Your strip only needs a 3.6V DC power supply.
It appears you have one 20Ω resistor for each LED.
This would power each LED with 30 mA or 90 mW per LED (3Vf x 0.03 A).
If this is true there would be about 25 LEDs on the strip (2.3W / 90 mW = 25.5).
A strip that requires a controller will have IC(s) on the strip.
Or have RGB LEDs that look something like this.
how can I use it without a controller, just a battery?
You can likely use just the + (red circle) and - (green circle) pins to power. I'm not sure about the 3rd pin. Not likely an earth ground as there are no lethal voltages. It may be unused. Most likely it is an additional pin for the 3.6V.
3.6V Batteries do exist. The most common being an 18650 Li-ion cell.
These batteries start at about 4.2V and discharge to 3.0V.
More likely it is meant to be powered with a 3.6V AC-DC adapter.
Battery powered LEDs dim as the battery voltage drops as it discharges.
source eemb.com LIR18650 datasheet
I did find someone (in France) selling these but they appear to be 12V.
NiMH will much better than alkaline.
Notice at the end of charge the voltage is still 1.1V.
It is best not to allow to discharge to zero volts but is much more forgiving than LI-ion.
source: Panasonic HHR120AA NiMH datasheet