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

2023/01/21 by Duván Cardona, Cardona, Duván, Julio Delgado +5 · 1 citation
Engineering · Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #Dynamical Systems (math.DS) #FOS: Mathematics #Functional Analysis (math.FA) #Numerical methods in inverse problems #Optimization and Control (math.OC) #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.2301.08999

openalex publication_date 2023/01/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let A and B be invariant linear operators with respect to a decomposition \H_j\_{j∈ ℕ} of a Hilbert space 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)=u₀, in terms of the (global) matrix-valued symbols σ_A and σ_B of A and B, respectively, associated to the decomposition \H_j\_{j∈ 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.

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