Optimization of the one-endpoint boundary control of vibrations for the case in which the other endpoint is fixed and energy is bounded (Q1874886)

The author considers string vibrations governed by the equation \[ u_{tt}(x,t)-u_{xx}(x,t)=0,\quad 0\leq x\leq l,\;0\leq t\leq T, \] under the condition that the string endpoint \(x=l\) is fixed and the control is applied at the endpoint \(x=0\). He solves the problem on the existence and closed-form representation of a left-endpoint boundary control \(\mu_*(t)\), that brings the vibration process from the state \(\{u(x,0)=\phi(x),\;u_t(x,0)=\psi(x)\}\) to a state \(\{u(x,T)=\{\phi_*(x),\;u_t(x,0)=\psi_*(x)\}\) with minimum deviation from the unattainable state \(\{\phi_1(x),\psi_1(x)\}\) in the norm of the space \(\mathcal H=W_2^1[0,l]\times L_2[0,l]\). The case \(l\leq T\leq 2l\) is considered. The existence and uniqueness of the corresponding minimization problem \(J(\mu)\to \inf _{\mu\in \mathcal M}\) is verified. The closed form of a solution is derived.