Uniform stabilization of spherical shells by boundary dissipation (Q1913895)

The authors study a system of the form \[ u_{tt}+ {\textstyle{e\over R}}v_{tt}-L(u)- {\textstyle{e\over R}}L(v)+ {\textstyle{{{(1+v)}\over R}}}w_\rho=0, \] \[ w_{tt}- {\textstyle{e\over\rho}} [v_{tt}\rho]_\rho+ {\textstyle{e\over\rho}} [L(v)\rho]_\rho- {\textstyle{{{(1+\nu)}\over{\rho R}}}}(u\rho)_\rho+ {\textstyle{{{2(1+\nu)}\over R^2}}} w=0, \] where \(v\equiv(u/R)+w_\rho\), \(L(u)\equiv u_{\rho\rho}+ (u_\rho/\rho)-(u/\rho^2)\), the unknown \((u,w)\) being functions of \(\rho\), \(t\) and subjected to the boundary conditions: \(u=w_\rho=L(v)=0\), \(\rho=0\), \(t>0\), and \[ u_\rho- {\textstyle{{{(1+\nu)}\over R}}}w+\nu {\textstyle{{u\over \rho_0}}}+ u_t+u=0, \qquad ev_\rho+v_t=0, \qquad eL(v)-ev_{tt}=w_t \] for \(\rho=\rho_0\), \(t>0\). It is shown that this system generates a semigroup on a natural function space and that the corresponding energy functional decays exponentially with growing time. As a consequence, a corresponding exact controllability result is obtained. The usual energy method combined with the Carleman type estimates are used in the proofs.