Bilinear control for global controllability of the semilinear parabolic equations with superlinear terms (Q2717282)

This is an interesting and well-written paper on controllability for the semilinear heat equation. The control system is NEWLINE\[NEWLINE \frac{\partial u}{\partial t}=\Delta u+k(t)u-f(x,t,u,\nabla u)+v(x,t)\chi_{\omega} (x) \text{ in} \quad Q_T=\Omega\times (0,T),NEWLINE\]NEWLINE NEWLINE\[NEWLINEu=0 \text{ in} \quad\Sigma_T =\partial \Omega \times (0,T), u(0)=u_0\in L^2 (\Omega).NEWLINE\]NEWLINE The controls are \(k\in L^{\infty}(0,T)\) and \(v\in L^2 (Q_T)\). The main specific feature of this setting is the fact that the nonlinearity \(f\) is superlinear in both the third and the fourth variables. In this case there is a well-known lack of controllability when the control \(v\) acts only. This is why the control \(k\) appears here. Due to the possible nonuniqueness of solutions, two approximate controllability concepts are defined and two results are stated and proved. For the one dimensional problem, a controllability result is proved in case the control \(v\) depends on \(t\) only.NEWLINENEWLINEFor the entire collection see [Zbl 0959.00047].