This　article　is　concerned　with　the　stability　of　the　closed-loop　system　using　various　control　policies　generated　by　value　iteration.　Some　stability　properties　involving　admissibility　criteria,　the　attraction　domain,　and　so　forth,　are　investigated.　An　offline　integrated　value　iteration　(VI)　scheme　with　a　stability　guarantee　is　developed　by　combining　the　advantages　of　VI　and　policy　iteration,　which　is　convenient　to　obtain　admissible　control　policies.　Also,　based　on　the　concept　of　attraction　domain,　an　online　adaptive　dynamic　programming　algorithm　using　immature　control　policies　is　developed.　Remarkably,　it　is　ensured　that　the　state　trajectory　under　the　online　algorithm　converges　to　the　origin.　Particularly,　for　linear　systems,　the　online　ADP　algorithm　with　a　general　scheme　possesses　more　enhanced　stability　property.　The　theoretical　results　reveal　that　the　stability　of　the　linear　system　can　be　guaranteed　even　if　the　control　policy　sequence　includes　finite　unstable　elements.　The　numerical　results　verify　the　effectiveness　of　the　present　algorithms.