Portable speaker power supply

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Im making a portable solar stereo with the help of the podzuma design. But I want to use it in the house with out the battery, just running off of the power supply. Can I leave it plugged in while not in use? I was wondering with this schematic will the battery stop charging when its full or will it over charge and damage the battery. And where would I wire a 1.8 watt solar panel in here. I know that theres a diode and a resister running to the battery. Any help would be awesome. Thanks
An externally hosted image should be here but it was not working when we last tested it.
 
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