The wave equation with a boundary control at one endpoint and with the other endpoint being fixed (Q2703855)

The author considers the string vibrations described by the wave equation \(u_{tt}(x,t)- u_{xx}(x, t)= 0\) where \(t\in [0,T]\), \(x\in[0,l]\).NEWLINENEWLINENEWLINEThe displacement and the velocity of the string points satisfy equalities \(u(x,0)= \phi_0(x)\), \(u_t(x,0)= \psi_0(x)\) at the initial time \(t= 0\) and \(u(x, T)= \phi_1(x)\), \(u_t(x,T)= \psi_1(x)\) at the final time \(t= T\) where \(\phi_0,\phi_1,\psi_0,\psi_1\in W^2_2[0, l]\) and \(\phi_0(l)= \psi_0(l)= \phi_1(l)= \psi_1(l)= 0\). The paper deals with the natural question of existence of the boundary control \(u(0,t)= \mu(t)\) applied to the left endpoint of the string which brings the vibrational process from the state \(\{\phi_0(x), \psi_0(x)\}\) at \(t= 0\) to the state \(\{\phi_1(x), \psi_1(x)\}\) at \(t= T\).NEWLINENEWLINENEWLINEIn the paper the explicit analytical formula for the control \(\mu(t)\) is given. This result seems to be interesting and may have technical as well as physical applications.