This　paper　studies　a　class　of　continuous-time　scalar-state　stochastic　Linear–Quadratic　(LQ)　optimal　control　problems　with　the　linear　control　constraints.　Using　the　state　separation　theorem　induced　from　its　special　structure,　we　derive　the　analytical　solution　for　this　class　of　problems.　The　revealed　optimal　control　policy　is　a　piece-wise　affine　function　of　system　state.　This　control　policy　can　be　computed　efficiently　by　solving　two　Riccati　equations　off-line.　Under　some　mild　conditions,　the　stationary　optimal　control　policy　can　be　also　achieved　for　this　class　of　problems　over　an　infinite　horizon.　Examples　are　presented　to　shed　light　on　the　theoretical　results　established.　©　2020　Elsevier　Ltd