Tracking Controllability for the Heat Equation

• 成果类型: Article
• 署名作者: Barcena-Petisco, Jon Asier; Zuazua, Enrique
• 署名单位: University of Basque Country; University of Erlangen Nuremberg; University of Deusto; Autonomous University of Madrid
• 刊物名称: IEEE TRANSACTIONS ON AUTOMATIC CONTROL
• ISSN/ISSBN: 0018-9286
• DOI: 10.1109/TAC.2024.3476174
• 发表日期: 2025
• 页码: 1935-1940
• 关键词: Controllability; mathematical models; Target tracking; propagation; Space heating; Aerospace electronics; observability; Transforms; COSTS; trajectory; Linear system observers; Linear systems; optimal control; tracking controllability

摘要

In this article, we study the tracking or sidewise controllability of the heat equation. More precisely, we seek controls that, acting on part of the boundary of the domain where the heat process evolves, aim to assure that the normal trace or flux on the complementary set tracks a given trajectory. The dual equivalent observability problem is identified. It consists of estimating the boundary sources, localized on a given subset of the boundary, out of boundary measurements on the complementary subset. Classical unique continuation and smoothing properties of the heat equation allow us proving approximate tracking controllability properties and the smoothness of the class of trackable trajectories. We also develop a new transmutation method that allows to transfer known results on the sidewise controllability of the wave equation to the tracking controllability of the heat one. Using the flatness approach, we also give explicit estimates on the cost of approximate tracking control. The analysis is complemented with a discussion of some possible variants of these results and a list of open problems.