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This article presents a novel control strategy for zero-voltage-switching (ZVS) quasi-square-wave boost converter. The controller is based on an exact, closed-form analytical solution of the converter, which is obtained under the assumption that the duration of the valley resonant transition corresponds to a $90^\circ$ angle. A zero-voltage-detection (ZVD) circuit is used to tune the switching frequency to a value that results in the $90^\circ$ valley resonant transition. The controller is implemented digitally, and it is not affected by the propagation delay of the ZVD signal. ZVS of the main device is guaranteed by the closed-loop ZVD frequency tuning, and accurate current control is achieved with the exact analytical solution of the converter. Contrary to commonly used approaches that rely on high-frequency current measurement, this current control is realized without measuring the current. The controller is implemented in a power factor correction circuit, in conjunction with an outer voltage loop. The approach is validated experimentally on a 5-kW prototype, using GaN devices, and featuring a power density of 120 W/in${^{3}}$, peak efficiency greater than 99%, and a total harmonic distortion well below the specified requirements, across the entire load range.