Transformation operators in controllability problems for the degenerate wave equation with variable coefficients - Fardigola, L. V. (Ukrains’kyi Matematychnyi Zhurnal, 2018)
We study the control system w_tt = 1ρ (kw_x) x + γ w, w(0, t) = u(t), x (0, l), t (0, T), in special modified spaces of the Sobolev type. Here, ρ , k, and γ are given functions on [0, l); u L^∞ (0, T) is a control, and T > 0 is a constant. The functions ρ and k are positive on [0, l) and may tend to zero or to infinity as x → l. The growth of distributions from these spaces is determined by the growth of ρ and k as x → l. Applying the method of transformation operators, we establish necessary and sufficient conditions for the L^∞ -controllability and approximate L^∞ -controllability at a given time and at a free time.