If you’re simply looking for guidance on how to build a voltage-controlled oscillator, this circuit can establish a linear relationship from voltage to frequency conversion. By modifying the formula, you can achieve “Gain.” Duty control can be accomplished by using V(Q) as a clock signal to drive a sawtooth waveform for PWM with an additional circuit [or you can take V(vp) as sawtooth, which will be more simple]. I believe it shouldn’t be too challenging to modify this circuit to fulfill the purpose mentioned in your post. But perhaps you are seeking something different.
The voltage and frequency relationship in your first post (numerical section) seems not linear, but this method is linear v to freq. This is another uncertainty from your first post.
