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arXiv · 2301.08999

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

Duván Cardona·Julio Delgado·Brian Grajales·Michael Ruzhansky
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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 H\"ormander sub-Laplacians on compact Lie groups. We also give conditions for the controllibility of wave and Schr\"odinger equations in these settings.

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Duván Cardona, Julio Delgado, Brian Grajales, Michael Ruzhansky. 2023-01-21. Control of the Cauchy problem on Hilbert spaces: A global approach via symbol criteria. https://arxiv.org/abs/2301.08999

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