I bought a Chinese desk lamp (LED). As the battery discharges, I've observed the intensity of the lights reduce. I was thinking of tweaking the circuit such that the intensity remains constant as long as the battery is sufficiently charged and it shuts down when the battery is properly discharged or at least grows very dim.
I was thinking of using a 7805 to provide a constant supply to the LEDs. Which lead me to the title question that if the battery supply reduces a little too much with discharging will this harm the internal circuits of the 7805 or will 7805 somehow reduce the battery life?
I am yet to analyse the LED circuit and I know my prime target would most probably require more than a simple regulator. I just want to have some practical insights here.
I am uploading the circuit image here just for reference.

 

You've told us nothing about the battery type and voltage .
What does that circuit (just visible through the gloom) actually do?
Methinks a 7805 will only make the problem worse, since it needs to drop a couple of volts in order to function.
What you're probably looking for is a DC-DC boost converter.

 

What you're probably looking for is a DC-DC boost converter.

Agreed. The 7805 will just make things worse. It won't be damaged as the battery voltage sags, but the output voltage will sag along with it, and will be lower than the battery voltage as @Alec_t just noted.

That in your lamp circuit looks primitive and is likely not very efficient. You may want to just replace the whole thing with a constant-current, DC-DC boost converter meant for driving LEDs. This efficient device can squeeze the energy out of the battery while maintaining a near-constant brightness. To shop for one, you'll need to know the input voltage range from the battery, the minimum voltage needed to light the LED, and the required LED current. You may have to measure that current with a meter.

 

You've told us nothing about the battery type and voltage .
What does that circuit (just visible through the gloom) actually do?
Methinks a 7805 will only make the problem worse, since it needs to drop a couple of volts in order to function.
What you're probably looking for is a DC-DC boost converter.

Actually a buck convertor. A boost voltage convertor increases voltage.

You can find 5v buck convertors in those cheap USB chargers. Usually they are a 34063. Old technology and not the most efficient chip out there but very versatile and you will be able to find tons of example circuits and a number of calculators. Basically lots of support.

Repurposing them from a charger makes your job even easier. Assuming it is in the configuration you want then you won't have to run out and buy the sensing resistors.

 

There is also the LM2675 series from TI. They come in a variety of configurations including 5V. Your problem there is making sure you isolate the coil from the chip. Otherwise very easy to use.