Optimization of the Boundary Control of an Elastic Force at One Endpoint of a String with the Other Endpoint Fixed

For a large time interval $T$, we study the boundary control of an elastic force at the endpoint $x=0$ of a string under the condition that the second endpoint $x=l$ of the string is fixed. We determine and present in an analytical form the optimal boundary control $u_x(0,t)=\mu (t)$ that minimizes the elastic boundary energy integral $\int _0^T\mu ^2(t)\,dt$ over the set of all functions $\mu (t)$ from the class $L_2[0,T]$ under the condition that the oscillation process transfers the string from an arbitrary given initial state into an arbitrary given final state.