In　this　work,　the　generalized　value　iteration　with　a　discount　factor　is　developed　for　optimal　control　of　discrete-time　nonlinear　systems,　which　is　initialized　with　a　positive　definite　value　function　rather　than　zero.　The　convergence　analysis　of　the　discounted　value　function　sequence　is　provided.　The　condition　for　the　discount　factor　is　given　to　guarantee　the　stability　of　the　controlled　plants.　With　this　operation,　the　iterative　control　policy　that　asymptotically　stabilizes　the　closed-loop　system　can　be　determined.　The　introduction　of　a　discount　factor　has　eased　some　conditions　that　the　system　dynamics　and　the　initialization　of　the　generalized　value　iteration　need　to　fulfill.　It　is　not　required　that　the　initial　control　policy　is　stabilizing.　In　the　iteration　process,　under　some　conditions,　if　the　system　is　asymptotically　stable　at　the　current　iteration,　then　it　can　be　guaranteed　that　the　iterative　control　policies　after　this　current　iteration　step　also　are　stabilizing.　It　is　convenient　and　practical　to　evaluate　the　asymptotic　stability　of　the　closed-loop　system　using　the　iterative　control　policy.　A　numerical　example　with　physical　background　is　carried　out　to　validate　the　present　results.　©　2020　Elsevier　B.V.