555 timer control voltage operation
If you establish a certain voltage Vc at the control pin, either with a stiff voltage source or some kind of network of resistors or whatever, then the capacitor charges from that Vc/2 up to Vc and discharges back to Vc/2. Note that the resistor values are not well specified, 5K is a 'typical' value, but they are well-matched to each other.
(Looking into the control pin it looks like a voltage source of Vcc*(2/3) with a source resistance of roughly 5K||10K = 3.3K. Some variants of the original NE555 such as the TLC555 use 100K nominal resistors so it would be more like 67K.)
The consequence of this is that changing Vc by means of external resistors (rather than, say, an op-amp output with feedback) generally will significantly degrade the initial accuracy and the temperature stability of the output frequency since the internal resistors are not very stable with temperature nor necessarily all that close to nominal to begin with.



simulate this circuit &ndash; Schematic created using CircuitLab
If we simulate this with the control voltage swept from 0 to Vcc over 1 second you can see the capacitor voltage extremes change (orange/yellow trace) , and the duty cycle of the output (blue trace) changes.