This　paper　is　concerned　with　the　optimal　boundary　control　of　a　non-dimensional　non-linear　parabolic　system　consisting　of　the　Kuramoto-Sivashinsky-Korteweg-de　Vries　equation　and　a　heat　equation.　By　the　Dubovitskii　and　Milyutin　functional　analytical　approach,　first　in　the　fixed　final　horizon　case　we　prove　the　Pontryagin　maximum　principle　of　the　optimal　control　problem　of　this　coupled　system.　Then　under　weaker　additional　conditions,　we　study　the　controlled　system　in　the　free　final　horizon　case　and　present　further　investigational　results　of　current　interests.　The　necessary　optimality　conditions　are　established　for　optimal　control　problems　in　these　two　cases.　Finally,　a　remark　on　how　to　utilize　the　obtained　results　is　also　made　for　illustration.