Controllability problems for the heat equation on a half-axis with a bounded control in the Neumann boundary condition

Abstract

In the paper, the problems of controllability and approximate controllability are studied for the control system wt = wxx, wx(0, •) = u, x > 0, t ∈ (0, T ), where u ∈ L ∞ (0, T ) is a control. It is proved that each initial state of the system is approximately controllable to each target state in a given time T . A necessary and sufficient condition for controllability in a given time T is obtained in terms of solvability of a Markov power moment problem. It is also shown that there is no initial state which is null-controllable in a given time T . Orthogonal bases are constructed in H 1 and H 1 . Using these bases, numerical solutions to the approximate controllability problem are obtained. The results are illustrated by examples.

Controllability Problems for the Heat Equation with Variable Coefficients on a Half-Axis Controlled by the Neumann Boundary Condition

“…If an initial state of control system is controllable to the origin then the initial state is also the origin (see Theorem 2.6). In the case of constant coefficients (ρ = k = 1, γ = 0), the result of this theorem has been obtained earlier in [10]. This result is similar to that of the paper [22].…”

“…It is proved that each initial state of the control system is approximately controllable to any target state in a given time T > 0 (Theorem 2.7). In the case of constant coefficients (ρ = k = 1, γ = 0), the result of this theorem has been obtained earlier in [10]. In the case of variable coefficients, this result is similar to those of papers [5][6][7][8] for the wave equation with variable coefficients on a half-axis controlled either by the Dirichlet or by the Neumann boundary condition.…”

“…Control system (2.8), (2.9) has been investigated in [10]. In particular, it has been proved therein that Z x (0 + , •) = u, a.e.…”

“…Controllability problems for the heat equation with constant and variable coefficients were studied in a number of papers (see, e.g., [1-4, 9, 10, 13, 14, 17, 21-27]). However, there are only a few papers where these problems were investigated for the heat equation with constant coefficients on unbounded domains (see, e.g., [2,9,10,22,23]), and it seems these problems were investigated for the heat equation with variable coefficients on unbounded domains only in [11].…”

Controllability problems for the heat equation with variable coefficients on a half-axis

“…Controllability problems for the heat equation with constant and variable coefficients were studied in a number of papers (see, e.g., [1-4, 9-12, 14, 18-22, 24, 25]). However, there are much fewer papers where these problems were investigated for the heat equation with constant coefficients in unbounded domains (see, e.g., [2,9,10,19,20,23]), and it seems there are no papers where these problems were investigated for the heat equation with variable coefficients in unbounded domains.…”