Optimization of boundary controls of string vibrations

For a large time interval $T$ boundary controls of string vibrations are optimized in the following seven boundary-control problems: displacement control at one end (with the other end fixed or free); displacement control at both ends; elastic force control at one end (with the other end fixed or free); elastic force control at both ends; combined control (displacement control at one end and elastic force control at the other). Optimal boundary controls in each of these seven problems are sought as functions minimizing the corresponding boundary-energy integral under the constraints following from the initial and terminal conditions for the string at $t=0$ and $t=T$, respectively. For all seven problems, the optimal boundary controls are written out in closed analytic form.