This　paper　proposes　a　novel　Krasovskii–LaSalle　theorem　for　discrete-time　switched　linear　time-invariant　systems.　The　result　is　utilized　to　guarantee　exponential　convergence　of　both　leaderless　and　leader-following　consensus　of　discrete-time　multi-agent　systems　under　jointly　connected　switching　network　topology.　Different　from　the　continuous-time　multi-agent　systems　under　switching　networks,　where　the　magnitude　of　the　dwell　time　of　switching　signals　is　irrelevant,　in　the　discrete-time　case,　this　value　is　critical　for　consensus　reaching.　A　special　index,　called　the　constrained　controllability　index,　is　defined.　This　index　can　be　easily　calculated　and　provide　a　tighter　lower　bound　of　the　dwell　time　when　compared　to　those　given　in　the　literature.　With　our　novel　method,　the　controllability　condition　on　the　system　matrix　is　relaxed　to　the　stabilizability　condition,　and　the　proposed　results　not　only　expand　the　existing　discussions　to　balanced　switching　networks　but　also　cover　some　unbalanced　networks.　Finally,　numerical　examples　and　simulations　are　provided　to　verify　the　theoretical　results.　©　2018　Elsevier　Ltd