The paper under discussion refines some work done by the reviewer [SIAM J. Control 4, 276--294 (1966; Zbl 0166.35805)]. It is concerned with control of the vibrating string, represented by the partial differential equation \[ \rho \frac{\partial^{2} w}{\partial t^{2}}-\tau \frac{\partial^{2} w}{\partial x^{2}}=0, \quad 0 \leq x \leq l, \quad 0 \leq t \leq T \] The control \( f(t) \) appears in the boundary conditions \( w(0, t)=0, w(l, t)=f(t) \). This system can be transformed into \[ \rho \frac{\partial^{2} y}{\partial t^{2}}-\tau \frac{\partial^{2} y}{\partial x^{2}}=\rho \frac{x}{l} u(t), \quad 0 \leq x \leq l, \quad 0 \leq t \leq T, y(0, t) \equiv 0, \quad y(l, t) \equiv 0, \] where \( u(t)=f^{\prime \prime}(t) \). The control constraints are \( |u(t)| \leq 1,0 \leq t \leq T \). The objective is to find \( u(t) \), measurable and satisfying the indicated constraints, in such a way as to transfer the initial state \( y(x, 0), \partial y / \partial t(x, 0) \) into a terminal state \( y({x}, {T}), \partial {y} / \partial t({x}, {T}) \) with least possible energy, this energy being given by \[ \frac{1}{2} \int_{0}^{1}\left(p\left(\frac{\partial y}{\partial t}(x, T)\right)^{2}+\tau\left(\frac{\partial y}{\partial x}(x, T)\right)^{2}\right) d x \] Necessary conditions for optimality of \( u(t) \) in this sense were developed by the reviewer [loc. cit.]. However, those conditions are expressed in terms of the unknown terminal state and therefore leave much to be desired. The author of the present paper succeeds in expressing the necessary conditions in terms of the, presumably known, initial state. Using Fourier methods the terminal state is expressed as a linear function of the initial state and the control \( u(t) \). The terminal energy is then quadratic in the initial state and the control. Hilbert space variational techniques are applied to obtain the necessary conditions for optimality. It is shown, e.g., that the optimal control \( u(t) \) is periodic (the period is the fundamental period for the uncontrolled string), and achieves extremal values \(\pm 1\) in a subset of \( [0, T] \) having positive measure, unless the minimal achievable energy is zero, in which case such extremal values may never be reaIized. The work is highly detailed and explicit and no doubt the last word on this particular question.