Geometric bounds on the growth rate of null-controllability cost for the heat equation in small time

2003/07/11 by Luc Miller, Miller, Luc
Engineering · Mathematics · #35B37 (Primary) #35K05 (Secondary) #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Numerical methods in inverse problems #Optimization and Control (math.OC) #Stability and Controllability of Differential Equations #math.AP #math.OC #msc:35B37 #msc:35K05

paper · pdf · doi:10.48550/arxiv.math/0307158

26 pages, uses elsart.sty, typos and section 5.3 corrected

openalex publication_date 2003/07/11 · arxiv created 2003/11/05 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Given a control region Ω on a compact Riemannian manifold M, we consider the heat equation with a source term g localized in Omega. It is known that any initial data in L²(M) can be stirred to 0 in an arbitrarily small time T by applying a suitable control g in L²([0,T]xOmega), and, as T tends to 0, the norm of g grows like e^(C/T) times the norm of the data. We investigate how C depends on the geometry of Omega. We prove C≥ d²/4 where d is the largest distance of a point in M from Ω. When M is a segment of length L controlled at one end, we prove C≤ alpha L² for some alpha < 2. Moreover, this bound implies C≤ alpha L_Omega² where L_Omega is the length of the longest generalized geodesic in M which does not intersect Ω. The control transmutation method used in proving this last result is of a broader interest.

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