Connected　cruise　control　(CCC),　as　an　approach　to　regulate　vehicle＇s　longitudinal　motion　in　car-following　platoon　scenarios,　can　improve　the　safety　and　efficiency　of　the　traffic　flow.　In　order　to　make　it　available　for　extensive　information　sharing　among　vehicles,　wireless　vehicle-to-vehicle　(V2V)　communication　is　exploited　in　the　vehicular　platoon.　However,　time　delays　caused　by　wireless　communication　may　result　in　significant　degradation　of　system　stability.　In　this　article,　considering　arbitrary　V2V　communication　delays,　an　optimal　longitudinal　control　algorithm　is　proposed　for　the　CCC　vehicle　with　the　goal　of　minimizing　the　deviations　of　vehicle＇s　headway　and　velocity.　In　the　proposed　scheme,　the　vehicular　platoon　is　modeled　as　a　discrete-time　system　based　on　vehicle＇s　error　dynamics,　and　the　optimal-state　feedback　control　problem　is　designed　as　a　cost　function　represented　by　a　sum　of　platoon　states　in　a　quadratic　form.　A　backward　recursion　method　is　used　to　iteratively　derive　the　optimal　control　strategy,　namely　the　acceleration　for　CCC　vehicle,　based　on　the　current　platoon　states　and　previous　control　signals.　In　addition,　the　stability　analysis　shows　that　the　system　is　asymptotically　mean　square　stable　under　the　control　of　the　proposed　algorithm.　Finally,　numerical　simulations　indicate　that　the　proposed　algorithm　provides　better　system　performance　and　stability.