On the nonlinear dependence of the optimal boundary control on the initial and terminal conditions

We consider the optimal boundary control of string vibrations and study the case of boundary force control at one end of the string, the other end being free. We show that if the initial and terminal data are arbitrary, then the dependence of the optimal boundary control on these data may be nonlinear. Conditions under which the dependence is linear are indicated. An example is considered in which the optimal control can be found in closed form.