Boundary control of oscillations at one endpoint with the other endpoint fixed in terms of a finite-energy generalized solution of the wave equation (Q5951124)

In this paper, in terms of a finite-energy generalized solution of the wave equation \(u_{tt}(x,t)-u_{xx}(x,t)=0\), the author studies the problem on the boundary control of oscillations at the endpoint \(x=0\) provided that the endpoint \(x=l\) is fixed. For an arbitrary time interval \(T>0\), the author proves necessary and sufficient conditions for the existence of a boundary control \(u (0,t)= \mu(t)\) bringing the vibration process from the state \(\{u(x,0)= \phi (x)\), \(u_t(x,0)= \psi(x)\}\) to the state \(\{u(x,T)= \phi_1(x)\), \(u_t(x,T)= \psi_1(t)\}\) and obtains an explicit analytic form of the control.