Boundary control of vibrations at one endpoint with the other endpoint free in terms of a finite-energy generalized solution of the wave equation (Q1603140)

The unidimensional wave equation \(u_{tt}(x,t)-u_{xx}(x,t) =0\) is analysed here from the boundary control point of view, in terms of finite energy generalised solutions, when the point \(x=0\) is fixed and the point \(x=\ell\) is free. The equation is analysed in the following initial and boundary conditions: \(u(x,0)= \varphi(x)\), \(u_t(x,0)= \psi(x)\), \(u(x,T)= \varphi_1 (x)\), \(u_t(x,T)= \psi_1(x)\), \(\varphi(x)\) and \(\varphi_1(x)\in W^1_2 (0,\ell)\), \(\psi(x)\) and \(\psi_1(x)\in L_2(0,\ell)\). The authors establish necessary and sufficient conditions for the vibration process in two different cases: a. \(0<T\leq 2\ell\), b. \(T>2\ell\). For each case one presents the general forms of this boundary control, which contains two arbitrary functions defined on the interval \(t\in [0,T-2\ell]\), from the class \(W^1_2\) on these intervals and taking given values at the endpoints of these intervals.