On the reachable space for parabolic equations

2025/07/21 by Ervedoza, Sylvain, Tendani-Soler, Adrien
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2507.15407

Abstract

In this article, we provide a description of the reachable space for the heat equation with various lower order terms, set in the euclidean ball of ℝ^d centered at 0 and of radius one and controlled from the whole external boundary. Namely, we consider the case of linear heat equations with lower order terms of order 0 and 1, and the case of a semilinear heat equations. In the linear case, we prove that any function which can be extended as an holomorphic function in a set of the form Ω_α= \ z∈ℂ^d | |\Re(z)| + α|\Im(z)| < 1\ for some α∈ (0,1) and which admits a continuous extension up to Ω_α belongs to the reachable space. In the semilinear case, we prove a similar result for sufficiently small data. Our proofs are based on well-posedness results for the heat equation in a suitable space of holomorphic functions over Ω_α for α> 1.

Citations

• Exact controllability of anisotropic 1D partial differential equations in spaces of analytic functions
• Reachable space of the Hermite heat equation with boundary control
• Analytic properties of heat equation solutions and reachable sets
• Reachable states and holomorphic function spaces for the 1-D heat\n equation
• On the reachable states for the boundary control of the heat equation
• Local controllability to trajectories for non-homogeneous 2-d incompressible Navier-Stokes equations

Related

• Analytic properties of heat equation solutions and reachable sets
• The reachable space of the heat equation for a finite rod as a\n Reproducing Kernel Hilbert Space
• Reachable states and holomorphic function spaces for the 1-D heat\n equation
• Reachable space of the Hermite heat equation with boundary control
• Exact Controllability of Nonlinear Heat Equations in Spaces of Analytic Functions