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1.8 watts is not nearly enough to charge the battery back up, but anything going back in to the battery will help prolong the battery life, a 20 watt panel would be more reasonable as it would be charging at nearly 2 amps but would need nearly 20 sun hours to fully charge a dead battery so it would take nearly a week so you would need to still plug in at home, but it would be a slow enough charge that it wouldn't boil the battery, lead acid batteries like to remain active, so any energy going back in is going to help keep the lead sulphate from hardening on the lead plates of the battery and causing permanent damage.
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