Exact controllability of semilinear plate equations (Q2774038)

The internal control of the semilinear plate equation NEWLINE\[NEWLINE \begin{aligned} u_{tt}+ \Delta^2y+ f(y) & = \nabla\cdot \bigl(\chi_\omega(x) \nabla\pi (t,x)\bigr),\\ y & = \Delta y=0,\\ y(0) & = y_0,\;y_t(0)= y_1\end{aligned}NEWLINE\]NEWLINE is considered (together with a similar boundary control problem) where \(f\) is superlinear. More precisely, \(f\) is assumed to satisfy NEWLINE\[NEWLINE\lim_{s\to\infty} {f(s) \over s\sqrt {\log|s|}}=0.NEWLINE\]NEWLINE The proof is based on Lions' Hilbert uniqueness method and this requires a delicate observability estimate to be proved. The latter is obtained by first deriving a pointwise estimate for an associated Schrödinger equation. This equation appears after making the transformation \(u(t,s,x)= \int^t_sy(z,x)dz\) so that \(u\) satisfies the equation NEWLINE\[NEWLINEu_{tt}+u_{ss}+ \Delta^2u= \int^t_sq(z,x) u_t(z,s,x)dz.NEWLINE\]NEWLINE Then putting \(w=-iu_t +iu_s+ \Delta u\) we obtain the Schrödinger equation NEWLINE\[NEWLINE-iw_t+iw_s+\Delta w= \int^t_sq(z,x) u_t(z,s,x)dz.NEWLINE\]NEWLINE This summary gives the general idea of the proof which is somewhat lost in the technical details.