Approximate Controllability Problems for the Heat Equation in a Half-Plane Controlled by the Dirichlet Boundary Condition with a Bounded Control

2025/06/12 by L. V. Fardigola, Fardigola, Larissa, Kateryna Khalina +1
Engineering · Mathematics · #Analysis of PDEs (math.AP) #Differential Equations and Boundary Problems #FOS: Mathematics #Nonlinear Differential Equations Analysis #Optimization and Control (math.OC) #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.2506.10466

openalex publication_date 2025/06/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In the paper, the problems of approximate controllability are studied for the control system w_t=Δw, w(0,x₂,t)=u(x₂,t), x₁∈\mathbb R₊=(0,+∞), x₂∈\mathbb R, t∈(0,T), where u is a control belonging to a special subset of L^∞(\mathbb R× (0,T))∩ L²(\mathbb R× (0,T)). It is proved that each initial state belonging to L²(\mathbb R₊×\mathbb R) is approximately controllable to an arbitrary end state belonging to L²(\mathbb R₊×\mathbb R) by applying these controls. A numerical algorithm of solving the approximate controllability problem for this system is given. The results are illustrated by an example.

Citations

• Controllability Problems for the Heat Equation in a Half-Plane Controlled by the Neumann Boundary Condition with a Point-Wise Control
• Controllability Problems for the Heat Equation with Variable Coefficients on a Half-Axis Controlled by the Neumann Boundary Condition
• Observability inequalities for the heat equation with bounded potentials on the whole space
• Reachability and Controllability Problems for the Heat Equation on a\n Half-Axis
• Handbook of Mathematical Functions With Formulas, Graphs and Mathematical Tables (National Bureau of Standards Applied Mathematics Series No. 55)

Related

• Controllability Problems for the Heat Equation in a Half-Plane Controlled by the Neumann Boundary Condition with a Point-Wise Control
• Reachability and Controllability Problems for the Heat Equation on a\n Half-Axis
• Controllability Problems for the Heat Equation with Variable Coefficients on a Half-Axis Controlled by the Neumann Boundary Condition
• A different look at controllability
• The cost of approximate controllability for semilinear heat equations in one space dimension