Control of the Cauchy problem on Hilbert spaces: A global approach via symbol criteria

Abstract

Let $A$ and $B$ be invariant linear operators with respect to a decomposition $\{H_j{\}}_{j\in \mathbb{N}}$ of a Hilbert space $\mathcal{H}$ in subspaces of finite dimension. We give necessary and sufficient conditions for the controllability of the Cauchy problem $$u_t=Au+Bv,u(0)=u_0,$$ in terms of the (global) matrix-valued symbols ${\sigma }_A$ and ${\sigma }_B$ of $A$ and $B,$ respectively, associated to the decomposition $\{H_j{\}}_{j\in \mathbb{N}}$. Then, we present some applications including the controllability of the Cauchy problem on compact manifolds for elliptic operators and the controllability of fractional diffusion models for Hörmander sub-Laplacians on compact Lie groups. We also give conditions for the controllibility of wave and Schrödinger equations in these settings.

38 Pages