Stability:
Unstable. Gradually decomposes losing oxygen. Decomposes more rapidly at higher temperatures. Stability decreases in the presence of moisture.

I have got a way to recycle and reuse persulfate-type copper etchant infinitely. Its an electochemical setup that will cost you a lot when you set it up for the first time. But it does not require any kind of maintainence. You dont have to add any chemicals later. You can reuse the same solution forever.

THEORY : First, I will discuss the theoritical principles involved.

sodium/ammonium persulfate (Na2S2O8 / (NH3)2S2O8) etches copper in the following way :-

Na2S2O8(aq) + Cu(s) ----> Na2SO4(aq) + CuSO4(aq)

The electrochemical mechanism reverses this in the following way :-

CuSO4 ---> (Cu)2+ + (SO4)2-

At cathode (-ve electrode) :

(Cu)2+ + 2e ---> Cu (copper gets deposited on the electrode itself, thus removing copper ions from the solution)

At anode (+ve electrode) :

2 (SO4)2- ---> (S2O8)2- + 2e

Thus the persulfate ions ,ie. the primary oxidiser, is regenerated from the sulfate ions.

REQUIREMENTS : You will need the following components to set up your etching apparatus.

1) a platinum electrode (1 pcs) for the anode : This will be the most expensive component in the whole apparatus. Also note, there are two varieties of Pt electrodes. A) The platinized platium electrode , which is black or grey in colour because it is coated with Platinum Black (an allotrope of platinum), and, B) The shiny platinum electrode, without the coating, which has a shiny silver-like appearance. You are supposed to use the shiny-type Pt electrode. The black one won't work.
2) a copper electrode for the cathode : you may use a length of a copper wire (after striping off the insulation) or a spare bit of copperclad pcb. Make sure this electrode has a large surface area.
3) a glass/plastic container to hold the solution and act as your etching tray
4) sodium persulfate (Na2S2O8) : Alternatively, sodium sulfate (Na2SO4) may be used to begin with and sodium persulfate can be generated from that by electrolysis.
5) a little bit of conc. sulfuric acid (H2SO4) to lower the pH a bit : an acidified solution helps to speed up the etchant-regeneration process.
6) water : ordinary tap water would do but distilled water is better.
7) Power Supply - a mechanism to supply a lot of current (about 5-10 Amps) at controlled voltages to the electrodes : A simple 9-volt battery with a potentiometer might do. But a 9-volt battery can't supply sufficient amount of current. So the regeneration process may be too slow to be acceptable. A better way is to draw the power from the house-electric supply. Use a 12-volt DC wall transformer/adapter to get a 12V DC source of current. Then prepare a custom voltage-regulation circuit that uses the 12V DC output and can provide 5-10 Amps at a voltage that you can control precisely.
(Note : the required output voltages are typically between 0.5V to 4.5V DC with a control precision of atleast 0.1V)
8) You can also add an agitator mechanism (aquarium pump) and a heater mechanism (immersion heater / aquarium thermostat) to the apparatus to speed up the etching process.

SETUP THE APPARATUS :-

Prepare a sodium persulfate solution at a concentration of 230 gram/litre. If you are starting with sodium sulfate instead of sodium persulfate, mix at a concentration of 460 gram/litre. Add to it about 10 ml of conc. sulfuric acid per litre of the solution. Your etching solution is ready.

Now pour your etching solution in your etching tray and drop the electrodes in it. (the copper electrode is supposed to be the cathode (-ve) and the platinum electrode is to be anode(+ve). Make sure you connect them correctly.)

Now, if you mixed your solution with sodium persulfate, you are ready to go. But if you used sodium sulfate, you need to run a current through the solution so that sodium persulfate can be generated within the solution. Calibrate the voltage of the regulator and let the current run for some time (about one hour). (The voltage calibration process is being given below, as a part of the etching process.)

THE ETCHING PROCESS :-

Drop the pcb, which is to be etched, in the solution and start a current in the circuit.

Then you need to adjust the potentiometer or the v-reg to supply the correct voltage.
The goal would be to set the voltage as large as possible but just short of when a vigourous bubbling starts at the platinum anode.

(Note, if you are are using an aquarium air-pump to agitate the solution, you must turn it off during this voltage calibration process. Otherwise you wont be able to distinguish the bubbling at the anode. After calibration, you can turn on the pump and leave the apparatus to etch your pcb.)

Ideally, You must begin with a voltage as low as possible and increase the voltage until you can see a lot of bubbling at the anode. Once you find the threshold point, you must set the voltage to slightly less that.

POST-PROCESSING OF ETCHANT SOLUTION :-

After etching is completed, remove the pcb from the solution and let the current run for some more time. This will allow all copper ions dissolved in the solution to get deposited on the cathode and clean-up the solution. (It is easy to detect presence of copper ions in the solution. Copper ions impart a blue colour to the solution. So wait till the blue colour fades away)

After this process completes, you must remove both the electrodes from the solution before you remove power from the electrodes. If you remove power from the electrodes while they are in the solution, the cathode which is made of copper, turns into an ordinary copper rod, and the etchant immediately starts eating away copper from the electrode. So if you leave the cathode in the solution without power supply, the etchant will essentially get used up and you will need to regenerate it again before you can use it.
Note that, although the solution doesn't react with the platinum electrode, but if left in it for a long time, it can corrode the suface. And you don't want the most expensive part of your etching apparatus destroyed.

Best of Luck with your new etching system