and the nonlinear function \(f : [o,\tau] \times \mathbb R \times \mathbb R\;\rightarrow \mathbb R\) is smooth enough, and there are \(a,b,c \in \mathbb R, R > 0\) and \(\frac {1}{2} \leq \beta < 1\) such that \(|f(t,z,u) -az| \leq c|u|^\beta\) for all \(u,z \in \mathbb R\), \(|u|\), \(|z| \geq R\). 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 from an initial state \(z_0\) to an \(\epsilon\) neighborhood of the final state \(z_1\) at time \(\tau \geq 0\).