Uniform stabilization of spherical shells by boundary dissipation

Abstract

Under homogeneous boundary conditions, the "energy" remains constant in time (conservative problem). We then introduce suitable dissipative feedback controls on the boundary (forces, shears, moments) and show that

(i) the resulting closed loop feedback problem generates a s.c. semigroup of contractions on a natural function space;

the corresponding "energy" (norm of the semigroup solutions) decays exponentially in the uniform topology.

As a consequence of the above uniform stabilization result, we obtain---via a result of [15]---a corresponding exact controllability result by explicit boundary controls, which improves upon the recent result of [4, 5]. Energy (multipliers) methods are used, along with semigroup methods. In the process of absorbing lower-order terms to obtain the final energy inequality, we establish a unique continuation result for the system of strongly coupled equations describing the dynamics of the shell, which is of interest in its own right. To this end, Carleman estimates are used after a suitable change of variable.