flat charging with 12v

Hi guys, I have a deep cycle car battery hooked up in my trailer to power various electronics. I also have an 8 amp 12V Power Supply (PS) that I use when I have access to mains. Currently I simply switch between the battery and PS manually. If we stop right here my question would be: Would there be any damage to the battery if I leave the 12V PS hooked up the battery indefinitely? From my understanding the following would happen, and please correct me where I am wrong: I start with a fully charged battery which will be around 12.6V. Now the PS will not supply any current and all the current will come straight from the battery because the Voltage of the battery is higher than what the PS provides. As soon as the battery has discharged to about 50% (~12V) the PS would start to supply power and output up to its maximum current until the current draw is so high (>8Amps) that the battery would kick in again further discharging the battery. Once the high current draw stops the battery would remain at its current level because 12V is really not enough for the cells to charge. To complicate the situation there is also a smart solar charger in the mix that charges the battery on sunny days (with voltages up to around 14.7 Volts for a full charge). I have no clue how or if this would affect the circuit but I am hoping that even with the 12V supply hooked up at all times the smart charger would charge the battery. Any advice would be appreciated as I want to preserve my batter life as much as possible.

Provided the (powered or unpowered) PS tolerates having a higher voltage source connected to its output (ie, no back discharge), it should be fine. Normally battery chargers would be used, but I suspect I understand your situation. And check that 14.7V is not too high for your battery; lead-acids should be 14.4V max long-term.