In　this　paper,　a　strict　linear　Lyapunov　function　is　developed　in　order　to　investigate　the　exponential　stability　of　a　linear　hyperbolic　partial　differential　equation　with　positive　boundary　conditions.　Based　on　the　method　of　characteristics,　some　properties　of　the　positive　solutions　are　derived　for　the　hyperbolic　initial　boundary　value　problems.　The　dissipative　boundary　condition　in　terms　of　linear　inequalities　is　proven　to　be　not　only　sufficient　but　also　necessary　under　an　extra　assumption　on　the　velocities　of　the　hyperbolic　systems.　An　application　to　control　of　the　freeway　traffic　modeled　by　the　Aw-Rascle　traffic　flow　equation　illustrates　and　motivates　the　theoretical　results.　The　boundary　control　strategies　are　designed　by　integrating　the　on-ramp　metering　with　the　mainline　speed　limit.　Finally,　the　proposed　feedback　laws　are　tested　under　simulation,　first　in　the　free-flow　case　and　then　in　the　congestion　mode,　which　show　adequate　performance　to　stabilize　the　local　freeway　traffic.