Could this be a feasible fuel system? First off, please forgive me, I am a chemistry novice!

Overall Process


Fuel for the system is distilled water
Split water into Hydrogen and Oxygen (electrolysis or similar process)
Exhaust Oxygen
Burn Hydrogen to power a generator/engine
Generator/engine charges battery
Battery powers Step 2 as well as whatever else we want to power


Problems


Energy gained needs to be greater than the sum of lost energy in all steps, as well as the energy required to perform electrolysis
We could compress the exhaust oxygen and use that as an energy source?
Is there a danger of running out of hydrogen? (Forgive my naivety, but I've heard about a helium shortage...)
"Energy gained needs to be greater than the sum of lost energy in all steps"

http://www.lhup.edu/~dsimanek/museum/unwork.htm 
 Perpetual motion machines 
http://news.sciencemag.org/chemistry/2014/01/rhubarb-battery-could-store-energy-future 
 Flow battery, which works.

1) You cannot win (First Law of thermodynamics from Noether's theorems and the isotropy of time) 
2) You can only break even on a very cold day (Second Law of thermodynamics from Carnot cycle and the law of large numbers). 
3) It never gets that cold (Third Law of thermodynamics.   If there were an entropy difference at absolute zero, T = 0 K could be reached in a finite number of steps. However, at T = 0 K there is no entropy difference, so an infinite number of steps are needed).