A symmetry formula for the spectral fractional Laplacian, and applications to boundary controllability for plate equation with structural damping

Let $\mathrm{\Delta }$ be the Dirichlet Laplacian on a bounded domain $\mathrm{\Omega }\subset {\mathbb{R}}^N$, and let $(-\mathrm{\Delta }{)}^{\alpha }$ be the associated spectral fractional Laplacian with $\alpha \le 1,\rho <2$. For general bounded domains with $C^2$ boundary, we prove a symmetry formula for $\alpha <1\mathrm{/}2$, extending a result previously proven on rectangles for $\alpha <1$. As a consequence of this formula, well-posedness results are proven for the structurally damped plate equation $$u_{tt}+{\mathrm{\Delta }}^{2}u+(-\mathrm{\Delta }{)}^{\alpha }{u}_t=0$$ subject to Dirichlet or moment boundary control. For rectangular domains with $\alpha <1$, we prove boundary null-controllability results. For $\alpha <1\mathrm{/}2,\rho \le 2$, Dirichlet null controllability is proved for the unit disk in ${\mathbb{R}}^2$. This analysis then extended to the classical case, $\alpha =1$, on rectangles, where higher regularity is required for Dirichlet control.