Can i claim my children on my tax returns, if their mother claimed them and not me on her foodstamps?

so i don't reap the benefits of her food stamps but she has our two boys on it. i do pay for all the need of my boys, other than their food. they do live with me and her equally. but she is not my dependent, just them. can i claim them as my dependents on my tax returns without her getting penalized?