The boundary energy control problem for the sine-Gordon and the nonlinear Klein–Gordon equations is posed. Two control laws solving this problem based on the speed-gradient method with smooth and nonsmooth goal functions are proposed. The control law obtained via a nonsmooth goal function is proved to steer the system to any required nonzero energy level in finite time. The results of the numerical evaluation of the proposed algorithm for an undamped nonlinear elastic string demonstrate attainability of the control goal for the cases of both decreasing and increasing systems’ energy and show high rate of vanishing of the control error.