Transformation Operators and Modified Sobolev Spaces in Controllability Problems on a Half-Axis

Abstract

In the paper, the control system wₜₜ =1/ρ(kwₓ)ₓ + γw, wₓ(0, t) = u(t), x > 0, t belongs (0, T), is considered in special modified spaces of Sobolev type Here ρ, k, and γ are given functions on [0, +∞); u belongs L∞(0, ∞) is a control; T > 0 is a constant. The growth of distributions from these spaces depends on the growth of ρ and k. With the aid of some transformation operators, it is proved that the control system replicates the controllability properties of the auxiliary system zₜₜ = zξξ − q²z, zξ(0, t) = v(t), ξ > 0, t belongs (0, T), and vise versa. Here q ≥ 0 is a constant and v belongs L∞(0, ∞) is a control. For the main system, necessary and sufficient conditions of the L∞-controllability and the approximate L∞-controllability are obtained from those known for the auxiliary system.