Problem 5.5 Exact controllability of the semi-linear wave equation

In (1), (y(t, ·), yt(t, ·)) is the state and u(t, ·) is the control which acts on the system through the subset ω of Ω. In what follows, we choose the state space and the control space as H 0 (Ω)× L(Ω) and L((0, T )× Ω), respectively. Of course, the choice of these spaces is not unique. But this one is very natural in the context of the wave equation. The space H 0 (Ω)× L(Ω) is often referred to as the energy space. The exact (internal) controllability problem for (1) (in H 0 (Ω)×L2(Ω)) may be formulated as follows: for any given (y0, y1), (z0, z1) ∈ H 0 (Ω) × L(Ω), to find (if possible) a control u ∈ L((0, T ) × Ω) such that the weak solution y of