Control Estimates for 0th-Order Pseudodifferential Operators

Abstract

Abstract We introduce the control conditions for 0th-order pseudodifferential operators $\textbf{P}$ whose real parts satisfy the Morse–Smale dynamical condition. We obtain microlocal control estimates under the control conditions. As a result, we show that there are no singular profiles in the solution to the evolution equation $(i\partial _{t}-\textbf{P})u=f$ when $\textbf{P}$ has a damping term that satisfies the control condition and $f\in C^{\infty }$. This is motivated by the study of a microlocal model for the damped internal waves.