Boundary control of the displacement at one end with the other end free for a process described by the telegraph equation with a variable coefficient

摘要

We study the boundary control of the displacement at x = 0 for a process described by the telegraph equation with a variable coefficient the homogeneous condition ux(l, t) = 0 set at x = l. Here, Q_T = [0 ≤ x ≤ l] × [0 ≤ t ≤ T] and the coefficient q(x, t) is bounded and measurable in QT. The boundary control problem consists of constructing a function μ(t) for which the process with the condition u(0, t) = μ(t) passes from the initial state {u(x, 0) = ?(x), u_t(x, 0) = ψ(x)} at t = 0 to the terminal state {u(x, T) = ?_1(x), u_t(x, T) = ψ_1(x)} at t = T.