We consider the problem
u_{tt} − a(x) u_{xx} − b(x) u_x = 0, (t, x) ∈ Q,
u(t, 0) = 0, t ∈ [0, +∞) ,
u(0, x) = u_0(x) , u_t(0, x) = u_1(x) , x ∈ (0, 1),
where Q = (0, +∞) × (0, 1), f ∈ L_{loc}^2[0, +∞), a, b ∈ C^0[0, 1], a > 0 on (0, 1]
and a(0) = 0. If we are interested in a controllability problem we assume
u(t, 1) = f(t) for t ∈ [0, +∞), thus the function f acts as a boundary control and it is used to drive the solution to 0 at a given time T.
Otherwise, if we are interested in the stabilization problem we consider the
damping boundary condition u_t(t, 1) + η u_x(t, 1) + β u(t, 1) = 0 for t ∈ [0, +∞) , where η is a given function and β is a nonnegative constant.
Partial Differential Equations @DipMat 