Progress in Partial Differential Equations
In the present paper we study the boundary control by the third boundary condition on the left end of a string, the right end being fixed. An optimality criterion based on the minimization of an integral of a linear combination of the control itself and its antiderivative raised to an arbitrary power p≥1 is established. A method is developed permitting one to find a control satisfying this optimality criterion and write it out in the explicit form. The optimal control for p>1 is proved. Thereby proposed optimality criterion uniquely determines the optimal solution of boundary control problem under consideration.