The paper describes oscillators where the amplitude is controlled by the restoration of initial conditions at the energy storage elements. If these elements are capacitors, the control circuit includes a window comparator, a polarity comparator, an AND gate, a switch and a source of reference voltage. This control circuit momentarily connects (each oscillation period) the oscillator capacitor to the source of reference voltage at the instant when energy in another storage element (a capacitor or an inductor) is absent. If the reference voltage is modulated, the new value of the oscillation amplitude is established without an amplitude transient in the oscillators with two energy storage elements (with a possible delay of up to one oscillation period). The oscillators with more than two storage elements will have an amplitude transient which can be calculated period by period. In steady-state oscillations the output voltage wave is the repetition of a damped sinusoid (with the addition of small exponential components in the oscillators with three and more energy storage elements). The wave shape is calculated in the closed form and its harmonic content can be established. The experimental waveforms (steady state and amplitude transients) are given for the Wien-bridge and twin-TbridgeRC-oscillators.