A dependence of the cost of fast controls for the heat equation on the support of initial datum

2021/02/24 by Hoài-Minh Nguyên, Nguyen, Hoai-Minh
Computer Science · Engineering · Mathematics · #35A23 #35B30 #93B05 #93B07 #93C20 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical methods in inverse problems #Optimization and Control (math.OC) #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.2102.12172

openalex publication_date 2021/02/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The controllability cost for the heat equation as the control time T goes to 0 is well-known of the order e^C/T for some positive constant C, depending on the controlled domain and for all initial datum. In this paper, we prove that the constant C can be chosen to be arbitrarily small if the support of the initial data is sufficiently close to the controlled domain, but not necessary inside the controlled domain. The proof is in the spirit on Lebeau and Robbiano's approach in which a new spectral inequality is established. The main ingredient of the proof of the new spectral inequality is three-sphere inequalities with partial data.

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