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

Abstract

Let $\Delta$ be the Dirichlet Laplacian on a bounded domain $\Omega \subset \mathbb{R}^{N}$, and let $(-\Delta)^\alpha$ be the associated spectral fractional Laplacian with $\alpha \leq 1, \ \rho <2$. For general bounded domains with $C^2$ boundary, we prove a symmetry formula for $\alpha <1/2$, 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 $$u_{tt}+\Delta^2u+(-\Delta)^\alpha u_t=0$$ subject to Dirichlet or moment boundary control. For rectangular domains with $\alpha <1$, we prove boundary null-controllability results. For $\alpha <1/2, \ \rho \leq 2$, 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.