Simple methods for switching a boost converter, to get a stable output voltage?
Use TL494 that has a reference and feedback loop to stabilize the voltage.
http://www.ti.com/lit/an/slva001e/slva001e.pdf

If you can use microcontroller, use its PWM channel and ADC to achieve this.
Any MCU can be used. For example, AtTiny series from Atmel.
You need to program the MCU with what you want.
MCU can also help you achieve temperature compensation of the charging voltage and many other things e.g. over-voltage and over-current protection.

Contrary to what has been said above, 555 is not unreliable and not inefficient.
It can be used as a regulated booster than a simple booster (as it is in your case) but you need to modify your design.
In your current design add a fixed reference voltage source and an op-amp. Let op-amp compare the voltage at the output to the reference voltage. If reference voltage is 1V, you need to divide the output voltage by 14.4 (using voltage divider network) and then let op-amp compare the voltages. Output of the op-amp should manipulate (increase or decrease) the duty factor so that the output voltage is fixed at 14.4V regardless of load or input voltage variations.